Finite Group Databasebeta

← #33069956

Subgroups — #33069956

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988934C211yesmaximal, characteristic, complemented
2#22619418C211yesmaximal, characteristic, complemented
3#22619411C211yesmaximal, characteristic
4#22609375C214yesmaximal, complemented
5#22609368C214yesmaximal, complemented
6#22608690C211yesmaximal, characteristic, complemented
7#22608602C211yesmaximal, characteristic, complemented
8#22608598C211yesmaximal, characteristic, complemented
9C8×#41×C73C211yesmaximal, characteristic, complemented
10C2×#14556621—21no—
11C2×#14556616—21no—
12C2×#14556591C2211yescharacteristic
13C2×#14556588C2211yescharacteristic
14#17988934—21no—
15#17988934—21no—
16#17988934C2214yes—
17#17988797—21no—
18#17988797C2212yescomplemented
19#17988792—21no—
20#17988792C2212yescomplemented
21#17988787—21no—
22#17988787C2212yes—
23#17988782—21no—
24#17988782C2212yes—
25#17988611C2211yescharacteristic
26#17988611C2211yescharacteristic
27#17988595—21no—
28#17988591—21no—
29C2×#14556513—22no—
30#17988340C2214yescomplemented
31#17988332C2214yescomplemented
32#17988330C2214yes—
33C4×#41×C73C2211yescharacteristic
34C8×Q8×C73C2212yescomplemented
35C8×D4×C73—22no—
36C8×D4×C73C2212yescomplemented
37#431×C73—21no—
38#431×C73C2211yescharacteristic
39C2×C4×C8×C73—21no—
40C2×C4×C8×C73C2211yescharacteristic
41C22×Dic146—21no—
42C22×D292—21no—
43C22×D292—21no—
44C22×D292—21no—
45#14556621—44no—
46#14556616—44no—
47C2×#11924018—22no—
48C2×#11924016—22no—
49C2×#11924015—22no—
50C2×#11924013—22no—
51C2×#11924012—21no—
52C2×#11924012—21no—
53#14556591—21no—
54#14556591—21no—
55#14556591—21no—
56#14556591C2312yes—
57#14556588—21no—
58#14556588—21no—
59#14556588—21no—
60#14556588C2312yes—
61C2×C4⋊Dic73—21no—
62C2×C4⋊Dic73—21no—
63C4⋊#10596831—21no—
64C4⋊#10596831—21no—
65C4⋊#10596831C2312yes—
66C4⋊#10596831C2312yes—
67#14556514—22no—
68#14556514—22no—
69#14556513—24no—
70#14556513—24no—
71#14556511—22no—
72#14556511—22no—
73C2×#41×C73D411yescharacteristic
74C22×C8×C73—22no—
75C22×C8×C73D412yes—
76C4×Q8×C73D411yescharacteristic
77C4×Q8×C73C2311yescharacteristic
78C4×D4×C73—21no—
79C4×D4×C73—21no—
80C4×D4×C73D411yescharacteristic
81C4×D4×C73C2311yescharacteristic
82#117×C73—21no—
83#117×C73D411yescharacteristic
84C2×C42×C73—21no—
85C2×C42×C73C2311yescharacteristic
86C4⋊C8×C73—22no—
87C4⋊C8×C73C2312yes—
88C22⋊C8×C73—21no—
89C22⋊C8×C73—22no—
90C4×C8×C73—22no—
91C4×C8×C73—21no—
92C4×C8×C73D411yescharacteristic
93C4×C8×C73D411yescharacteristic
94C4×C8×C73C2312yes—
95C23×D73—41no—
96C2×Dic146—22no—
97C2×Dic146—42no—
98C2×Dic146—42no—
99C2×Dic146—41no—
100C2×D292—22no—
101C2×D292—22no—
102C2×D292—22no—
103C2×D292—44no—
104C2×D292—42no—
105C2×D292—42no—
106C2×D292—41no—
107C2×D292—41no—
108C2×D292—41no—
109C2×D292—82no—
110C22×Dic73—41no—
111#11924018—44no—
112#11924016—44no—
113#11924015—44no—
114#11924013—44no—
115#11924012—42no—
116#11924012—42no—
117C2×#10596831—22no—
118C2×#10596831—22no—
119C4⋊Dic73—22no—
120C4⋊Dic73—22no—
121C4⋊Dic73—41no—
122C4⋊Dic73—41no—
123#41×C73—22no—
124#41×C73D814yes—
125C2×Q8×C73C2×D411yescharacteristic
126C2×D4×C73—21no—
127C2×D4×C73C2×D411yescharacteristic
128C22×C4×C73—21no—
129C22×C4×C73—21no—
130C22×C4×C73—21no—
131C22×C4×C73C2×D411yescharacteristic
132C22×C4×C73C2×D411yescharacteristic
133C22×C4×C73#4111yescharacteristic
134C2×C8×C73—22no—
135C2×C8×C73—42no—
136C2×C8×C73—44no—
137C2×C8×C73C2×D412yes—
138C2×C8×C73C2×D412yes—
139C4⋊C4×C73—21no—
140C4⋊C4×C73—21no—
