Finite Group Databasebeta

← #33105225

Subgroups — #33105225

Order 18688.

990 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×Dic73C211yesmaximal, characteristic
2#22612879C211yesmaximal, characteristic, complemented
3D4×#11924007C211yesmaximal, characteristic, complemented
4#22610617C211yesmaximal, characteristic, complemented
5#22610611C211yesmaximal, characteristic, complemented
6#22610503C211yesmaximal, characteristic, complemented
7#22610501C211yesmaximal, characteristic, complemented
8#22610281C211yesmaximal, characteristic
9#22610057C211yesmaximal, characteristic, complemented
10#22609929C211yesmaximal, characteristic, complemented
11#22609929C211yesmaximal, characteristic, complemented
12#22609792C211yesmaximal, characteristic, complemented
13#102×Dic73C211yesmaximal, characteristic, complemented
14C4×#14556512C211yesmaximal, characteristic, complemented
15C73×#1879C211yesmaximal, characteristic
16C2×D4×Dic73C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19C2×#14556547—21no—
20C22×#11924007—21no—
21D8×Dic73C418yescomplemented
22C2×#14556525C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C4×#10596831C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988490C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988483C2211yescharacteristic
34#17988481—21no—
35#17988475—21no—
36#17988460—21no—
37#17988459—21no—
38#17988456C2211yescharacteristic
39#17988456C2211yescharacteristic
40#17988445—21no—
41#17988445—21no—
42C4⋊C4×Dic73C2211yescharacteristic
43C4×#11924007C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×D8×C73C2211yescharacteristic
54C2×C4×D4×C73C2211yescharacteristic
55C2×#102×C73C2211yescharacteristic
56C2×#102×C73C2211yescharacteristic
57#390×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C23×Dic73—21no—
61C23×Dic73—21no—
62C23×Dic73—21no—
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65D4×Dic73—22no—
66D4×Dic73—22no—
67D4×Dic73—22no—
68D4×Dic73—41no—
69D4×Dic73C2×C414yes—
70D4×Dic73C2×C414yescomplemented
71#14556547—44no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—44no—
83C2×#11924007D411yescharacteristic
84C2×#11924007D411yescharacteristic
85C2×C22⋊Dic73—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73C2311yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94#14556525C2×C414yescomplemented
95#14556520—22no—
96#14556512—22no—
97#14556512—41no—
98#14556512C2×C414yes—
99#14556512C2×C414yescomplemented
100C4×#10596831—41no—
101C8×Dic73—41no—
102C8×Dic73C2×C414yescomplemented
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498D411yescharacteristic
108#14556498D411yescharacteristic
109C22×D4×C73C2311yescharacteristic
110C22×D4×C73C2311yescharacteristic
111C23×C4×C73—21no—
112C2×D8×C73—22no—
113C2×D8×C73—22no—
114C2×D8×C73C2×C414yes—
115C22×C8×C73C2311yescharacteristic
116C22×C8×C73C2311yescharacteristic
117C4×D4×C73—22no—
118C4×D4×C73—41no—
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73—21no—
123C2×C22⋊C4×C73—21no—
124C2×C42×C73C2311yescharacteristic
125#102×C73—22no—
126#102×C73—41no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—24no—
130C22×Dic73—24no—
131C22×Dic73—21no—
132C22×Dic73—21no—
133C22×Dic73—42no—
134C22×Dic73—42no—
135C22×Dic73—42no—
136C22×Dic73—42no—
137C22×Dic73—42no—
138C22×Dic73—42no—
139C22×Dic73—81no—
140C22×Dic73C2×D411yescharacteristic
141C22×Dic73C2×D411yescharacteristic
142C22×Dic73C2×D411yescharacteristic
143C22×Dic73C2×D411yescharacteristic
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831C22×C412yes—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73C22×C412yes—
157C4⋊Dic73C22×C412yes—
158#11924007—22no—
159#11924007—42no—
160#11924007—42no—
161#11924007—42no—
162#11924007—42no—
163#11924007—81no—
164#11924007—81no—
165#11924007D814yescomplemented
166C22⋊Dic73—24no—
167C22⋊Dic73—24no—
168C22⋊Dic73—42no—
169C22⋊Dic73—42no—
170C22⋊Dic73—81no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C4×Dic73C22×C412yes—
176C4×Dic73#4112yes—
177C24×C73—21no—
178C24×C73—21no—
179C2×D4×C73—21no—
180C2×D4×C73—21no—
181C2×D4×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73—21no—
184C2×D4×C73—21no—
185C2×D4×C73—21no—
186C2×D4×C73—21no—
187C2×D4×C73C22×C412yes—
188C2×D4×C73C22×C412yes—
189C2×D4×C73#4112yes—
