Finite Group Databasebeta

← #33093132

Subgroups — #33093132

Order 18688.

972 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988912C211yesmaximal, characteristic
2#22612437C211yesmaximal, characteristic
3#117×Dic73C211yesmaximal, characteristic, complemented
4#22610557C211yesmaximal, characteristic, complemented
5#22610550C211yesmaximal, characteristic, complemented
6#22610522C211yesmaximal, characteristic, complemented
7#22610515C211yesmaximal, characteristic, complemented
8#22610253C211yesmaximal, characteristic
9#22610023C211yesmaximal, characteristic, complemented
10#107×Dic73C211yesmaximal, characteristic, complemented
11#106×Dic73C211yesmaximal, characteristic, complemented
12#22609745C211yesmaximal, characteristic, complemented
13#22609745C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic73C211yesmaximal, characteristic, complemented
15C73×#1854C211yesmaximal, characteristic, complemented
16C2×#14556565C2211yescharacteristic
17C2×#14556553C2211yescharacteristic
18C2×#14556551—21no—
19C2×#14556550C2211yescharacteristic
20C22×C4×Dic73C2211yescharacteristic
21#17988912C418yescomplemented
22C2×#14556525C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556513C2211yescharacteristic
25C2×#14556513C2211yescharacteristic
26C2×#14556511C2211yescharacteristic
27C2×#14556511C2211yescharacteristic
28C2×C22⋊#10596831C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988488C2211yescharacteristic
32#17988483C2211yescharacteristic
33#17988476—21no—
34#17988464—21no—
35C4⋊C4×Dic73C2211yescharacteristic
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C22⋊C4×Dic73—21no—
41C22⋊C4×Dic73—21no—
42#17988434—21no—
43#17988433—21no—
44C42×Dic73—21no—
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×#117×C73C2211yescharacteristic
54C2×#107×C73C2211yescharacteristic
55C2×#106×C73C2211yescharacteristic
56C2×C22⋊C8×C73C2211yescharacteristic
57#380×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C22⋊D73—21no—
62C23×Dic73D411yescharacteristic
63C23×Dic73D411yescharacteristic
64C22×C4×D73—21no—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556565—22no—
68#14556565C2×C414yes—
69#14556553—22no—
70#14556553—22no—
71#14556551—44no—
72#14556550—22no—
73#14556550C2×C414yescomplemented
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×#11924007—21no—
78C2×C22⋊Dic73—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×C4×Dic73—22no—
84C2×C4×Dic73—22no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73D411yescharacteristic
92C2×C4×Dic73D411yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95#14556525C2×C414yescomplemented
96#14556523C2×C414yescomplemented
97#14556513C2×C414yescomplemented
98#14556513C2×C414yescomplemented
99#14556511—22no—
100#14556511—22no—
101C22⋊#10596831—22no—
102C8×Dic73—22no—
103C8×Dic73—22no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110C23×C4×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113#117×C73—22no—
114#117×C73—22no—
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C42×C73—21no—
122#107×C73—22no—
123#106×C73—22no—
124C22⋊C8×C73C2×C414yescomplemented
125#95×C73—21no—
126C23×D73—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292—21no—
132C2×D292C22×C412yes—
133C2×C22⋊D73—24no—
134C2×C22⋊D73—42no—
135C2×C22⋊D73—42no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—22no—
140C22×Dic73—22no—
141C22×Dic73—22no—
142C22×Dic73—22no—
143C22×Dic73—22no—
144C22×Dic73—41no—
145C22×Dic73C2×D411yescharacteristic
146C22×Dic73C2×D411yescharacteristic
147C22×Dic73C2×D411yescharacteristic
148C22×Dic73C2×D411yescharacteristic
149C2×C4×D73—42no—
150C2×C4×D73—42no—
151C2×C4×D73—42no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73C22×C412yes—
162C4⋊Dic73C22×C412yes—
163C4⋊Dic73C22×C412yes—
164#11924007—42no—
165C22⋊Dic73—24no—
166#11924005—24no—
167#11924005—42no—
168#11924005—42no—
169#11924005—42no—
170C4×Dic73—24no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—22no—
174C4×Dic73—22no—
175C4×Dic73—42no—
176C4×Dic73—42no—
177C4×Dic73—42no—
178C4×Dic73—42no—
179C4×Dic73—42no—
180C4×Dic73#4112yes—
181C4×Dic73#4112yes—
182C24×C73C2×D411yescharacteristic
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73C22×C412yes—
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73#4112yes—
194C22×C4×C73C2411yescharacteristic