141C4⋊C4×C73C2×D411yescharacteristic
142C4⋊C4×C73C2×D411yescharacteristic
143C22⋊C4×C73—21no—
144C22⋊C4×C73—21no—
145C22⋊C4×C73—21no—
146C42×C73—21no—
147C42×C73—21no—
148C42×C73—21no—
149C42×C73C2×D411yescharacteristic
150C42×C73C2×D411yescharacteristic
151C42×C73#4111yescharacteristic
152C42×C73C2411yescharacteristic
153C23×C73—21no—
154C23×C73#12111yescharacteristic
155C22×D73—42no—
156C22×D73—82no—
157C22×D73—82no—
158C22×D73—82no—
159Q8×C73—21no—
160Q8×C73C2×D812yes—
161D4×C73—24no—
162D4×C73—21no—
163D4×C73C2×D812yes—
164C2×C4×C73—21no—
165C2×C4×C73—22no—
166C2×C4×C73—24no—
167C2×C4×C73—22no—
168C2×C4×C73—21no—
169C2×C4×C73—21no—
170C2×C4×C73—42no—
171C2×C4×C73—42no—
172C2×C4×C73#12111yescharacteristic
173C2×C4×C73#12111yescharacteristic
174C2×C4×C73#12111yescharacteristic
175C2×C4×C73C2×D812yes—
176C2×C4×C73#13512yes—
177C2×C4×C73C22×D411yescharacteristic
178C2×C4×C73C22×D411yescharacteristic
179C2×C4×C73C2×#4111yescharacteristic
180C2×Dic73—42no—
181C2×Dic73—82no—
182D292—42no—
183D292—42no—
184D292—44no—
185D292—82no—
186D292—82no—
187D292—82no—
188D292—82no—
189Dic146—42no—
190Dic146—42no—
191Dic146—82no—
192C8×C73—22no—
193C8×C73—22no—
194C8×C73—42no—
195#10596831—42no—
196#10596831—42no—
197C22×C73—21no—
198C22×C73—21no—
199C22×C73—22no—
200C22×C73—41no—
201C22×C73—41no—
202C22×C73#46612yes—
203C22×C73C2×#12111yescharacteristic
204C2×D73—82no—
205C2×D73—82no—
206C2×D73—82no—
207C2×D73—164no—
208C4×C73—21no—
209C4×C73—22no—
210C4×C73—21no—
211C4×C73—41no—
212C4×C73—41no—
213C4×C73#46612yes—
214C4×C73#47112yes—
215C4×C73C22×D811yescharacteristic
216C4×C73C2×#13511yescharacteristic
217C4×C73#57512yes—
218Dic73—82no—
219#19276—731nomaximal, Sylow
220C2×C73—22no—
221C2×C73—42no—
222C2×C73#304011yescharacteristic
223C2×C73C2×#46611yescharacteristic
224C2×C73C2×#47111yescharacteristic
225D73—162no—
226C2×#486—731no—
227#3039—731no—
228#3038—731no—
229C8×#41D7311yescharacteristic, complemented
230#1682—734no—
231#1681—734no—
232#1567—731no—
233#1546—731no—
234#1545—731no—
235C73#1927611yescharacteristic, complemented, Sylow
236C2×#137—1461no—
237C2×AGL(1,ℤ/8)—1461no—
238C2×C4⋊Q8—731no—
239C2×#127—731no—
240C4×#41C2×D7311yescharacteristic
241#486—734no—
242#486—1461no—
243#486—1461no—
244#462—732no—
245#462—1461no—
246#461—732no—
247#461—1461no—
248#460—732no—
249#460—1461no—
250#459—732no—
251#459—1461no—
252C8×Q8C2×D7312yescomplemented
253C8×D4—22no—
254C8×D4C2×D7312yescomplemented
255#431—21no—
256#431C2×D7311yescharacteristic
257#423—731no—
258#423—731no—
259#419—1461no—
260#418—1461no—
261C2×#102—1462no—
262C2×C4×C8—21no—
263C2×C4×C8C2×D7311yescharacteristic
264#335—734no—
265#332—734no—
266#331—734no—
267C2×#41D29211yescharacteristic
268C22×Q8—1461no—
269C22×D4—1461no—
270C22×D4—1461no—
271C22×D4—1461no—
272#137—2924no—
273AGL(1,ℤ/8)—2924no—
274C2×Q16—1462no—
275C2×SD16—1462no—
276C2×SD16—1462no—
277C2×D8—1462no—
278C2×M16—1461no—
279C2×M16—1461no—
280C22×C8—22no—
281C22×C8C22⋊D7312yes—
282C4⋊Q8—732no—
283C4⋊Q8—1461no—
284C4⋊Q8—1461no—
285C4⋊Q8—1461no—
286#127—732no—
287#127—1461no—
288#127—1461no—
289#127—1461no—
290C4×Q8D29211yescharacteristic
291C4×Q8C22×D7311yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4D29211yescharacteristic
295C4×D4C22×D7311yescharacteristic
296#117—21no—
297#117D29211yescharacteristic
298C2×C4⋊C4—1461no—
299C2×C4⋊C4—1461no—
300C2×C42—21no—
301C2×C42C22×D7311yescharacteristic
302C4⋊C8—22no—
303C4⋊C8—732no—
304C4⋊C8—732no—
305C4⋊C8—1461no—
