Finite Group Databasebeta

← #33107997

Subgroups — #33107997

Order 18688.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613031C211yesmaximal, characteristic, complemented
2D4×C4⋊Dic73C211yesmaximal, characteristic
3#22610901C211yesmaximal, characteristic, complemented
4#22610695C211yesmaximal, characteristic, complemented
5#22610670C211yesmaximal, characteristic, complemented
6#22610660C211yesmaximal, characteristic, complemented
7#22610518C211yesmaximal, characteristic, complemented
8#22610504C211yesmaximal, characteristic, complemented
9#22610194C211yesmaximal, characteristic
10#22610194C211yesmaximal, characteristic
11#22609808C211yesmaximal, characteristic, complemented
12#22609758C211yesmaximal, characteristic
13#22609756C211yesmaximal, characteristic
14#102×Dic73C211yesmaximal, characteristic
15C2×#478×C73C211yesmaximal, characteristic
16C2×D4×Dic73C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C22×C4⋊Dic73—21no—
19C2×#14556523C2211yescharacteristic
20C2×#14556523C2211yescharacteristic
21C2×#14556512C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556502C2211yescharacteristic
24C2×#14556501C2211yescharacteristic
25C2×C8×Dic73C2211yescharacteristic
26#17988486—21no—
27#17988486C2211yescharacteristic
28#17988486C2211yescharacteristic
29#17988486C2211yescharacteristic
30#17988486C2211yescharacteristic
31#17988473—21no—
32#17988473—21no—
33#17988462—21no—
34#17988462—21no—
35#17988457C2211yescharacteristic
36#17988447—21no—
37C4⋊C4×Dic73—21no—
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×#125×C73C2211yescharacteristic
53C2×C4×D4×C73C2211yescharacteristic
54#478×C73C418yescomplemented
55C2×#107×C73C2211yescharacteristic
56C2×#106×C73C2211yescharacteristic
57C2×C4⋊C8×C73C2211yescharacteristic
58C2×#102×C73C2211yescharacteristic
59C2×#102×C73C2211yescharacteristic
60C23×Dic73—21no—
61C23×Dic73—21no—
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64D4×Dic73—22no—
65D4×Dic73—22no—
66D4×Dic73—41no—
67D4×Dic73—41no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—44no—
75C2×C4⋊Dic73D411yescharacteristic
76C2×C4⋊Dic73D411yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C22⋊Dic73—21no—
82C2×C22⋊Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73C2311yescharacteristic
87C2×C4×Dic73C2311yescharacteristic
88#14556523—41no—
89#14556523—41no—
90#14556512—22no—
91#14556512Q814yes—
92#14556502—22no—
93#14556501—41no—
94C8×Dic73—41no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498D411yescharacteristic
103#14556498D411yescharacteristic
104C22×D4×C73C2311yescharacteristic
105C23×C4×C73—21no—
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108#125×C73—22no—
109#125×C73C2×C414yescomplemented
110C4×D4×C73—22no—
111C4×D4×C73C2×C414yes—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C22⋊C4×C73—21no—
118C2×C42×C73C2311yescharacteristic
119#107×C73C2×C414yescomplemented
120#106×C73C2×C414yescomplemented
121C4⋊C8×C73C2×C414yescomplemented
122#102×C73C2×C414yes—
123#102×C73C2×C414yes—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—42no—
129C22×Dic73—42no—
130C22×Dic73—42no—
131C22×Dic73—42no—
132C22×Dic73—81no—
133C22×Dic73—81no—
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831C2×Q812yes—
144C4⋊Dic73—21no—
145C4⋊Dic73—22no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—41no—
153C4⋊Dic73—42no—
154C4⋊Dic73—42no—
155C4⋊Dic73—42no—
156C4⋊Dic73—42no—
157C4⋊Dic73—81no—
158C4⋊Dic73—81no—
159C4⋊Dic73C2×Q812yes—
160C22⋊Dic73—42no—
161C22⋊Dic73—42no—
162C4×Dic73—21no—
163C4×Dic73—21no—
164C4×Dic73—21no—
165C4×Dic73—21no—
166C4×Dic73—42no—
167C4×Dic73—81no—
168C4×Dic73—81no—
169C24×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73C22×C412yes—
174C2×D4×C73C2×Q812yes—
175C2×D4×C73#4112yes—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—24no—
180C22×C4×C73—42no—
181C22×C4×C73—42no—
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2411yescharacteristic
