Finite Group Databasebeta

← #34028289

Subgroups — #34028289

Order 18688.

766 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989636C211yesmaximal, characteristic, complemented
2C2×#17989611C211yesmaximal, characteristic
3#22626104C211yesmaximal, characteristic, complemented
4#22626029C211yesmaximal, characteristic, complemented
5#22626004C211yesmaximal, characteristic
6#22626001C211yesmaximal, characteristic
7#22626001C211yesmaximal, characteristic
8#22626000C211yesmaximal, characteristic
9#22625990C211yesmaximal, characteristic, complemented
10#22625984C211yesmaximal, characteristic, complemented
11#22625731C211yesmaximal, characteristic, complemented
12#22625731C211yesmaximal, characteristic, complemented
13#22625723C211yesmaximal, characteristic, complemented
14#22625661C211yesmaximal, characteristic
15#22613206C211yesmaximal, characteristic
16#17989636—24no—
17#17989611—24no—
18C2×#14556658C2211yescharacteristic
19C2×#14556658C2211yescharacteristic
20C2×#14556658C2211yescharacteristic
21C2×#14556658C2211yescharacteristic
22C2×C4⋊#10596833C2211yescharacteristic
23C2×C4⋊#10596833C2211yescharacteristic
24C2×C22⋊#10596833C2211yescharacteristic
25C2×C22⋊#10596833C2211yescharacteristic
26C2×C22⋊#10596833C2211yescharacteristic
27C2×C22⋊#10596833C2211yescharacteristic
28C2×#14556646C2211yescharacteristic
29C2×#14556646C2211yescharacteristic
30C2×#14556646C2211yescharacteristic
31C2×C4×#10596833C2211yescharacteristic
32C2×C4×#10596833C2211yescharacteristic
33C2×C4×#10596833C2211yescharacteristic
34C2×C4×#10596833C2211yescharacteristic
35C2×C4×#10596833C2211yescharacteristic
36#17989453C2211yescharacteristic
37#17989453C2211yescharacteristic
38#17989453C2211yescharacteristic
39#17989453C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44#17989453C2211yescharacteristic
45#17989453C2211yescharacteristic
46C2×#14556556C2211yescharacteristic
47C2×#14556547C2211yescharacteristic
48#17988528C411yescharacteristic
49#17988523C411yescharacteristic
50#17988520C411yescharacteristic
51#17988517C411yescharacteristic
52#17988510C411yescharacteristic
53#17988507C411yescharacteristic
54#17988499C411yescharacteristic
55C4⋊C4×Dic73C2211yescharacteristic
56C4⋊C4×Dic73C2211yescharacteristic
57#17988438C2211yescharacteristic
58C22⋊C4×Dic73C2211yescharacteristic
59C22⋊C4×Dic73C2211yescharacteristic
60#396×C73C411yescharacteristic
61C22×#10596833C2311yescharacteristic
62C22×#10596833C2311yescharacteristic
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67C22×#10596833C2311yescharacteristic
68C22×#10596833C2311yescharacteristic
69#14556658—22no—
70#14556658—22no—
71#14556658—22no—
72#14556658D414yes—
73C4⋊#10596833—22no—
74C4⋊#10596833—22no—
75C22⋊#10596833—22no—
76C22⋊#10596833—22no—
77C22⋊#10596833—22no—
78C22⋊#10596833—22no—
79#14556646—22no—
80#14556646—22no—
81#14556646—22no—
82C4×#10596833—22no—
83C4×#10596833—22no—
84C4×#10596833—22no—
85C4×#10596833—22no—
86C4×#10596833—22no—
87C22×Dic146C2×C411yescharacteristic
88C23×Dic73C2311yescharacteristic
89#14556556—22no—
90#14556556—22no—
91#14556547—22no—
92#14556547—22no—
93C2×C4⋊Dic73C2×C411yescharacteristic
94C2×C4⋊Dic73C2×C411yescharacteristic
95C2×#11924007C2×C411yescharacteristic
96C2×#11924007C2×C411yescharacteristic
97C2×#11924007C2×C411yescharacteristic
98C2×#11924007C2×C411yescharacteristic
