Finite Group Databasebeta

← #33103259

Subgroups — #33103259

Order 18688.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988595C211yesmaximal, characteristic
2C22⋊C4×Dic146C211yesmaximal, characteristic
3#22610620C211yesmaximal, characteristic, complemented
4#22610606C211yesmaximal, characteristic, complemented
5#22610572C211yesmaximal, characteristic, complemented
6#22610565C211yesmaximal, characteristic, complemented
7#22610198C211yesmaximal, characteristic, complemented
8#22610184C211yesmaximal, characteristic, complemented
9#22610026C211yesmaximal, characteristic
10#22609962C211yesmaximal, characteristic
11#22609865C211yesmaximal, characteristic
12#22609764C211yesmaximal, characteristic, complemented
13C4×#14556514C211yesmaximal, characteristic, complemented
14#22609715C211yesmaximal, characteristic
15C73×#2347C211yesmaximal, characteristic
16C23×Dic146C2211yescharacteristic
17C22×#11924012C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22#17988595C418yescomplemented
23C2×#14556514C2211yescharacteristic
24C2×#14556514C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×C22⋊#10596831C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33C2×C4×#10596831C2211yescharacteristic
34#17988493C2211yescharacteristic
35#17988493C2211yescharacteristic
36#17988491—21no—
37#17988491—21no—
38#17988447C2211yescharacteristic
39#17988445—21no—
40C22⋊C4×Dic73—21no—
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49C2×#117×C73C2211yescharacteristic
50#389×C73C2211yescharacteristic
51#389×C73C2211yescharacteristic
52#388×C73—21no—
53#388×C73—21no—
54#383×C73—21no—
55#380×C73C2211yescharacteristic
56C4×C4⋊C4×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C4×C22⋊C4×C73C2211yescharacteristic
60C22×Dic146—22no—
61C22×Dic146—22no—
62C22×Dic146—21no—
63C22×Dic146—22no—
64C22×Dic146—41no—
65C22×Dic146—41no—
66C22×Dic146C2×C414yes—
67C22×Dic146C2311yescharacteristic
68C22×Dic146C2311yescharacteristic
69C23×Dic73—21no—
70C2×#11924012—41no—
71C2×#11924012—41no—
72C2×#11924012C2×C414yes—
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75C22×#10596831C2311yescharacteristic
76C22×#10596831C2311yescharacteristic
77C4×Dic146—22no—
78C4×Dic146—22no—
79C4×Dic146—41no—
80C4×Dic146—41no—
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×C22⋊Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88#14556524—41no—
89#14556522—41no—
90#14556514—22no—
91#14556514—22no—
92#14556514—41no—
93#14556514—41no—
94#14556514C2×C414yes—
95#14556514C2×C414yes—
96#14556514C2×C414yes—
97#14556514C2×C414yes—
98C22⋊#10596831—22no—
99#14556502—41no—
100C4×#10596831—41no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106C23×C4×C73C2311yescharacteristic
107#117×C73—22no—
108#117×C73C2×C414yes—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C42×C73—21no—
120C2×C42×C73—21no—
121C2×C42×C73—21no—
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#95×C73—21no—
125#95×C73—21no—
126#95×C73—21no—
127#95×C73—21no—
128#95×C73—21no—
129#95×C73D411yescharacteristic
130#95×C73D411yescharacteristic
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146—28no—
139C2×Dic146—24no—
140C2×Dic146—21no—
141C2×Dic146—44no—
142C2×Dic146—44no—
143C2×Dic146—44no—
144C2×Dic146—42no—
145C2×Dic146—42no—
146C2×Dic146C22×C412yes—
147C2×Dic146C22×C412yes—
148C2×Dic146#4112yes—
149C2×Dic146#4112yes—
150C22×Dic73—21no—
151C22×Dic73—24no—
152C22×Dic73—21no—
153C22×Dic73—21no—
154C22×Dic73—21no—
155C22×Dic73—42no—
156C22×Dic73—42no—
157#11924012—44no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831C22×C412yes—
169C2×#10596831C22×C412yes—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174#11924007—42no—
175#11924007—42no—
176C22⋊Dic73—42no—
177C4×Dic73—42no—
178C4×Dic73—42no—
179C24×C73C2×D411yescharacteristic
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73C22×C412yes—
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73#4112yes—
195C22×C4×C73C2411yescharacteristic