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73C22×C412yes—
190C2×C8×C73C22×C412yes—
191C4⋊C4×C73—24no—
192C4⋊C4×C73—24no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73C22×C412yes—
197C4⋊C4×C73C22×C412yes—
198C4⋊C4×C73C22×C412yes—
199C22⋊C4×C73—24no—
200C42×C73C22×C412yes—
201C42×C73#4112yes—
202C23×C73—22no—
203C23×C73—22no—
204C23×C73—22no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73C22×D411yescharacteristic
210D4×C73—22no—
211D4×C73—22no—
212D4×C73—22no—
213D4×C73C4×Q814yes—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—48no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73C4×D412yes—
229C2×C4×C73C4×D412yes—
230C2×C4×C73C23×C411yescharacteristic
231C2×C4×C73C22×Q811yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C2×C4×C73H(5,𝔽2)11yescharacteristic
235C2×C4×C732−1+411yescharacteristic
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—22no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—84no—
260C2×Dic73—84no—
261C2×Dic73AGL(1,ℤ/8)12yes—
262C8×C73—22no—
263C8×C73—22no—
264#10596831—22no—
265#10596831—41no—
266C22×C73—24no—
267C22×C73—42no—
268C22×C73—42no—
269C22×C73—42no—
270C22×C73—42no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73C2×C4×D411yescharacteristic
274C22×C73#54611yescharacteristic
275C22×C73D4×Q811yescharacteristic
276C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C73#57511yescharacteristic
278C22×C73#57711yescharacteristic
279C22×C73#57711yescharacteristic
280C4×C73—22no—
281C4×C73—22no—
282C4×C73—42no—
283C4×C73—42no—
284C4×C73—42no—
285C4×C73C2×C4×Q811yescharacteristic
286C4×C73#52011yescharacteristic
287C4×C73#55211yescharacteristic
288C4×C73#55411yescharacteristic
289Dic73—22no—
290Dic73—42no—
291Dic73—41no—
292Dic73—41no—
293Dic73—41no—
294Dic73—81no—
295Dic73—81no—
296Dic73—81no—
297#22978—731nomaximal, Sylow
298C2×C73—44no—
299C2×C73#236011yescharacteristic
300C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C73#295211yescharacteristic
302C2×C73#330711yescharacteristic
303C2×C73#331011yescharacteristic
304C2×C73#338411yescharacteristic
305C2×C73#338811yescharacteristic
306C2×#478D7311yescharacteristic
307#2370—731no—
308D4×C4⋊C4—731no—
309#1992—731no—
310#1948—731no—
311#1943—731no—
312#1941—731no—
313#1920—731no—
314#1918—731no—
315#1864—731no—
316#1864—731no—
317#1771—731no—
318#1762—731no—
319#1762—731no—
320C4×#102—731no—
321C73#2297811yescharacteristic, complemented, Sylow
322C2×#125C2×D7311yescharacteristic
323C2×C4×D4—731no—
324C2×C4×D4—731no—
325C2×C4×D4C2×D7311yescharacteristic
326C22×C4⋊C4—1461no—
327#478Dic7318yescomplemented
328C2×#107C2×D7311yescharacteristic
329C2×#106—731no—
330C2×#106—731no—
331C2×#106C2×D7311yescharacteristic
332C2×C4⋊C8C2×D7311yescharacteristic
333C2×#102—731no—
334C2×#102—731no—
335C2×#102C2×D7311yescharacteristic
336C2×#102C2×D7311yescharacteristic
337C2×#97—731no—
338C2×#97—731no—
339C2×C4×C8—731no—
340#389—731no—
341#389—731no—
342#389—731no—
343#389—731no—
344#389—1461no—
345#387—1461no—
346#387—1461no—
347#385—1461no—
348#385—1461no—
349#384—731no—
350#380—1461no—
351C4×C4⋊C4—731no—
352C4×C4⋊C4—731no—
353C4×C4⋊C4—731no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—1461no—
356#340—731no—
357#340—731no—
358#340—731no—
359#340—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#336—731no—
365#336—731no—
366C22×D4C22×D7311yescharacteristic
367C23×C4—21no—
368C23×C4—1461no—
369C23×C4—1461no—
370C22×C8—731no—
371C22×C8—731no—
372C22×C8C22×D7311yescharacteristic
373C22×C8C22×D7311yescharacteristic
374#125—22no—
375#125C2×Dic7314yescomplemented
376C4×D4—22no—
377C4×D4—2921no—
378C4×D4—2921no—
379C4×D4—1462no—
380C4×D4—1462no—
381C4×D4C2×Dic7314yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—2924no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1461no—
402C2×C22⋊C4—1461no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—1461no—
406C2×C42—1461no—
407C2×C42—1461no—
