Finite Group Databasebeta

← #33093983

Subgroups — #33093983

Order 18688.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612673C211yesmaximal, characteristic, complemented
2#22612644C211yesmaximal, characteristic
3#22610901C211yesmaximal, characteristic, complemented
4#22610748C211yesmaximal, characteristic
5#22610630C211yesmaximal, characteristic
6#22610623C211yesmaximal, characteristic, complemented
7#22610616C211yesmaximal, characteristic
8#22610609C211yesmaximal, characteristic, complemented
9#22610059C211yesmaximal, characteristic, complemented
10#22609763C211yesmaximal, characteristic, complemented
11#22609758C211yesmaximal, characteristic
12#103×Dic73C211yesmaximal, characteristic, complemented
13#102×Dic73C211yesmaximal, characteristic
14#22609617C211yesmaximal, characteristic, complemented
15C2×#488×C73C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×D4×Dic73C2211yescharacteristic
18C2×#14556515C2211yescharacteristic
19C2×#14556515C2211yescharacteristic
20C2×#14556512C2211yescharacteristic
21C2×#14556512C2211yescharacteristic
22C2×#14556502C2211yescharacteristic
23C2×#14556501C2211yescharacteristic
24C2×#14556501C2211yescharacteristic
25C2×C8×Dic73C2211yescharacteristic
26C2×C8×Dic73C2211yescharacteristic
27#17988494C2211yescharacteristic
28#17988490C2211yescharacteristic
29#17988486—21no—
30#17988486—21no—
31#17988486C2211yescharacteristic
32#17988486C2211yescharacteristic
33#17988480—21no—
34#17988480—21no—
35#17988473—21no—
36#17988473—21no—
37#17988462—21no—
38#17988462—21no—
39#17988457C2211yescharacteristic
40#17988457C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43#17988433C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52C2×#125×C73C2211yescharacteristic
53C2×#124×C73C2211yescharacteristic
54#488×C73C418yescomplemented
55C2×#103×C73C2211yescharacteristic
56C2×#103×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58C2×#102×C73C2211yescharacteristic
59C2×C4×C8×C73C2211yescharacteristic
60C23×Dic73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63Q8×Dic73—22no—
64Q8×Dic73—22no—
65D4×Dic73—22no—
66D4×Dic73—22no—
67C2×C4⋊Dic73—21no—
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⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73C2311yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86#14556515—22no—
87#14556515—22no—
88#14556512—22no—
89#14556512—22no—
90#14556502—22no—
91#14556501—22no—
92#14556501—22no—
93C8×Dic73—22no—
94C8×Dic73—22no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498D411yescharacteristic
102#14556498D411yescharacteristic
103#14556498D411yescharacteristic
104#14556498D411yescharacteristic
105C22×Q8×C73C2311yescharacteristic
106C22×D4×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C22×C8×C73C2311yescharacteristic
109#125×C73—22no—
110#125×C73C2×C414yescomplemented
111#124×C73—22no—
112#124×C73C2×C414yes—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73—21no—
119C2×C42×C73C2311yescharacteristic
120#103×C73C2×C414yescomplemented
121#103×C73C2×C414yescomplemented
122#102×C73C2×C414yes—
123#102×C73C2×C414yes—
124C4×C8×C73C2×C414yescomplemented
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—42no—
130C22×Dic73—81no—
131C22×Dic73—81no—
132C22×Dic73C2×D411yescharacteristic
133C22×Dic73C2×D411yescharacteristic
134C22×Dic73C2×D411yescharacteristic
135C22×Dic73C2×D411yescharacteristic
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831#4112yes—
141C2×#10596831#4112yes—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—42no—
151C4⋊Dic73—81no—
152C4⋊Dic73—81no—
153C4⋊Dic73—81no—
154C4⋊Dic73—81no—
155C4⋊Dic73—81no—
156C4⋊Dic73—81no—
157C22⋊Dic73—42no—
158C22⋊Dic73—81no—
159C22⋊Dic73—81no—
160C4×Dic73—21no—
161C4×Dic73—21no—
162C4×Dic73—42no—
163C4×Dic73—81no—
164C4×Dic73—81no—
165C4×Dic73#4112yes—
166C4×Dic73#4112yes—
167C24×C73—21no—
168C2×Q8×C73—21no—
169C2×Q8×C73—21no—
170C2×Q8×C73—21no—
171C2×Q8×C73—21no—
172C2×Q8×C73—21no—
