Finite Group Databasebeta

← #33144209

Subgroups — #33144209

Order 18688.

736 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614952C211yesmaximal, characteristic, complemented
2#22611305C211yesmaximal, characteristic
3#22611305C211yesmaximal, characteristic
4#22611077C211yesmaximal, characteristic
5#22611067C211yesmaximal, characteristic
6#22610888C211yesmaximal, characteristic, complemented
7#22610883C211yesmaximal, characteristic, complemented
8#22610692C211yesmaximal, characteristic, complemented
9#22610655C211yesmaximal, characteristic, complemented
10#22610516C211yesmaximal, characteristic, complemented
11#22610509C211yesmaximal, characteristic, complemented
12#22610066C211yesmaximal, characteristic, complemented
13#22609941C211yesmaximal, characteristic
14#22609756C211yesmaximal, characteristic
15C73×#2354C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556583C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×C4⋊#10596831C2211yescharacteristic
22C2×C4⋊#10596831C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556502C2211yescharacteristic
28#17988520—21no—
29#17988507—21no—
30#17988491—21no—
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988484—21no—
34#17988453C2211yescharacteristic
35C4×C4⋊Dic73C2211yescharacteristic
36#17988349C2211yescharacteristic
37#17988349C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×C4×D4×C73C2211yescharacteristic
49C2×#117×C73—21no—
50#390×C73C2211yescharacteristic
51#390×C73C2211yescharacteristic
52#388×C73—21no—
53#388×C73—21no—
54#384×C73C2211yescharacteristic
55#380×C73—21no—
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C4⋊C4×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64#14556591—22no—
65#14556591—41no—
66#14556583—22no—
67#14556583—41no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C4×Dic73—21no—
78#14556522—22no—
79#14556522—22no—
80C4⋊#10596831—22no—
81C4⋊#10596831—41no—
82C4⋊#10596831—41no—
83#14556512—22no—
84#14556512—22no—
85#14556512—41no—
86#14556512—41no—
87#14556502—22no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92#14556498—21no—
93C22×D4×C73C2311yescharacteristic
94C23×C4×C73—21no—
95C4×D4×C73—22no—
96C4×D4×C73Q814yescomplemented
97#117×C73—44no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73D411yescharacteristic
101C2×C4⋊C4×C73D411yescharacteristic
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C22⋊C4×C73—21no—
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C22⋊C4×C73—21no—
110C2×C42×C73—21no—
111C2×C42×C73—21no—
112C2×C42×C73C2311yescharacteristic
113C2×C42×C73C2311yescharacteristic
114C2×C42×C73C2311yescharacteristic
115#95×C73—21no—
116#95×C73—21no—
117#95×C73D411yescharacteristic
118#95×C73D411yescharacteristic
119C2×Dic146—42no—
120C2×Dic146—42no—
121C2×Dic146—81no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—42no—
145C4⋊Dic73#4112yes—
146C4⋊Dic73#4112yes—
147#11924007—42no—
148#11924007—42no—
149#11924007—42no—
150#11924007—81no—
151C4×Dic73—81no—
152C24×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73—21no—
156C2×D4×C73—21no—
157C2×D4×C73—21no—
158C2×D4×C73C2×Q812yes—
159C22×C4×C73—21no—
160C22×C4×C73—24no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—42no—
164C22×C4×C73—42no—
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2×D411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2411yescharacteristic
172C4⋊C4×C73—22no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—41no—
179C4⋊C4×C73—42no—
180C4⋊C4×C73—81no—
181C4⋊C4×C73C2×Q812yes—