408C2×C42C22×D7311yescharacteristic
409#107C2×Dic7314yescomplemented
410#106—2921no—
411#106—2921no—
412#106C2×Dic7314yescomplemented
413C4⋊C8C2×Dic7314yescomplemented
414#102—734no—
415#102—1462no—
416#102C2×Dic7314yes—
417#102C2×Dic7314yes—
418#97—2921no—
419#97—1462no—
420C4×C8—2921no—
421#95—731no—
422#95—731no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C22×Dic7312yes—
435C2×D4Q8×D7312yes—
436C2×D4#1192402912yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—2922no—
444C22×C4—2922no—
445C22×C4—2922no—
446C22×C4—2922no—
447C22×C4—5841no—
448C22×C4—5841no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—731no—
452C22×C4—731no—
453C22×C4—1461no—
454C22×C4—1461no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4D4×D7311yescharacteristic
458C22×C4D4×D7311yescharacteristic
459C22×C4C23×D7311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—732no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8C22×Dic7312yes—
471C2×C8C22×Dic7312yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—2922no—
478C4⋊C4—2921no—
479C4⋊C4—2922no—
480C4⋊C4—2922no—
481C4⋊C4—2922no—
482C4⋊C4—5841no—
483C4⋊C4—5841no—
484C4⋊C4—732no—
485C4⋊C4—1462no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4C22×Dic7312yes—
494C4⋊C4C22×Dic7312yes—
495C4⋊C4C22×Dic7312yes—
496C22⋊C4—24no—
497C22⋊C4—2922no—
498C22⋊C4—2922no—
499C42—2922no—
500C42—5841no—
501C42—5841no—
502C42—1461no—
503C42—1461no—
504C42—1461no—
505C42—1461no—
506C42C22×Dic7312yes—
507C42#1192402712yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D7311yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4Q8×Dic7314yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—48no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—5844no—
546C2×C4—5844no—
547C2×C4—732no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1462no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4D4×Dic7312yes—
561C2×C4D4×Dic7312yes—
562C2×C4C23×Dic7311yescharacteristic
563C2×C4C2×Q8×D7311yescharacteristic
564C2×C4C2×#1192402711yescharacteristic
565C2×C4C2×#1192402911yescharacteristic
566C2×C4#1455663711yescharacteristic
567C2×C4#1455663811yescharacteristic
568C8—22no—
569C8—22no—
570C8—2921no—
571C8—1462no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×D4×Dic7311yescharacteristic
580C22#1798922811yescharacteristic
581C22#1798925011yescharacteristic
582C22AGL(1,ℤ/8)×D7311yescharacteristic
583C22#1798938311yescharacteristic
584C22#1798939411yescharacteristic
585C22#1798939411yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—2922no—
592C4—2921no—
593C4—2921no—
594C4—2921no—
595C4—5841no—
596C4—5841no—
597C4—5841no—
598C4—1462no—
599C4C2×Q8×Dic7311yescharacteristic
600C4#1798907011yescharacteristic
601C4#1798927911yescharacteristic
602C4#1798929511yescharacteristic
603C2—44no—
604C2#2261294011yescharacteristic
605C2AGL(1,ℤ/8)×Dic7311yescharacteristic
606C2#2261881511yescharacteristic
607C2#2262190411yescharacteristic
608C2#2262194911yescharacteristic
609C2#2262283311yescharacteristic
610C2#2262287311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#478×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC2×#478D73
hypocenterC73#22978
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #22613031C2semidirect
3. #22610901C2semidirect
4. #22610695C2semidirect
5. #22610670C2semidirect
6. #22610660C2semidirect
7. #22610518C2semidirect
8. #22610504C2semidirect
11. #22609808C2semidirect
54. #478×C73C4semidirect
109. #125×C73C2×C4semidirect
119. #107×C73C2×C4semidirect
120. #106×C73C2×C4semidirect
121. C4⋊C8×C73C2×C4semidirect
321. C73#22978semidirect
327. #478Dic73semidirect
375. #125C2×Dic73semidirect
409. #107C2×Dic73semidirect
412. #106C2×Dic73semidirect
413. C4⋊C8C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
#226130312
D4×C4⋊Dic732
#226109012
#226106952
#226106702
#226106602
#226105182
#226105042
#226101942
#226098082
#226097582
#226097562
#102×Dic732
C2×#478×C732
#2297873