Finite Group Databasebeta

← #33068814

Subgroups — #33068814

Order 18688.

696 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620324C211yesmaximal, characteristic, complemented
2C2×#17988877C211yesmaximal, characteristic, complemented
3D4×#11924012C211yesmaximal, characteristic, complemented
4#22609504C212yesmaximal, complemented
5#22609462C212yesmaximal, complemented
6#22609380C212yesmaximal, complemented
7#22609338C212yesmaximal, complemented
8#22608684C211yesmaximal, characteristic, complemented
9#22608611C211yesmaximal, characteristic, complemented
10#22608608C211yesmaximal, characteristic, complemented
11C73×#3100C211yesmaximal, characteristic
12C22×#11924012—21no—
13C2×#14556591C2211yescharacteristic
14#17988946—22no—
15#17988937C2212yescomplemented
16#17988933C2212yescomplemented
17#17988877—21no—
18#17988877—21no—
19#17988877C2212yes—
20#17988877C2212yes—
21#17988665C2212yes—
22#17988659—22no—
23D4×#10596831C2212yes—
24C2×#14556522—21no—
25#17988611C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27#17988588—21no—
28#17988588—21no—
29C2×#14556512—21no—
30#17988551—21no—
31C4×#11924012C2211yescharacteristic
32#17988339C2212yescomplemented
33#17988337C2212yescomplemented
34#17988335C2212yescomplemented
35#17988335C2212yescomplemented
36#17988333C2212yescomplemented
37#17988331C2212yescomplemented
38#532×C73C2211yescharacteristic
39C2×C4×D4×C73C2211yescharacteristic
40#480×C73—21no—
41#474×C73C2211yescharacteristic
42#474×C73C2211yescharacteristic
43#459×C73C2211yescharacteristic
44#459×C73C2211yescharacteristic
45#447×C73—21no—
46C4×D8×C73—21no—
47C4×D8×C73—21no—
48#431×C73C2211yescharacteristic
49C22×Dic146—21no—
50C2×#11924012—22no—
51C2×#11924012—21no—
52C2×#11924012—44no—
53C2×#11924012D411yescharacteristic
54C2×#11924012D411yescharacteristic
55C22×#10596831—22no—
56C22×#10596831—21no—
57#14556591—21no—
58#14556591—21no—
59#14556591—21no—
60#14556591C2312yes—
61C2×C4⋊Dic73—21no—
62C2×C4⋊Dic73—21no—
63#14556522—22no—
64#14556522—22no—
65C4⋊#10596831—21no—
66C4⋊#10596831—21no—
67C4⋊#10596831C2312yes—
68C4⋊#10596831C2312yes—
69#14556515—22no—
70#14556515—22no—
71#14556512—22no—
72#14556512—22no—
73#14556512—22no—
74#14556512—22no—
75C22⋊#10596831—22no—
76C22⋊#10596831—22no—
77#14556502D412yes—
78C4×#10596831C2312yes—
79C2×#41×C73—21no—
80C22×D4×C73D411yescharacteristic
81C23×C4×C73—21no—
82C2×D8×C73—21no—
83C2×D8×C73—21no—
84C4⋊Q8×C73C2311yescharacteristic
85#127×C73C2311yescharacteristic
86#124×C73—21no—
87#121×C73—21no—
88#121×C73—21no—
89C4×D4×C73—21no—
90C4×D4×C73—21no—
91C4×D4×C73—21no—
92C4×D4×C73—21no—
93C4×D4×C73D411yescharacteristic
94C4×D4×C73D411yescharacteristic
95C4×D4×C73C2311yescharacteristic
96C4×D4×C73C2311yescharacteristic
97C2×C4⋊C4×C73D411yescharacteristic
98C2×C22⋊C4×C73—21no—
99C2×C42×C73C2311yescharacteristic
100#107×C73—21no—
101#107×C73—21no—
102C4⋊C8×C73C2311yescharacteristic
103C4⋊C8×C73C2311yescharacteristic
104#102×C73—21no—
105#102×C73—21no—
106#102×C73—21no—
107#102×C73—21no—
108C22⋊C8×C73—21no—
109C4×C8×C73—21no—
110C2×Dic146—22no—
111C2×Dic146—41no—
112C2×Dic146—82no—
113C22×Dic73—41no—
114#11924012—22no—
115#11924012—22no—
116#11924012—44no—
117#11924012—42no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—44no—
121C2×#10596831—42no—
122C2×#10596831C2×D412yes—
123C2×#10596831C2×D412yes—
124C4⋊Dic73—22no—
125C4⋊Dic73—22no—
126C4⋊Dic73—41no—
127C4⋊Dic73—41no—
128C24×C73—21no—
129#41×C73—42no—
130#41×C73—42no—
131C2×Q8×C73—21no—
132C2×D4×C73—22no—
133C2×D4×C73—21no—
134C2×D4×C73—21no—
135C2×D4×C73—21no—
136C2×D4×C73—21no—
137C2×D4×C73—41no—
138C2×D4×C73—41no—
139C2×D4×C73C2×D411yescharacteristic
140C2×D4×C73C2×D411yescharacteristic
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—42no—
147C22×C4×C73—42no—
148C22×C4×C73—41no—
149C22×C4×C73C2×D411yescharacteristic