173C2×Q8×C73C22×C412yes—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73—21no—
177C2×D4×C73—21no—
178C2×D4×C73—21no—
179C2×D4×C73C22×C412yes—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2411yescharacteristic
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73C22×C412yes—
191C2×C8×C73C22×C412yes—
192C4⋊C4×C73—24no—
193C4⋊C4×C73—24no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73C22×C412yes—
197C4⋊C4×C73C22×C412yes—
198C22⋊C4×C73—24no—
199C22⋊C4×C73—24no—
200C42×C73C22×C412yes—
201C42×C73#4112yes—
202C23×C73—22no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73C22×D411yescharacteristic
210Q8×C73—22no—
211Q8×C73—22no—
212Q8×C73—41no—
213Q8×C73—41no—
214D4×C73—22no—
215D4×C73—22no—
216D4×C73—41no—
217D4×C73—41no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73C4×D412yes—
232C2×C4×C73C4×D412yes—
233C2×C4×C73C23×C411yescharacteristic
234C2×C4×C73C2×#4111yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
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—41no—
260C2×Dic73—41no—
261C2×Dic73—84no—
262C2×Dic73#12412yes—
263C2×Dic73#12412yes—
264C8×C73—22no—
265C8×C73—22no—
266#10596831—22no—
267#10596831—22no—
268C22×C73—42no—
269C22×C73—42no—
270C22×C73—42no—
271C22×C73—81no—
272C22×C73—81no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73C2×C4×D411yescharacteristic
276C22×C73C2×#12411yescharacteristic
277C22×C73#53211yescharacteristic
278C22×C73#57611yescharacteristic
279C22×C73#57711yescharacteristic
280C22×C73#57711yescharacteristic
281C22×C73#57811yescharacteristic
282C4×C73—22no—
283C4×C73—22no—
284C4×C73—42no—
285C4×C73—42no—
286C4×C73—42no—
287C4×C73C4×#4111yescharacteristic
288C4×C73C4×#4111yescharacteristic
289C4×C73#52911yescharacteristic
290C4×C73#52911yescharacteristic
291Dic73—22no—
292Dic73—22no—
293Dic73—41no—
294Dic73—41no—
295Dic73—81no—
296Dic73—81no—
297Dic73—81no—
298Dic73—81no—
299#21643—731nomaximal, Sylow
300C2×C73—82no—
301C2×C73C4×#12411yescharacteristic
302C2×C73#295111yescharacteristic
303C2×C73#295211yescharacteristic
304C2×C73#313611yescharacteristic
305C2×C73#313711yescharacteristic
306C2×C73#317811yescharacteristic
307C2×C73#317911yescharacteristic
308C2×#488D7311yescharacteristic
309#2330—731no—
310#2326—731no—
311#1992—731no—
312#1959—731no—
313#1936—731no—
314#1935—731no—
315#1934—731no—
316#1933—731no—
317#1834—731no—
318#1763—731no—
319#1762—731no—
320C4×#103—731no—
321C4×#102—731no—
322#1726—731no—
323C73#2164311yescharacteristic, complemented, Sylow
324C2×#125C2×D7311yescharacteristic
325C2×#124C2×D7311yescharacteristic
326C2×C4×Q8—731no—
327C2×C4×D4—731no—
328#488Dic7318yescomplemented
329C2×#103—731no—
330C2×#103—731no—
331C2×#103C2×D7311yescharacteristic
332C2×#103C2×D7311yescharacteristic
333C2×#102—731no—
334C2×#102—731no—
335C2×#102C2×D7311yescharacteristic
336C2×#102C2×D7311yescharacteristic
337C2×#97—731no—
338C2×#97—731no—
339C2×#97—731no—
340C2×C4×C8—731no—
341C2×C4×C8—731no—
342C2×C4×C8C2×D7311yescharacteristic
343#391—731no—
344#390—731no—
345#389—731no—
346#389—731no—
347#389—1461no—
348#389—1461no—
349#388—1461no—
350#388—1461no—
351#387—1461no—
352#387—1461no—
353#385—1461no—
354#385—1461no—
355#384—731no—
356#384—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4—731no—
359#376—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368C22×Q8C22×D7311yescharacteristic
369C22×D4C22×D7311yescharacteristic
370C23×C4—1461no—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8C22×D7311yescharacteristic
374C22×C8C22×D7311yescharacteristic
375#125—22no—
376#125C2×Dic7314yescomplemented
377#124—22no—
378#124C2×Dic7314yes—
379C4×Q8—1462no—
380C4×Q8—1462no—
381C4×D4—1462no—
382C4×D4—1462no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—731no—
387C2×C4⋊C4—731no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4—1461no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1461no—
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#103—1462no—
410#103—1462no—