99C2×#11924007C2×C411yescharacteristic
100C2×#11924007C2×C411yescharacteristic
101C2×C22⋊Dic73C2×C411yescharacteristic
102C2×C22⋊Dic73C2×C411yescharacteristic
103C2×C22⋊Dic73C2×C411yescharacteristic
104C2×C4×Dic73C2311yescharacteristic
105C2×C4×Dic73C2311yescharacteristic
106C2×C4×Dic73C2311yescharacteristic
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2311yescharacteristic
109C2×C4×Dic73C2311yescharacteristic
110#14556498C2×C411yescharacteristic
111#14556498C2×C411yescharacteristic
112#14556498C2×C411yescharacteristic
113#14556498C2×C411yescharacteristic
114#14556498C2×C411yescharacteristic
115#14556498C2×C411yescharacteristic
116#14556498C2×C411yescharacteristic
117#14556498C2×C411yescharacteristic
118#14556498C2×C411yescharacteristic
119C2×C4⋊C4×C73C2×C411yescharacteristic
120C2×C4⋊C4×C73C2×C411yescharacteristic
121C2×C4⋊C4×C73C2×C411yescharacteristic
122C2×C22⋊C4×C73C2×C411yescharacteristic
123C2×C22⋊C4×C73C2×C411yescharacteristic
124C2×C22⋊C4×C73C2×C411yescharacteristic
125#95×C73C2×C411yescharacteristic
126C2×#10596833—21no—
127C2×#10596833—21no—
128C2×#10596833—21no—
129C2×#10596833—21no—
130C2×#10596833—21no—
131C2×#10596833—21no—
132C2×#10596833—21no—
133C2×#10596833—21no—
134C2×#10596833—21no—
135C2×#10596833—21no—
136C2×#10596833—21no—
137C2×#10596833—21no—
138C2×#10596833—21no—
139C2×#10596833—21no—
140C2×#10596833—21no—
141C2×#10596833—21no—
142C2×#10596833—21no—
143C2×#10596833—21no—
144C2×#10596833—21no—
145C2×#10596833—21no—
146C2×#10596833C2×D412yes—
147C2×#10596833C2×D412yes—
148C2×#10596833#4112yes—
149C2×#10596833#4112yes—
150C2×Dic146—22no—
151C2×Dic146—22no—
152C2×Dic146—22no—
153C22×Dic73—21no—
154C22×Dic73—21no—
155C22×Dic73—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73C22×C411yescharacteristic
160C22×Dic73C22×C411yescharacteristic
161C22×Dic73C22×C411yescharacteristic
162C22×Dic73C22×C411yescharacteristic
163C22×Dic73C22×C411yescharacteristic
164C22×Dic73C22×C411yescharacteristic
165C22×Dic73C22×C411yescharacteristic
166C22×Dic73C2411yescharacteristic
167C4⋊Dic73—22no—
168C4⋊Dic73—41no—
169#11924007—22no—
170#11924007—22no—
171#11924007—22no—
172#11924007—22no—
173#11924007—22no—
174#11924007—22no—
175C22⋊Dic73—22no—
176C22⋊Dic73—22no—
177C22⋊Dic73—22no—
178C4×Dic73—21no—
179C4×Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C4×Dic73—21no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73C2×D412yes—
189C4×Dic73#4112yes—
190C24×C73C22×C411yescharacteristic
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C22×C411yescharacteristic
193C22×C4×C73C22×C411yescharacteristic
194C22×C4×C73C22×C411yescharacteristic
195C22×C4×C73C22×C411yescharacteristic
196C22×C4×C73C22×C411yescharacteristic
197C4⋊C4×C73—22no—
198C4⋊C4×C73—22no—
199C4⋊C4×C73—41no—
200C22⋊C4×C73—22no—
201C22⋊C4×C73—22no—
202C22⋊C4×C73—41no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73C23×C411yescharacteristic
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73C4×D412yes—
228C2×C4×C73#13112yes—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
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—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73C4×D412yes—
252C2×Dic73#13112yes—
253C2×Dic73#13112yes—
254C2×Dic73C22×D411yescharacteristic
255C2×Dic73C2×#4111yescharacteristic
256C2×Dic73C2×#4111yescharacteristic