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73#4111yescharacteristic
152D8×C73—42no—
153D8×C73—42no—
154C2×C8×C73—21no—
155C2×C8×C73—21no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73C2×D411yescharacteristic
160C4⋊C4×C73C2×D411yescharacteristic
161C22⋊C4×C73—21no—
162C22⋊C4×C73—21no—
163C22⋊C4×C73—21no—
164C22⋊C4×C73—21no—
165C22⋊C4×C73—41no—
166C22⋊C4×C73—41no—
167C42×C73C2×D411yescharacteristic
168C42×C73C2×D411yescharacteristic
169C42×C73#4111yescharacteristic
170C42×C73C2411yescharacteristic
171C23×C73—21no—
172C23×C73—21no—
173C23×C73—42no—
174C23×C73—42no—
175C23×C73—41no—
176C23×C73—81no—
177C23×C73#12111yescharacteristic
178Q8×C73—41no—
179Q8×C73—41no—
180D4×C73—21no—
181D4×C73—21no—
182D4×C73—21no—
183D4×C73—42no—
184D4×C73—41no—
185D4×C73—41no—
186D4×C73—41no—
187D4×C73—41no—
188D4×C73—81no—
189D4×C73—82no—
190D4×C73#13712yes—
191C2×C4×C73—21no—
192C2×C4×C73—21no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—41no—
196C2×C4×C73—42no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—42no—
202C2×C4×C73—42no—
203C2×C4×C73—41no—
204C2×C4×C73—82no—
205C2×C4×C73#12111yescharacteristic
206C2×C4×C73#12111yescharacteristic
207C2×C4×C73#12111yescharacteristic
208C2×C4×C73#13512yes—
209C2×C4×C73AGL(1,ℤ/8)12yes—
210C2×C4×C73C22×D411yescharacteristic
211C2×C4×C73C22×D411yescharacteristic
212C2×C4×C73C2×#4111yescharacteristic
213C2×Dic73—42no—
214C2×Dic73—82no—
215Dic146—82no—
216Dic146—82no—
217C8×C73—41no—
218C8×C73—41no—
219#10596831—22no—
220#10596831—22no—
221#10596831—42no—
222#10596831—42no—
223C22×C73—21no—
224C22×C73—21no—
225C22×C73—22no—
226C22×C73—41no—
227C22×C73—41no—
228C22×C73—41no—
229C22×C73—41no—
230C22×C73—41no—
231C22×C73—84no—
232C22×C73—81no—
233C22×C73—81no—
234C22×C73—81no—
235C22×C73#46312yes—
236C22×C73C2×#12111yescharacteristic
237C4×C73—21no—
238C4×C73—21no—
239C4×C73—21no—
240C4×C73—41no—
241C4×C73—41no—
242C4×C73—81no—
243C4×C73#46312yes—
244C4×C73C2×#13511yescharacteristic
245C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
246C4×C73C2×#13711yescharacteristic
247C4×C73#57511yescharacteristic
248Dic73—82no—
249#19168—731nomaximal, Sylow
250C2×C73—22no—
251C2×C73—42no—
252C2×C73—81no—
253C2×C73—161no—
254C2×C73C2×#46311yescharacteristic
255C2×C73#305611yescharacteristic
256C2×C73#306811yescharacteristic
257#3141—731no—
258#3100D7311yescharacteristic
259C2×#476—731no—
260D4×M16—731no—
261#1703—732no—
262#1697—732no—
263#1683—732no—
264#1677—732no—
265#1565—731no—
266#1548—731no—
267#1547—731no—
268C73#1916811yescharacteristic, complemented, Sylow
269C22×M16—1461no—
270#532C2×D7311yescharacteristic
271C2×C4⋊Q8—731no—
272C2×C4×D4C2×D7311yescharacteristic
273#489—1462no—
274#487—732no—
275#486—732no—
276#480—21no—
277#476—732no—
278#476—732no—
279#476—1461no—
280#476—1461no—
281#474C2×D7311yescharacteristic
282#474C2×D7311yescharacteristic
283#459C2×D7311yescharacteristic
284#459C2×D7311yescharacteristic
285#447—21no—
286C4×D8—21no—
287C4×D8—21no—
288#436—732no—
289#435—1462no—
290C8×D4—732no—
291#431C2×D7311yescharacteristic
292C2×#106—1461no—
293#423—731no—
294C2×C4⋊C8—731no—
295#417—1461no—
296#417—1461no—
297C2×#102—1461no—
298#407—1461no—
299C4×M16—731no—
300#335—732no—
301#334—732no—
302#333—732no—
303#333—732no—
304#332—732no—
305#331—732no—
306C2×#41—21no—
307C22×Q8—1461no—
308C22×D4C22⋊D7311yescharacteristic
309C23×C4—21no—
310C2×D8—21no—
311C2×D8—21no—
312C2×M16—2924no—
313C2×M16—731no—
314C2×M16—731no—
315C2×M16—1462no—
316C2×M16—1461no—
317C22×C8—1462no—
318C22×C8—1461no—
319C4⋊Q8—732no—
320C4⋊Q8—1461no—
321C4⋊Q8—1461no—
322C4⋊Q8—1461no—
323C4⋊Q8C22×D7311yescharacteristic
324#127C22×D7311yescharacteristic
325#124—21no—
326#121—21no—
327#121—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4C22⋊D7311yescharacteristic
333C4×D4C22⋊D7311yescharacteristic