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—42no—
203C4⋊C4×C73—42no—
204C4⋊C4×C73C22×C412yes—
205C4⋊C4×C73C22×C412yes—
206C22⋊C4×C73—24no—
207C22⋊C4×C73—22no—
208C22⋊C4×C73—41no—
209C22⋊C4×C73—81no—
210C42×C73—24no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—81no—
216C42×C73—81no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—21no—
222C23×C73—21no—
223C23×C73C4×D412yes—
224C23×C73C22×D411yescharacteristic
225C2×C4×C73—21no—
226C2×C4×C73—21no—
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—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73C4×D412yes—
255C2×C4×C73#13712yes—
256C2×C4×C73C23×C411yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×C4×C73H(5,𝔽2)11yescharacteristic
261C2×C4×C73H(5,𝔽2)11yescharacteristic
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—22no—
265C2×Dic73—22no—
266C2×Dic73—48no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275Dic146—22no—
276Dic146—22no—
277Dic146—22no—
278Dic146—22no—
279Dic146—22no—
280Dic146—48no—
281Dic146—41no—
282Dic146—41no—
283Dic146#11714yes—
284#10596831—22no—
285#10596831—22no—
286#10596831—41no—
287#10596831—41no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73C2×C4×D411yescharacteristic
296C22×C73#54611yescharacteristic
297C22×C73#54611yescharacteristic
298C22×C73C2×#13711yescharacteristic
299C22×C73#57511yescharacteristic
300C22×C73#57711yescharacteristic
301C22×C73#57811yescharacteristic
302C4×C73—22no—
303C4×C73—22no—
304C4×C73—41no—
305C4×C73—42no—
306C4×C73—42no—
307C4×C73—41no—
308C4×C73—41no—
309C4×C73—81no—
310C4×C73—81no—
311C4×C73C2×#11711yescharacteristic
312C4×C73#51811yescharacteristic
313C4×C73#55111yescharacteristic
314C4×C73#55111yescharacteristic
315Dic73—24no—
316Dic73—42no—
317Dic73—81no—
318Dic73—81no—
319#22581—731nomaximal, Sylow
320C2×C73—42no—
321C2×C73#234211yescharacteristic
322C2×C73C4×#13711yescharacteristic
323C2×C73#294711yescharacteristic
324C2×C73#331311yescharacteristic
325C2×C73#332011yescharacteristic
326C2×C73#332611yescharacteristic
327C2×C73#332811yescharacteristic
328C2×#419—731no—
329#2347D7311yescharacteristic
330Q8×C22⋊C4—731no—
331#1935—731no—
332#1933—731no—
333#1928—731no—
334#1927—731no—
335#1865—731no—
336#1863—731no—
337#1825—731no—
338#1809—731no—
339#1787—731no—
340#1763—731no—
341C4×#103—731no—
342#1753—731no—
343C73#2258111yescharacteristic, complemented, Sylow
344C23×Q8—731no—
345C22×M16—731no—
346C2×C4×Q8—731no—
347C2×C4×Q8—731no—
348C2×#117C2×D7311yescharacteristic
349C2×#107—731no—
350C2×#106—731no—
351#419—738no—
352C2×#103—731no—
353C2×#103—731no—
354C2×#103—731no—
355C2×#103—731no—
356C2×#103—731no—
357C2×#103—731no—
358C2×#103—731no—
359C2×#103—731no—
360C2×C22⋊C8—731no—
361C2×#97—731no—
362C2×C4×C8—731no—
363#391—731no—
364#391—731no—
365#391—1461no—
366#391—1461no—
367#389C2×D7311yescharacteristic
368#389C2×D7311yescharacteristic
369#388—21no—
370#388—21no—
371#383—21no—
372#380—731no—
373#380—1461no—
374#380C2×D7311yescharacteristic
375C4×C4⋊C4—21no—
376C4×C4⋊C4C2×D7311yescharacteristic
377C4×C4⋊C4C2×D7311yescharacteristic
378C4×C22⋊C4—1461no—
379C4×C22⋊C4C2×D7311yescharacteristic
380#340—731no—
381#340—731no—
382#340—731no—
383#340—731no—
384#340—731no—
385#340—731no—
386#336—731no—
387#336—731no—
388C22×Q8—2921no—
389C22×Q8—2921no—
390C22×Q8—734no—
391C22×Q8—731no—
392C22×Q8—731no—
393C22×Q8—1462no—
394C22×Q8—1462no—
395C22×Q8—1461no—
396C22×Q8—1462no—
397C23×C4—1461no—
398C23×C4C22×D7311yescharacteristic
399C2×M16—2921no—
400C2×M16—2921no—
401C2×M16—734no—
402C22×C8—731no—
403C22×C8—731no—
404C22×C8—731no—
405C22×C8—731no—
406C4×Q8—2921no—
407C4×Q8—2921no—
408C4×Q8—1462no—
409C4×Q8—1462no—
410#117—22no—
411#117C4×D7314yes—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—1461no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4C22×D7311yescharacteristic
419C2×C4⋊C4C22×D7311yescharacteristic
420C2×C4⋊C4C22×D7311yescharacteristic
421C2×C4⋊C4C22×D7311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1461no—