257C2×Dic73C2×#4111yescharacteristic
258C2×Dic732−1+411yescharacteristic
259C2×Dic732−1+411yescharacteristic
260Dic146—28no—
261#10596833—22no—
262#10596833—22no—
263#10596833—22no—
264#10596833—22no—
265#10596833—41no—
266#10596833—41no—
267#10596833—41no—
268#10596833—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73C2×C4×D411yescharacteristic
277C22×C73C4×#4111yescharacteristic
278C22×C73#51911yescharacteristic
279C22×C73C2×#13111yescharacteristic
280C22×C73C2×#13111yescharacteristic
281C22×C73C2×#13111yescharacteristic
282C22×C73#56811yescharacteristic
283C4×C73—22no—
284C4×C73—22no—
285C4×C73—41no—
286C4×C73—41no—
287C4×C73—41no—
288C4×C73—41no—
289Dic73—22no—
290Dic73—22no—
291Dic73—22no—
292Dic73—41no—
293Dic73—41no—
294Dic73—41no—
295Dic73—41no—
296Dic73#54711yescharacteristic
297Dic73#54811yescharacteristic
298Dic73#55211yescharacteristic
299Dic73#55611yescharacteristic
300#23056—731nomaximal, Sylow
301C2×C73—81no—
302C2×C73#236211yescharacteristic
303C2×C73#292911yescharacteristic
304C2×C73#293511yescharacteristic
305C2×C73#296611yescharacteristic
306C2×C73#296911yescharacteristic
307C2×C73#298511yescharacteristic
308C2×C73#298611yescharacteristic
309C2×C8×Q8—731no—
310C2×#433—731no—
311#2396—731no—
312#1985—731no—
313#1939—731no—
314#1924—731no—
315#1923—731no—
316#1923—731no—
317#1923—731no—
318#1917—731no—
319#1916—731no—
320#1771—731no—
321#1771—731no—
322C8×C4⋊C4—731no—
323#1750—731no—
324C73#2305611yescharacteristic, complemented, Sylow
325C2×C4×Q8—731no—
326C2×#117—731no—
327C8×Q8—1464no—
328#433—1464no—
329C2×C4⋊C8—731no—
330C2×C4⋊C8—731no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—731no—
333C2×C4⋊C8—731no—
334C2×C4⋊C8—731no—
335C2×C22⋊C8—731no—
336C2×C22⋊C8—731no—
337C2×C22⋊C8—731no—
338C2×C22⋊C8—731no—
339C2×#97—731no—
340C2×#97—731no—
341C2×#97—731no—
342C2×C4×C8—731no—
343C2×C4×C8—731no—
344C2×C4×C8—731no—
345C2×C4×C8—731no—
346C2×C4×C8—731no—
347#400—731no—
348#399—731no—
349#398—731no—
350#398—731no—
351#396—731no—
352#396#6440811yescharacteristic
353#395—731no—
354#393—731no—
355C4×C4⋊C4—731no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C22⋊C4—731no—
359C4×C22⋊C4—731no—
360#336—731no—
361#336—731no—
362#336—731no—
363#336—731no—
364#336—731no—
365#336—731no—
366#336—731no—
367#336—731no—
368#336—731no—
369#336—731no—
370C22×Q8—731no—
371C23×C4—731no—
372C22×C8—731no—
373C22×C8—731no—
374C22×C8—731no—
375C22×C8—731no—
376C22×C8—731no—
377C22×C8—731no—
378C22×C8—731no—
379C22×C8—731no—
380C4×Q8—1462no—
381C4×Q8—1462no—
382#117—1462no—
383#117—1462no—
384C2×C4⋊C4—731no—
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—731no—
392C2×C4⋊C4C2×#6440811yescharacteristic
393C2×C4⋊C4C2×#6440811yescharacteristic
394C2×C4⋊C4C2×#6440811yescharacteristic
395C2×C22⋊C4—731no—
396C2×C22⋊C4—731no—
397C2×C22⋊C4—731no—
398C2×C22⋊C4C2×#6440811yescharacteristic
399C2×C22⋊C4C2×#6440811yescharacteristic
400C2×C22⋊C4C2×#6440811yescharacteristic
401C2×C42—731no—
402C2×C42—731no—
403C2×C42—731no—
404C2×C42—731no—
405C2×C42—731no—
406C2×C42—731no—
407C4⋊C8—734no—
408C4⋊C8—1462no—
409C4⋊C8—1462no—
410C4⋊C8—1462no—
411C4⋊C8—1462no—
412C4⋊C8—1462no—
413C22⋊C8—1462no—
414C22⋊C8—1462no—
415C22⋊C8—1462no—
416C22⋊C8—1462no—
417#97—1462no—
418#97—1462no—
419#97—1462no—
420C4×C8—1462no—
421C4×C8—1462no—
422C4×C8—1462no—