411#103C2×Dic7314yescomplemented
412#103C2×Dic7314yescomplemented
413#102—1462no—
414#102—1462no—
415#102C2×Dic7314yes—
416#102C2×Dic7314yes—
417#97—1462no—
418#97—1462no—
419#97—1462no—
420C4×C8—1462no—
421C4×C8—1462no—
422C4×C8C2×Dic7314yescomplemented
423#95—731no—
424#95—731no—
425#95—731no—
426#95—731no—
427#95—1461no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433C24—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C22×Dic7312yes—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C22×Dic7312yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—2922no—
450C22×C4—5841no—
451C22×C4—5841no—
452C22×C4—731no—
453C22×C4—731no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4D4×D7311yescharacteristic
461C22×C4D4×D7311yescharacteristic
462C22×C4C23×D7311yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—732no—
468C2×C8—732no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8C22×Dic7312yes—
474C2×C8C22×Dic7312yes—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—2922no—
480C4⋊C4—5841no—
481C4⋊C4—5841no—
482C4⋊C4—5841no—
483C4⋊C4—5841no—
484C4⋊C4—5841no—
485C4⋊C4—5841no—
486C4⋊C4—1461no—
487C4⋊C4—1461no—
488C4⋊C4—1461no—
489C4⋊C4—1461no—
490C4⋊C4—1461no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4C22×Dic7312yes—
495C4⋊C4C22×Dic7312yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—2922no—
499C22⋊C4—5841no—
500C22⋊C4—5841no—
501C42—2922no—
502C42—5841no—
503C42—5841no—
504C42—732no—
505C42—732no—
506C42—1461no—
507C42—1461no—
508C42C22×Dic7312yes—
509C42#1192402712yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D7311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—5844no—
552C2×C4—732no—
553C2×C4—732no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4—1461no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4D4×Dic7312yes—
565C2×C4D4×Dic7312yes—
566C2×C4C23×Dic7311yescharacteristic
567C2×C4#41×D7311yescharacteristic
568C2×C4#41×D7311yescharacteristic
569C2×C4#41×D7311yescharacteristic
570C2×C4#41×D7311yescharacteristic
571C2×C4C2×#1192402711yescharacteristic
572C8—22no—
573C8—22no—
574C8—1462no—
575C8—1462no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C2×D4×Dic7311yescharacteristic
584C22#124×D7311yescharacteristic
585C22#1798913611yescharacteristic
586C22#1798938911yescharacteristic
587C22#1798939411yescharacteristic
588C22#1798939411yescharacteristic
589C22#1798940111yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—2921no—
596C4—2921no—
597C4—5841no—
598C4—5841no—
599C4—5841no—
600C4—5841no—
601C4—1462no—
602C4—1462no—
603C4#41×Dic7311yescharacteristic
604C4#41×Dic7311yescharacteristic
605C4#1798911611yescharacteristic
606C4#1798911611yescharacteristic
607C2—82no—
608C2#124×Dic7311yescharacteristic
609C2#2261880711yescharacteristic
610C2#2261881511yescharacteristic
611C2#2262028411yescharacteristic
612C2#2262029511yescharacteristic
613C2#2262059811yescharacteristic
614C2#2262061311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#488×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC2×#488D73
hypocenterC73#21643
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#488

Complements

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

NH ≅ G/Ntype
1. #22612673C2semidirect
3. #22610901C2semidirect
6. #22610623C2semidirect
8. #22610609C2semidirect
9. #22610059C2semidirect
10. #22609763C2semidirect
12. #103×Dic73C2semidirect
14. #22609617C2semidirect
54. #488×C73C4semidirect
110. #125×C73C2×C4semidirect
120. #103×C73C2×C4semidirect
121. #103×C73C2×C4semidirect
124. C4×C8×C73C2×C4semidirect
323. C73#21643semidirect
328. #488Dic73semidirect
376. #125C2×Dic73semidirect
411. #103C2×Dic73semidirect
412. #103C2×Dic73semidirect
422. C4×C8C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
#226126732
#226126442
#226109012
#226107482
#226106302
#226106232
#226106162
#226106092
#226100592
#226097632
#226097582
#103×Dic732
#102×Dic732
#226096172
C2×#488×C732
#2164373