190C2×D4×C73#4112yes—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—42no—
194C22×C4×C73—42no—
195C22×C4×C73—81no—
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2411yescharacteristic
199D8×C73—28no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73C22×C412yes—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C22⋊C4×C73—42no—
211C22⋊C4×C73—42no—
212C22⋊C4×C73—81no—
213C42×C73—21no—
214C42×C73—21no—
215C23×C73—22no—
216C23×C73—22no—
217C23×C73—22no—
218C23×C73—22no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73—41no—
229C23×C73C22×D411yescharacteristic
230D4×C73—22no—
231D4×C73—22no—
232D4×C73—22no—
233D4×C73—22no—
234D4×C73—22no—
235D4×C73—41no—
236D4×C73—41no—
237D4×C73#11714yescomplemented
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—84no—
250C2×C4×C73C2×D812yes—
251C2×C4×C73C23×C411yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73H(5,𝔽2)11yescharacteristic
256C2×C4×C732−1+411yescharacteristic
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—48no—
271C2×Dic73—48no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—84no—
279C2×Dic73C4×D412yes—
280C2×Dic73C4×D412yes—
281C2×Dic73C2×D812yes—
282C2×Dic73C2×D812yes—
283C2×Dic73#13512yes—
284C8×C73—22no—
285C8×C73—41no—
286#10596831—22no—
287#10596831—41no—
288C22×C73—24no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—42no—
292C22×C73—42no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73—42no—
296C22×C73—42no—
297C22×C73—42no—
298C22×C73—81no—
299C22×C73—81no—
300C22×C73—81no—
301C22×C73—81no—
302C22×C73C2×C4×D411yescharacteristic
303C22×C73#54611yescharacteristic
304C22×C73#54811yescharacteristic
305C22×C73C22×D811yescharacteristic
306C22×C73C2×#13511yescharacteristic
307C22×C73#57711yescharacteristic
308C22×C73#57711yescharacteristic
309C4×C73—22no—
310C4×C73—41no—
311C4×C73—81no—
312C4×C73—81no—
313C4×C73C2×#11711yescharacteristic
314C4×C73#52011yescharacteristic
315C4×C73#55211yescharacteristic
316C4×C73#55311yescharacteristic
317Dic73—22no—
318Dic73—22no—
319Dic73—42no—
320Dic73—42no—
321Dic73—41no—
322Dic73—81no—
323Dic73—81no—
324Dic73C4×D814yes—
325#22766—731nomaximal, Sylow
326C2×C73—44no—
327C2×C73—82no—
328C2×C73#235211yescharacteristic
329C2×C73C2×C4×D811yescharacteristic
330C2×C73#295211yescharacteristic
331C2×C73#329411yescharacteristic
332C2×C73#330311yescharacteristic
333C2×C73#335811yescharacteristic
334C2×C73#336011yescharacteristic
335C2×C4×D8—731no—
336#2354—731no—
337D4×C4⋊C4—731no—
338#1934—731no—
339#1934—731no—
340#1918—731no—
341#1918—731no—
342#1879—731no—
343#1879D7311yescharacteristic
344#1834—731no—
345#1802—731no—
346#1802—731no—
347C8×C4⋊C4—731no—
348C4×#102—731no—
349C4×#102—731no—
350C73#2276611yescharacteristic, complemented, Sylow
351C22×D8C2×D7311yescharacteristic
352C2×C4×D4—731no—
353C2×C4×D4—731no—
354C2×C4×D4—731no—
355C2×C4×D4C2×D7311yescharacteristic
356C2×#117—1461no—
357C22×C4⋊C4—1461no—
358C4×D8—738no—
359C2×#107—731no—
360C2×C4⋊C8—731no—
361C2×#102—731no—
362C2×#102—731no—
363C2×#102—731no—
364C2×#102—731no—
365C2×#102C2×D7311yescharacteristic
366C2×#102C2×D7311yescharacteristic
367C2×C4×C8—731no—
368C2×C4×C8—731no—
369C2×C4×C8—731no—
370#390—731no—
371#390C2×D7311yescharacteristic
372#389—731no—
373#389—731no—
374#388—1461no—
375#388—1461no—
376#385—1461no—
377#385—1461no—
378#384—731no—
379#384—731no—
380#380—1461no—
381#380—1461no—
382C4×C4⋊C4—731no—
383C4×C4⋊C4—731no—
384C4×C4⋊C4—731no—
385#340—731no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340C2×D7311yescharacteristic
393#336—731no—
394#336C2×D7311yescharacteristic
395C22×D4C22×D7311yescharacteristic
396C22×D4C22×D7311yescharacteristic
397C23×C4—21no—
398C23×C4—1461no—
399C23×C4—1461no—
400C23×C4—1461no—
401C2×D8—22no—
402C2×D8—22no—
403C2×D8C4×D7314yes—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8C22×D7311yescharacteristic
407C22×C8C22×D7311yescharacteristic
408C4×D4—22no—
409C4×D4—41no—
410C4×D4—2921no—
411C4×D4—734no—
412C4×D4—734no—
413C4×D4—1462no—
414C4×D4—1462no—
415C4×D4—1462no—
416#117—2924no—
417C2×C4⋊C4—2924no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4C22×D7311yescharacteristic
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1461no—