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73C22×C412yes—
200C2×C8×C73C22×C412yes—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73#4112yes—
208C4⋊C4×C73#4112yes—
209C22⋊C4×C73—24no—
210C22⋊C4×C73—42no—
211C42×C73—24no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73C4×D412yes—
219C23×C73C22×D411yescharacteristic
220C22×D73—22no—
221C22×D73—41no—
222C22×D73—41no—
223C22×D73—41no—
224C22×D73—41no—
225C22×D73—41no—
226C22×D73—41no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73C4×D412yes—
246C2×C4×C73C23×C411yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C22⋊D73—48no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—22no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—44no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—44no—
271C2×Dic73#12312yes—
272C2×Dic73#12312yes—
273C2×Dic73#13512yes—
274C2×Dic73#13512yes—
275C2×Dic73AGL(1,ℤ/8)12yes—
276D292—22no—
277D292—22no—
278D292—41no—
279D292—41no—
280C4×D73—84no—
281C8×C73—22no—
282C8×C73—22no—
283#10596831—41no—
284#10596831—41no—
285C22×C73—22no—
286C22×C73—22no—
287C22×C73—22no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73C2×C4×D411yescharacteristic
293C22×C73C2×#12311yescharacteristic
294C22×C73#52511yescharacteristic
295C22×C73C2×#13511yescharacteristic
296C22×C73C2×#13511yescharacteristic
297C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C73#57511yescharacteristic
299C2×D73—42no—
300C2×D73—42no—
301C2×D73—42no—
302C2×D73—81no—
303C2×D73—81no—
304C2×D73—81no—
305C2×D73—81no—
306C4×C73—22no—
307C4×C73—42no—
308C4×C73—42no—
309C4×C73—81no—
310C4×C73—81no—
311C4×C73C4×#4111yescharacteristic
312C4×C73C4×#4111yescharacteristic
313C4×C73#52911yescharacteristic
314C4×C73#52911yescharacteristic
315Dic73—22no—
316Dic73—22no—
317Dic73—41no—
318Dic73—42no—
319Dic73—42no—
320Dic73—42no—
321Dic73#48214yescomplemented
322C4×#482—731nomaximal, Sylow
323C2×C73—42no—
324C2×C73C4×#12311yescharacteristic
325C2×C73C4×#13511yescharacteristic
326C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
327C2×C73C2×#48211yescharacteristic
328C2×C73#309111yescharacteristic
329C2×C73#310211yescharacteristic
330C2×C73#310611yescharacteristic
331D73—82no—
332C2×#482—731no—
333C4×#121—731no—
334C4×#117—731no—
335#1926—731no—
336#1925—731no—
337#1921—731no—
338#1920—731no—
339#1875—731no—
340#1854D7311yescharacteristic, complemented
341#1825—731no—
342C4×#107—731no—
343C4×#106—731no—
344C4×#102—731no—
345C4×#102—731no—
346C4×C22⋊C8—731no—
347C73C4×#48211yescharacteristic, complemented, Sylow
348C2×#121—731no—
349C2×C4×D4—731no—
350C2×C4×D4—1461no—
351C2×#117—731no—
352C2×#117C2×D7311yescharacteristic
353C22×C42—731no—
354#482—738no—
355C2×#107—731no—
356C2×#107C2×D7311yescharacteristic
357C2×#106—731no—
358C2×#106C2×D7311yescharacteristic
359C2×#102—731no—
360C2×#102—731no—
361C2×#102—731no—
362C2×#102—731no—
363C2×C22⋊C8—731no—
364C2×C22⋊C8C2×D7311yescharacteristic
365C2×C4×C8—731no—
366C2×C4×C8—731no—
367#390—731no—
368#389—731no—
369#388—1461no—
370#386—1461no—
371#380C2×D7311yescharacteristic
372C4×C4⋊C4—731no—
373C4×C4⋊C4—731no—
374C4×C4⋊C4—731no—
375C4×C4⋊C4—731no—
376C4×C4⋊C4—731no—
377C4×C22⋊C4—1461no—
378C4×C22⋊C4—1461no—
379C4×C22⋊C4—1461no—
380#376—1461no—
381C43—1461no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340—731no—
386#340—731no—
387#340—731no—
388#340C2×D7311yescharacteristic
389#340C2×D7311yescharacteristic
390#336—731no—
391#336—731no—
392C22×D4—731no—
393C22×D4—1461no—
394C23×C4—731no—
395C23×C4—731no—
396C23×C4—1461no—
397C23×C4C22×D7311yescharacteristic
398C22×C8—731no—
399C22×C8—731no—
400C22×C8C22×D7311yescharacteristic
401C22×C8C22×D7311yescharacteristic
402#121—734no—
403#121—1462no—
404C4×D4—2924no—
405C4×D4—1462no—
406C4×D4—1462no—
407#117—22no—
408#117—22no—
409#117—734no—
410#117—1462no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4C22×D7311yescharacteristic
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C4⋊C4C22×D7311yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1461no—
422C2×C22⋊C4—1461no—
423C2×C22⋊C4—1461no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4—1461no—
426C2×C42—21no—
427C2×C42—731no—
428C2×C42—731no—
429C2×C42—731no—
430C2×C42—731no—
431C2×C42—1461no—
432C2×C42—1461no—
433C2×C42—1461no—
434C2×C42—1461no—
435C2×C42—1461no—