182C22⋊C4×C73—24no—
183C22⋊C4×C73—42no—
184C22⋊C4×C73—42no—
185C22⋊C4×C73—81no—
186C22⋊C4×C73—81no—
187C42×C73—21no—
188C42×C73—21no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73—42no—
192C42×C73—42no—
193C42×C73C2×Q812yes—
194C42×C73#4112yes—
195C23×C73—22no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73C22×D411yescharacteristic
203D4×C73—22no—
204D4×C73—22no—
205D4×C73—41no—
206D4×C73—41no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—22no—
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—48no—
224C2×C4×C73—41no—
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×C73—41no—
232C2×C4×C73C22⋊Q812yes—
233C2×C4×C73C22⋊Q812yes—
234C2×C4×C73AGL(1,ℤ/8)12yes—
235C2×C4×C73C22×Q811yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73H(5,𝔽2)11yescharacteristic
240C2×C4×C732−1+411yescharacteristic
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253Dic146—84no—
254#10596831—41no—
255#10596831—41no—
256#10596831—41no—
257#10596831—41no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73—81no—
265C22×C73C2×C22⋊Q811yescharacteristic
266C22×C73#54611yescharacteristic
267C22×C73#54811yescharacteristic
268C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C73#57511yescharacteristic
270C22×C73#57711yescharacteristic
271C22×C73#57711yescharacteristic
272C4×C73—22no—
273C4×C73—22no—
274C4×C73—22no—
275C4×C73—42no—
276C4×C73—41no—
277C4×C73—41no—
278C4×C73—41no—
279C4×C73—81no—
280C4×C73—81no—
281C4×C73#53311yescharacteristic
282C4×C73#54411yescharacteristic
283C4×C73#55511yescharacteristic
284C4×C73#55611yescharacteristic
285Dic73—42no—
286Dic73—42no—
287Dic73—81no—
288Dic73—81no—
289#26032—731nomaximal, Sylow
290C2×C73—82no—
291C2×C73#243911yescharacteristic
292C2×C73#307711yescharacteristic
293C2×C73#308411yescharacteristic
294C2×C73#330611yescharacteristic
295C2×C73#332111yescharacteristic
296C2×C73#336811yescharacteristic
297C2×C73#337611yescharacteristic
298#2550—731no—
299#2354D7311yescharacteristic
300#2065—731no—
301#2065—731no—
302#2025—731no—
303#2023—731no—
304#1989—731no—
305#1988—731no—
306#1948—731no—
307#1940—731no—
308#1920—731no—
309#1919—731no—
310#1836—731no—
311#1804—731no—
312#1762—731no—
313C73#2603211yescharacteristic, complemented, Sylow
314C2×C4⋊Q8—731no—
315C2×#125—731no—
316C2×C4×D4C2×D7311yescharacteristic
317C2×#117—21no—
318C2×#106—731no—
319C2×#106—731no—
320C2×C4⋊C8—731no—
321C2×C4⋊C8—731no—
322C2×C4⋊C8—731no—
323C2×#102—731no—
324C2×#102—731no—
325C2×#102—731no—
326C2×#102—731no—
327C2×#97—731no—
328#398—1461no—
329#395—1461no—
330#391—1461no—
331#390C2×D7311yescharacteristic
332#390C2×D7311yescharacteristic
333#389—731no—
334#389—731no—
335#389—1461no—
336#388—21no—
337#388—21no—
338#384C2×D7311yescharacteristic
339#382—731no—
340#380—21no—
341C4×C4⋊C4—731no—
342C4×C4⋊C4C2×D7311yescharacteristic
343C4×C4⋊C4C2×D7311yescharacteristic
344C4×C4⋊C4C2×D7311yescharacteristic
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340—731no—
351#340—731no—
352#340—731no—
353#340—731no—
354#340—731no—
355#336—731no—
356#336—731no—
357C22×Q8—1461no—
358C22×D4C22×D7311yescharacteristic
359C23×C4—21no—
360C22×C8—731no—
361C22×C8—731no—
362C22×C8—731no—
363C22×C8—731no—
364C4⋊Q8—2921no—
365C4⋊Q8—1462no—
366#125—2921no—
367#125—1462no—
368C4×D4—22no—
369C4×D4Dic14614yescomplemented
370#117—44no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—731no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4C22⋊D7311yescharacteristic