425C2×C22⋊C4C22⋊D7311yescharacteristic
426C2×C22⋊C4C22⋊D7311yescharacteristic
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—1461no—
431C2×C42—1461no—
432C2×C42C22×D7311yescharacteristic
433C2×C42C22×D7311yescharacteristic
434#107—2921no—
435#106—2921no—
436#103—2921no—
437#103—2921no—
438#103—734no—
439#103—734no—
440#103—734no—
441#103—734no—
442#103—1462no—
443#103—1462no—
444C22⋊C8—1462no—
445#97—2921no—
446C4×C8—2921no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95C22⋊D7311yescharacteristic
458#95C22⋊D7311yescharacteristic
459C24C2×C22⋊D7311yescharacteristic
460C2×Q8—2924no—
461C2×Q8—2924no—
462C2×Q8—2924no—
463C2×Q8—2922no—
464C2×Q8—2922no—
465C2×Q8—732no—
466C2×Q8—732no—
467C2×Q8—732no—
468C2×Q8—732no—
469C2×Q8—1468no—
470C2×Q8—1461no—
471C2×Q8—1461no—
472C2×Q8—1461no—
473C2×Q8—1461no—
474C2×Q8—1464no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—2922no—
488C22×C4—2922no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1464no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4C2×C4×D7312yes—
495C22×C4C2×C22⋊D7311yescharacteristic
496C22×C4C2×C22⋊D7311yescharacteristic
497C22×C4C2×C22⋊D7311yescharacteristic
498C22×C4C2×C22⋊D7311yescharacteristic
499C22×C4C2×C22⋊D7311yescharacteristic
500C22×C4#1192402812yes—
501C22×C4C23×D7311yescharacteristic
502M16—2924no—
503C2×C8—732no—
504C2×C8—732no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—2922no—
524C4⋊C4—2922no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4C2×C4×D7312yes—
530C4⋊C4C2×C4×D7312yes—
531C22⋊C4—24no—
532C22⋊C4—22no—
533C22⋊C4—41no—
534C22⋊C4—81no—
535C22⋊C4—2922no—
536C42—24no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—81no—
542C42—81no—
543C42—2922no—
544C42—2922no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C4×C22⋊D7312yes—
552C23C22×C22⋊D7311yescharacteristic
553Q8—2928no—
554Q8—2921no—
555Q8—2921no—
556Q8—734no—
557Q8—1462no—
558Q8—1462no—
559Q8—1462no—
560Q8—1462no—
561Q8—1462no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2928no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—1462no—
601C2×C4—1462no—
602C2×C4—1462no—
603C2×C4—1462no—
604C2×C4C4×C22⋊D7312yes—
605C2×C4#1455662112yes—
606C2×C4C22×C4×D7311yescharacteristic
607C2×C4#41×D7311yescharacteristic
608C2×C4#41×D7311yescharacteristic
609C2×C4C2×#1192402811yescharacteristic
610C2×C4#1455663711yescharacteristic
611C2×C4#1455663711yescharacteristic
612C8—2921no—
613C8—2921no—
614C8—1462no—
615C8—1462no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D7311yescharacteristic
624C22#1798922511yescharacteristic
625C22#1798922511yescharacteristic
626C22C2×#1455662111yescharacteristic
627C22#1798938511yescharacteristic
628C22#1798939811yescharacteristic
629C22#1798940211yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—2922no—
640C4—5841no—
641C4—5841no—
642C4—1464no—
643C4#117×D7311yescharacteristic
644C4#1798906111yescharacteristic
645C4#1798926111yescharacteristic
646C4#1798926111yescharacteristic
647C2—42no—
648C2#2261278111yescharacteristic
649C2C4×#1455662111yescharacteristic
650C2#2261877011yescharacteristic
651C2#2262199111yescharacteristic
652C2#2262209611yescharacteristic
653C2#2262217811yescharacteristic
654C2#2262220811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2347C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#2347D73
hypocenterC73#22581
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
3. #22610620C2semidirect
4. #22610606C2semidirect
5. #22610572C2semidirect
6. #22610565C2semidirect
7. #22610198C2semidirect
8. #22610184C2semidirect
12. #22609764C2semidirect
13. C4×#14556514C2semidirect
22. #17988595C4semidirect
343. C73#22581semidirect

Maximal subgroups

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

subgroupindex
C2×#179885952
C22⋊C4×Dic1462
#226106202
#226106062
#226105722
#226105652
#226101982
#226101842
#226100262
#226099622
#226098652
#226097642
C4×#145565142
#226097152
C73×#23472
#2258173