306C4⋊C8—1461no—
307C4⋊C8C22×D7312yes—
308#103—1462no—
309#103—1462no—
310#102—1464no—
311#102—1464no—
312#102—1462no—
313#102—1462no—
314C22⋊C8—21no—
315C22⋊C8—22no—
316C4×C8—22no—
317C4×C8—21no—
318C4×C8C22⋊D7311yescharacteristic
319C4×C8C22⋊D7311yescharacteristic
320C4×C8C22×D7312yes—
321C24—2921no—
322#41—22no—
323#41D58414yes—
324C2×Q8—2921no—
325C2×Q8—2922no—
326C2×Q8—2922no—
327C2×Q8—1462no—
328C2×Q8C2×D29211yescharacteristic
329C2×D4—21no—
330C2×D4—2924no—
331C2×D4—2921no—
332C2×D4—2921no—
333C2×D4—2921no—
334C2×D4—2922no—
335C2×D4—2922no—
336C2×D4—5842no—
337C2×D4—1462no—
338C2×D4—1462no—
339C2×D4—1462no—
340C2×D4C2×D29211yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—2921no—
345C22×C4C2×C22⋊D7311yescharacteristic
346C22×C4C2×D29211yescharacteristic
347C22×C4#1192402811yescharacteristic
348Q16—2924no—
349SD16—2924no—
350SD16—2924no—
351D8—2924no—
352M16—2922no—
353M16—2922no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—44no—
357C2×C8—1462no—
358C2×C8—1462no—
359C2×C8C2×C22⋊D7312yes—
360C2×C8C2×C22⋊D7312yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—2921no—
364C4⋊C4—2921no—
365C4⋊C4—1462no—
366C4⋊C4—1462no—
367C4⋊C4C2×D29211yescharacteristic
368C4⋊C4C2×D29211yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42C2×C22⋊D7311yescharacteristic
376C42C2×D29211yescharacteristic
377C42#1192402811yescharacteristic
378C42C23×D7311yescharacteristic
379C23—21no—
380C23—2922no—
381C23—5842no—
382C23—5842no—
383C23—5842no—
384C23#1455656511yescharacteristic
385Q8—21no—
386Q8—2922no—
387Q8—2922no—
388Q8—5842no—
389Q8C2×D58412yes—
390D4—24no—
391D4—21no—
392D4—2924no—
393D4—2922no—
394D4—2922no—
395D4—5842no—
396D4—5842no—
397D4—5842no—
398D4—5842no—
399D4C2×D58412yes—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—2922no—
409C2×C4—5842no—
410C2×C4#1455656511yescharacteristic
411C2×C4#1455656511yescharacteristic
412C2×C4#1455656511yescharacteristic
413C2×C4C2×D58412yes—
414C2×C4#1455661312yes—
415C2×C4C22×C22⋊D7311yescharacteristic
416C2×C4C22×D29211yescharacteristic
417C2×C4C2×#1192402811yescharacteristic
418C8—22no—
419C8—22no—
420C8—42no—
421C8—2922no—
422C8—2922no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—5842no—
429C22—5842no—
430C22—5842no—
431C22—11684no—
432C22#1798882312yes—
433C22C2×#1455656511yescharacteristic
434C4—21no—
435C4—22no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—5842no—
440C4#1798882312yes—
441C4#1798885012yes—
442C4C22×D58411yescharacteristic
443C4C2×#1455661311yescharacteristic
444C4#1798938512yes—
445C2—22no—
446C2—42no—
447C2—11682no—
448C2#2261942511yescharacteristic
449C2C2×#1798882311yescharacteristic
450C2C2×#1798885011yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC8×#41×C73C2
socleC22×C73C2×#121
prime radical#22619411C2
hypercenterC8×#41D73
hypocenterC73#19276
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. C2×#17988934C2semidirect
2. #22619418C2semidirect
4. #22609375C2semidirect
5. #22609368C2semidirect
6. #22608690C2semidirect
7. #22608602C2semidirect
8. #22608598C2semidirect
9. C8×#41×C73C2semidirect
18. #17988797C22semidirect
20. #17988792C22semidirect
30. #17988340C22semidirect
31. #17988332C22semidirect
34. C8×Q8×C73C22semidirect
36. C8×D4×C73C22semidirect
229. C8×#41D73semidirect
235. C73#19276semidirect
252. C8×Q8C2×D73semidirect
254. C8×D4C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179889342
#226194182
#226194112
#226093752
#226093682
#226086902
#226086022
#226085982
C8×#41×C732
#1927673