383C2×C4⋊C4C22⋊D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C4⋊C4C22×D7311yescharacteristic
386C2×C4⋊C4C22×D7311yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—1461no—
395C2×C42C22×D7311yescharacteristic
396C2×C42C22×D7311yescharacteristic
397C2×C42C22×D7311yescharacteristic
398#106—1462no—
399#106—1462no—
400C4⋊C8—2921no—
401C4⋊C8—2921no—
402C4⋊C8—1462no—
403#102—2921no—
404#102—2921no—
405#102—1462no—
406#102—1462no—
407#97—1462no—
408#95—21no—
409#95—21no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413#95—1461no—
414#95—1461no—
415#95C22⋊D7311yescharacteristic
416#95C22⋊D7311yescharacteristic
417C24—21no—
418C2×Q8—2922no—
419C2×Q8—2922no—
420C2×Q8—5841no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C2×Dic14612yes—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4C2×C22⋊D7311yescharacteristic
438C22×C4C2×C22⋊D7311yescharacteristic
439C22×C4C2×C22⋊D7311yescharacteristic
440C22×C4C2×C22⋊D7311yescharacteristic
441C22×C4C2×C22⋊D7311yescharacteristic
442C22×C4C2×C22⋊D7311yescharacteristic
443C22×C4C23×D7311yescharacteristic
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C2×C8—1461no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—2922no—
466C4⋊C4—2922no—
467C4⋊C4—2922no—
468C4⋊C4—2922no—
469C4⋊C4—5841no—
470C4⋊C4—732no—
471C4⋊C4—732no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4C2×Dic14612yes—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—42no—
489C42—42no—
490C42—5841no—
491C42C2×Dic14612yes—
492C42#1192402812yes—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23C22×C22⋊D7311yescharacteristic
501Q8—5844no—
502D4—22no—
503D4—22no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—1462no—
542C2×C4—1462no—
543C2×C4#1455657012yes—
544C2×C4#1455657012yes—
545C2×C4#1455661712yes—
546C2×C4C22×Dic14611yescharacteristic
547C2×C4#41×D7311yescharacteristic
548C2×C4#41×D7311yescharacteristic
549C2×C4C2×#1192402811yescharacteristic
550C2×C4#1455663711yescharacteristic
551C2×C4#1455663811yescharacteristic
552C8—2921no—
553C8—2921no—
554C8—2921no—
555C8—2921no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22C2×#1455657011yescharacteristic
564C22#1798922511yescharacteristic
565C22#1798924411yescharacteristic
566C22C2×#1455661711yescharacteristic
567C22#1798938511yescharacteristic
568C22#1798939511yescharacteristic
569C22#1798939811yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—2922no—
580C4—2922no—
581C4—5841no—
582C4—5841no—
583C4#1798914011yescharacteristic
584C4#1798921511yescharacteristic
585C4#1798930011yescharacteristic
586C4#1798930511yescharacteristic
587C2—82no—
588C2#2261361611yescharacteristic
589C2#2261972811yescharacteristic
590C2#2261979311yescharacteristic
591C2#2262188611yescharacteristic
592C2#2262211111yescharacteristic
593C2#2262265611yescharacteristic
594C2#2262275211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2354C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2354D73
hypocenterC73#26032
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
1. #22614952C2semidirect
6. #22610888C2semidirect
7. #22610883C2semidirect
8. #22610692C2semidirect
9. #22610655C2semidirect
10. #22610516C2semidirect
11. #22610509C2semidirect
12. #22610066C2semidirect
96. C4×D4×C73Q8semidirect
313. C73#26032semidirect
369. C4×D4Dic146semidirect

Maximal subgroups

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

subgroupindex
#226149522
#226113052
#226110772
#226110672
#226108882
#226108832
#226106922
#226106552
#226105162
#226105092
#226100662
#226099412
#226097562
C73×#23542
#2603273