436C2×C22⋊C4—1461no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C42—731no—
441C2×C42—731no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42C22×D7311yescharacteristic
445#107—734no—
446C4⋊C8—1462no—
447#102—22no—
448#102—41no—
449#102—2921no—
450#102—734no—
451#102—734no—
452#102—1462no—
453C4×C8—2921no—
454C4×C8—2921no—
455C4×C8—734no—
456#95—731no—
457#95—731no—
458#95—1461no—
459#95—1461no—
460#95—1461no—
461#95—1461no—
462C24—21no—
463C24—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D7312yes—
473C2×D4C2×C4×D7312yes—
474C2×D4#1192402712yes—
475C2×D4#1192402712yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—81no—
481C22×C4—2922no—
482C22×C4—2922no—
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—2922no—
486C22×C4—2922no—
487C22×C4—5841no—
488C22×C4—731no—
489C22×C4—731no—
490C22×C4—731no—
491C22×C4—731no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1464no—
495C22×C4—1464no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4D4×D7311yescharacteristic
499C22×C4D4×D7311yescharacteristic
500C22×C4C23×D7311yescharacteristic
501D8—28no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—732no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8C2×C4×D7312yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—2922no—
519C4⋊C4—2922no—
520C4⋊C4—2922no—
521C4⋊C4—2922no—
522C4⋊C4—5841no—
523C4⋊C4—5841no—
524C4⋊C4—734no—
525C4⋊C4—732no—
526C4⋊C4—732no—
527C4⋊C4—1462no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—81no—
537C22⋊C4—2922no—
538C22⋊C4—2922no—
539C22⋊C4—5841no—
540C22⋊C4—1464no—
541C22⋊C4—1464no—
542C42—21no—
543C42—21no—
544C42—2922no—
545C42—2922no—
546C42—732no—
547C42—732no—
548C42—1461no—
549C42—1461no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C2×D4×D7311yescharacteristic
565D4—22no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572D4#1455654714yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2928no—
590C2×C4—2928no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—5844no—
598C2×C4—732no—
599C2×C4—732no—
600C2×C4—732no—
601C2×C4—732no—
602C2×C4—732no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4D8×D7312yes—
613C2×C4C22×C4×D7311yescharacteristic
614C2×C4#41×D7311yescharacteristic
615C2×C4C2×#1192402711yescharacteristic
616C2×C4C2×#1192402711yescharacteristic
617C2×C4#1455663611yescharacteristic
618C2×C4#1455663811yescharacteristic
619C8—22no—
620C8—41no—
621C8—2921no—
622C8—1462no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C4×D4×D7311yescharacteristic
638C22#1798922411yescharacteristic
639C22#1798924311yescharacteristic
640C22C2×D8×D7311yescharacteristic
641C22#135×D7311yescharacteristic
642C22#1798939711yescharacteristic
643C22#1798939711yescharacteristic
644C4—22no—
645C4—41no—
646C4—81no—
647C4—81no—
648C4—2922no—
649C4—2922no—
650C4—2921no—
651C4—5841no—
652C4—5841no—
653C4—734no—
654C4—1462no—
655C4—1462no—
656C4C2×#1455654711yescharacteristic
657C4#1798906611yescharacteristic
658C4#1798927411yescharacteristic
659C4#1798928011yescharacteristic
660C2—44no—
661C2—82no—
662C2#2261285811yescharacteristic
663C2C4×D8×D7311yescharacteristic
664C2#2261881311yescharacteristic
665C2#2262173711yescharacteristic
666C2#2262185211yescharacteristic
667C2#2262256111yescharacteristic
668C2#2262257511yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1879C2
socleC23×C73C22×D4
prime radicalC22×D8×C73C22
cosocleC22×C4×C73C24
hypercenter#1879D73
hypocenterC73#22766
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
2. #22612879C2semidirect
3. D4×#11924007C2semidirect
4. #22610617C2semidirect
5. #22610611C2semidirect
6. #22610503C2semidirect
7. #22610501C2semidirect
9. #22610057C2semidirect
10. #22609929C2semidirect
11. #22609929C2semidirect
12. #22609792C2semidirect
13. #102×Dic73C2semidirect
14. C4×#14556512C2semidirect
21. D8×Dic73C4semidirect
70. D4×Dic73C2×C4semidirect
94. #14556525C2×C4semidirect
99. #14556512C2×C4semidirect
102. C8×Dic73C2×C4semidirect
165. #11924007D8semidirect
237. D4×C73#117semidirect
350. C73#22766semidirect
572. D4#14556547semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic732
#226128792
D4×#119240072
#226106172
#226106112
#226105032
#226105012
#226102812
#226100572
#226099292
#226097922
#102×Dic732
C4×#145565122
C73×#18792
#2276673