423C4×C8—1462no—
424C4×C8—1462no—
425#95—731no—
426#95—731no—
427#95—731no—
428#95—731no—
429#95—731no—
430#95—731no—
431#95—731no—
432#95—731no—
433#95—731no—
434#95C2×#6440811yescharacteristic
435C24C22×#6440811yescharacteristic
436C2×Q8—1462no—
437C2×Q8—1462no—
438C2×Q8—1462no—
439C22×C4—731no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—731no—
446C22×C4—731no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4C22×#6440811yescharacteristic
454C22×C4C22×#6440811yescharacteristic
455C22×C4C22×#6440811yescharacteristic
456C22×C4C22×#6440811yescharacteristic
457C22×C4C22×#6440811yescharacteristic
458C22×C4C22×#6440811yescharacteristic
459C2×C8—732no—
460C2×C8—732no—
461C2×C8—732no—
462C2×C8—732no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C2×C8—1461no—
473C2×C8—1461no—
474C2×C8—1461no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—41no—
486C4⋊C4—2921no—
487C4⋊C4—1462no—
488C4⋊C4—1462no—
489C4⋊C4—1462no—
490C4⋊C4—1462no—
491C4⋊C4—1462no—
492C4⋊C4—1462no—
493C4⋊C4—1462no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—41no—
497C22⋊C4—1462no—
498C22⋊C4—1462no—
499C22⋊C4—1462no—
500C42—732no—
501C42—732no—
502C42—1461no—
503C42—1461no—
504C42—1461no—
505C42—1461no—
506C42—1461no—
507C42—1461no—
508C42—1461no—
509C42—1461no—
510C42—1461no—
511C42—1461no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23C23×#6440811yescharacteristic
520Q8—1468no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—731no—
542C2×C4—731no—
543C2×C4—731no—
544C2×C4—731no—
545C2×C4—731no—
546C2×C4—731no—
547C2×C4—732no—
548C2×C4—732no—
549C2×C4—732no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4—1461no—
553C2×C4—1461no—
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×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4D4×#6440812yes—
569C2×C4#1455667812yes—
570C8—2921no—
571C8—2921no—
572C8—2921no—
573C8—2921no—
574C8—1462no—
575C8—1462no—
576C8—1462no—
577C8—1462no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×D4×#6440811yescharacteristic
586C22#41×#6440811yescharacteristic
587C22#1798965511yescharacteristic
588C22C2×#1455667811yescharacteristic
589C22D73⋊#13111yescharacteristic
590C22D73⋊#13111yescharacteristic
591C22#1798966511yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—2921no—
599C4—2921no—
600C4—2921no—
601C4—2921no—
602C4—731no—
603C4—731no—
604C4—731no—
605C4—731no—
606C4—1462no—
607C4—1462no—
608C4—1462no—
609C2—81no—
610C2#2262645711yescharacteristic
611C2#2262664511yescharacteristic
612C2#2262667511yescharacteristic
613C2#2262674911yescharacteristic
614C2#2262676211yescharacteristic
615C2#2262683711yescharacteristic
616C2#2262684011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#396×C73C4
prime radicalC24×C73C22×C4
cosocleC22×Dic73C24
hypercenter#396#64408
hypocenterC73#23056
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#17989636C2semidirect
3. #22626104C2semidirect
4. #22626029C2semidirect
9. #22625990C2semidirect
10. #22625984C2semidirect
11. #22625731C2semidirect
12. #22625731C2semidirect
13. #22625723C2semidirect
324. C73#23056semidirect

Maximal subgroups

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

subgroupindex
C2×#179896362
C2×#179896112
#226261042
#226260292
#226260042
#226260012
#226260002
#226259902
#226259842
#226257312
#226257232
#226256612
#226132062
#2305673