436C2×C42—1462no—
437C2×C42—1462no—
438C2×C42—1461no—
439#107—22no—
440#107—734no—
441#106—22no—
442#106—734no—
443#102—734no—
444#102—734no—
445#102—1462no—
446#102—1462no—
447C22⋊C8—1462no—
448C22⋊C8C4×D7314yescomplemented
449C4×C8—1462no—
450C4×C8—1462no—
451#95—21no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458C24—1461no—
459C24D4×D7311yescharacteristic
460C2×D4—2922no—
461C2×D4—2922no—
462C2×D4—732no—
463C2×D4—1464no—
464C2×D4—1461no—
465C2×D4—1461no—
466C2×D4—1461no—
467C2×D4—1461no—
468C2×D4—1461no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—2922no—
478C22×C4—2922no—
479C22×C4—2922no—
480C22×C4—2921no—
481C22×C4—731no—
482C22×C4—731no—
483C22×C4—731no—
484C22×C4—731no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4C2×C4×D7312yes—
494C22×C4D4×D7311yescharacteristic
495C22×C4#1192402912yes—
496C22×C4C23×D7311yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8C2×C4×D7312yes—
508C2×C8C2×C4×D7312yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—2922no—
516C4⋊C4—732no—
517C4⋊C4—732no—
518C4⋊C4—732no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4#1192402712yes—
523C4⋊C4#1192402712yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—2922no—
527C22⋊C4—2922no—
528C22⋊C4—2922no—
529C22⋊C4—1464no—
530C22⋊C4—1464no—
531C42—24no—
532C42—2922no—
533C42—2922no—
534C42—2922no—
535C42—2922no—
536C42—2922no—
537C42—732no—
538C42—732no—
539C42—1462no—
540C42—1462no—
541C42—1464no—
542C42—1461no—
543C42—1461no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—2921no—
551C23—2921no—
552C23—2921no—
553C23—2921no—
554C23—2921no—
555C23—2921no—
556C23—1462no—
557C23#1455655312yes—
558C23C2×D4×D7311yescharacteristic
559D4—2928no—
560D4—2921no—
561D4—2921no—
562D4—1462no—
563D4—1462no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2924no—
589C2×C4—2924no—
590C2×C4—5844no—
591C2×C4—732no—
592C2×C4—732no—
593C2×C4—732no—
594C2×C4—732no—
595C2×C4—732no—
596C2×C4—1462no—
597C2×C4—1462no—
598C2×C4—1462no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4#1455655312yes—
607C2×C4C22×C4×D7311yescharacteristic
608C2×C4#41×D7311yescharacteristic
609C2×C4#41×D7311yescharacteristic
610C2×C4C2×#1192402711yescharacteristic
611C2×C4C2×#1192402711yescharacteristic
612C2×C4C2×#1192402911yescharacteristic
613C8—22no—
614C8—22no—
615C8—2921no—
616C8—2921no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—2922no—
625C22—2922no—
626C22—2922no—
627C22—5841no—
628C22—5841no—
629C22—5841no—
630C22—5841no—
631C22C2×#1455655311yescharacteristic
632C22#123×D7311yescharacteristic
633C22#1798909311yescharacteristic
634C22#135×D7311yescharacteristic
635C22#135×D7311yescharacteristic
636C22AGL(1,ℤ/8)×D7311yescharacteristic
637C22#1798938611yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—2922no—
644C4—2922no—
645C4—2922no—
646C4—2921no—
647C4—734no—
648C4—1462no—
649C4—1462no—
650C4C4×#1192402711yescharacteristic
651C4C4×#1192402711yescharacteristic
652C4#1798912311yescharacteristic
653C4#1798912311yescharacteristic
654C2—42no—
655C2—5842no—
656C2#2261245011yescharacteristic
657C2#2261870211yescharacteristic
658C2#2261873811yescharacteristic
659C2#482×D7311yescharacteristic
660C2#2261984911yescharacteristic
661C2#2261995011yescharacteristic
662C2#2261998411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1854C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
cosocleC22×C4×C73C24
hypercenter#1854D73
hypocenterC73C4×#482
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #117×Dic73C2semidirect
4. #22610557C2semidirect
5. #22610550C2semidirect
6. #22610522C2semidirect
7. #22610515C2semidirect
9. #22610023C2semidirect
10. #107×Dic73C2semidirect
11. #106×Dic73C2semidirect
12. #22609745C2semidirect
13. #22609745C2semidirect
14. C22⋊C8×Dic73C2semidirect
15. C73×#1854C2semidirect
21. #17988912C4semidirect
73. #14556550C2×C4semidirect
95. #14556525C2×C4semidirect
96. #14556523C2×C4semidirect
97. #14556513C2×C4semidirect
98. #14556513C2×C4semidirect
124. C22⋊C8×C73C2×C4semidirect
321. Dic73#482semidirect
340. #1854D73semidirect
347. C73C4×#482semidirect
448. C22⋊C8C4×D73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179889122
#226124372
#117×Dic732
#226105572
#226105502
#226105222
#226105152
#226102532
#226100232
#107×Dic732
#106×Dic732
#226097452
C22⋊C8×Dic732
C73×#18542
C4×#48273