334C4×D4C22×D7311yescharacteristic
335C4×D4C22×D7311yescharacteristic
336C2×C4⋊C4—1461no—
337C2×C4⋊C4—1461no—
338C2×C4⋊C4C22⋊D7311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C42C22×D7311yescharacteristic
341#107—21no—
342#107—21no—
343#106—1462no—
344#106—1462no—
345C4⋊C8—732no—
346C4⋊C8—732no—
347C4⋊C8—1461no—
348C4⋊C8—1461no—
349C4⋊C8C22×D7311yescharacteristic
350C4⋊C8C22×D7311yescharacteristic
351#103—1462no—
352#103—1462no—
353#102—21no—
354#102—21no—
355#102—21no—
356#102—21no—
357#102—1462no—
358#102—1462no—
359#102—1462no—
360#102—1462no—
361C22⋊C8—21no—
362C22⋊C8—1462no—
363C22⋊C8—1462no—
364#97—732no—
365C4×C8—21no—
366C4×C8—732no—
367C24—21no—
368#41—42no—
369#41—42no—
370C2×Q8—21no—
371C2×Q8—2921no—
372C2×Q8—5842no—
373C2×Q8—1462no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4C2×C22⋊D7311yescharacteristic
382C2×D4C2×C22⋊D7311yescharacteristic
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—2921no—
392C22×C4D4×D7311yescharacteristic
393C22×C4C2×C22⋊D7311yescharacteristic
394C22×C4#1192402711yescharacteristic
395D8—42no—
396D8—42no—
397M16—2922no—
398M16—2924no—
399M16—1462no—
400M16—1462no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—2922no—
404C2×C8—2924no—
405C2×C8—732no—
406C2×C8—732no—
407C2×C8—1462no—
408C2×C8—1462no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—2921no—
413C4⋊C4—2921no—
414C4⋊C4—1462no—
415C4⋊C4—1462no—
416C4⋊C4C2×C22⋊D7311yescharacteristic
417C4⋊C4C2×C22⋊D7311yescharacteristic
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C42D4×D7311yescharacteristic
425C42C2×C22⋊D7311yescharacteristic
426C42#1192402711yescharacteristic
427C42C23×D7311yescharacteristic
428C23—21no—
429C23—21no—
430C23—42no—
431C23—42no—
432C23—41no—
433C23—81no—
434C23#1455656311yescharacteristic
435Q8—41no—
436Q8—41no—
437Q8—5842no—
438Q8—5842no—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—81no—
448D4—82no—
449D4#1455662112yes—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—82no—
464C2×C4—2922no—
465C2×C4—5842no—
466C2×C4#1455656311yescharacteristic
467C2×C4#1455656311yescharacteristic
468C2×C4#1455656311yescharacteristic
469C2×C4#1455661012yes—
470C2×C4#1455661512yes—
471C2×C4C2×D4×D7311yescharacteristic
472C2×C4C22×C22⋊D7311yescharacteristic
473C2×C4C2×#1192402711yescharacteristic
474C8—41no—
475C8—41no—
476C8—2922no—
477C8—2922no—
478C8—1462no—
479C8—1462no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—84no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22#1798880012yes—
493C22C2×#1455656311yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—81no—
500C4—5842no—
501C4#1798880012yes—
502C4C2×#1455661011yescharacteristic
503C4C2×#1455661511yescharacteristic
504C4C2×#1455662111yescharacteristic
505C4#1798938511yescharacteristic
506C2—22no—
507C2—42no—
508C2—81no—
509C2—161no—
510C2C2×#1798880011yescharacteristic
511C2#2261956311yescharacteristic
512C2#2261966111yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3100C2
socleC22×C73C2×#121
hypercenter#3100D73
hypocenterC73#19168
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #22620324C2semidirect
2. C2×#17988877C2semidirect
3. D4×#11924012C2semidirect
4. #22609504C2semidirect
5. #22609462C2semidirect
6. #22609380C2semidirect
7. #22609338C2semidirect
8. #22608684C2semidirect
9. #22608611C2semidirect
10. #22608608C2semidirect
15. #17988937C22semidirect
16. #17988933C22semidirect
32. #17988339C22semidirect
33. #17988337C22semidirect
34. #17988335C22semidirect
35. #17988335C22semidirect
36. #17988333C22semidirect
37. #17988331C22semidirect
268. C73#19168semidirect

Maximal subgroups

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

subgroupindex
#226203242
C2×#179888772
D4×#119240122
#226095042
#226094622
#226093802
#226093382
#226086842
#226086112
#226086082
C73×#31002
#1916873