Finite Group Databasebeta

← #33069938

Subgroups — #33069938

Order 18688.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988934C211yesmaximal, characteristic, complemented
2#22619746C211yesmaximal, characteristic, complemented
3#22619411C211yesmaximal, characteristic
4#22609347C214yesmaximal, complemented
5#22609340C214yesmaximal, complemented
6#22608687C211yesmaximal, characteristic, complemented
7#22608611C211yesmaximal, characteristic, complemented
8#22608598C211yesmaximal, characteristic, complemented
9C8×#41×C73C211yesmaximal, characteristic, complemented
10C2×#14556616—21no—
11C2×#14556591C2211yescharacteristic
12C2×#14556588C2211yescharacteristic
13#17988934—21no—
14#17988934—21no—
15#17988934C2214yes—
16#17988882C2212yescomplemented
17#17988877C2212yescomplemented
18#17988872—21no—
19#17988867—21no—
20#17988787—21no—
21#17988787C2212yes—
22#17988782—21no—
23#17988782C2212yes—
24#17988630—21no—
25#17988611C2211yescharacteristic
26#17988611C2211yescharacteristic
27#17988591—21no—
28#17988588—21no—
29C2×#14556513—22no—
30#17988338C2214yescomplemented
31#17988335C2214yescomplemented
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#14556616—44no—
46C2×#11924015—22no—
47C2×#11924013—22no—
48C2×#11924012—21no—
49C2×#11924012—21no—
50#14556591—21no—
51#14556591—21no—
52#14556591—21no—
53#14556591C2312yes—
54#14556588—21no—
55#14556588—21no—
56#14556588—21no—
57#14556588C2312yes—
58C2×C4⋊Dic73—21no—
59C2×C4⋊Dic73—21no—
60#14556524—22no—
61#14556522—22no—
62C4⋊#10596831—21no—
63C4⋊#10596831—21no—
64C4⋊#10596831C2312yes—
65C4⋊#10596831C2312yes—
66#14556515—22no—
67#14556513—24no—
68#14556513—24no—
69#14556512—22no—
70#14556511—22no—
71#14556511—22no—
72C2×#41×C73D411yescharacteristic
73C22×C8×C73—22no—
74C22×C8×C73D412yes—
75C4×Q8×C73D411yescharacteristic
76C4×Q8×C73C2311yescharacteristic
77C4×D4×C73—21no—
78C4×D4×C73—21no—
79C4×D4×C73D411yescharacteristic
80C4×D4×C73C2311yescharacteristic
81#117×C73—21no—
82#117×C73D411yescharacteristic
83C2×C42×C73—21no—
84C2×C42×C73C2311yescharacteristic
85C4⋊C8×C73—22no—
86C4⋊C8×C73C2312yes—
87C22⋊C8×C73—21no—
88C22⋊C8×C73—22no—
89C4×C8×C73—22no—
90C4×C8×C73—21no—
91C4×C8×C73D411yescharacteristic
92C4×C8×C73D411yescharacteristic
93C4×C8×C73C2312yes—
94C23×D73—41no—
95C2×Dic146—22no—
96C2×Dic146—41no—
97C2×Dic146—82no—
98C2×D292—22no—
99C2×D292—22no—
100C2×D292—22no—
101C2×D292—44no—
102C2×D292—42no—
103C2×D292—42no—
104C2×D292—41no—
105C2×D292—41no—
106C2×D292—41no—
107C2×D292—82no—
108C22×Dic73—41no—
109#11924015—44no—
110#11924013—44no—
111#11924012—42no—
112#11924012—42no—
113C2×#10596831—22no—
114C2×#10596831—22no—
115C4⋊Dic73—22no—
116C4⋊Dic73—22no—
117C4⋊Dic73—41no—
118C4⋊Dic73—41no—
119#41×C73—22no—
120#41×C73SD1614yes—
121C2×Q8×C73C2×D411yescharacteristic
122C2×D4×C73—21no—
123C2×D4×C73C2×D411yescharacteristic
124C22×C4×C73—21no—
125C22×C4×C73—21no—
126C22×C4×C73—21no—
127C22×C4×C73C2×D411yescharacteristic
128C22×C4×C73C2×D411yescharacteristic
129C22×C4×C73#4111yescharacteristic
130C2×C8×C73—22no—
131C2×C8×C73—42no—
132C2×C8×C73—44no—
133C2×C8×C73C2×D412yes—
134C2×C8×C73C2×D412yes—
135C4⋊C4×C73—21no—
136C4⋊C4×C73—21no—
137C4⋊C4×C73C2×D411yescharacteristic
138C4⋊C4×C73C2×D411yescharacteristic
139C22⋊C4×C73—21no—
140C22⋊C4×C73—21no—
141C22⋊C4×C73—21no—
142C42×C73—21no—
143C42×C73—21no—
144C42×C73—21no—
145C42×C73C2×D411yescharacteristic
146C42×C73C2×D411yescharacteristic
147C42×C73#4111yescharacteristic
148C42×C73C2411yescharacteristic
149C23×C73—21no—
150C23×C73#12111yescharacteristic
151C22×D73—42no—
152C22×D73—82no—
153C22×D73—82no—
154C22×D73—82no—
155Q8×C73—21no—
156Q8×C73C2×SD1612yes—
157D4×C73—24no—
158D4×C73—21no—
159D4×C73C2×SD1612yes—
160C2×C4×C73—21no—
161C2×C4×C73—22no—
162C2×C4×C73—24no—
163C2×C4×C73—22no—
164C2×C4×C73—21no—
165C2×C4×C73—21no—
166C2×C4×C73—42no—
167C2×C4×C73—42no—
168C2×C4×C73#12111yescharacteristic
169C2×C4×C73#12111yescharacteristic
170C2×C4×C73#12111yescharacteristic
171C2×C4×C73C2×SD1612yes—
172C2×C4×C73#13512yes—
173C2×C4×C73C22×D411yescharacteristic
174C2×C4×C73C22×D411yescharacteristic
175C2×C4×C73C2×#4111yescharacteristic
176C2×Dic73—42no—
177C2×Dic73—82no—
178D292—42no—
179D292—42no—
180D292—44no—
181D292—82no—
182D292—82no—
183D292—82no—
184D292—82no—
185Dic146—82no—
186Dic146—82no—
187C8×C73—22no—
188C8×C73—22no—
189C8×C73—42no—
190#10596831—42no—
191#10596831—42no—
192C22×C73—21no—
193C22×C73—21no—
194C22×C73—22no—
195C22×C73—41no—
196C22×C73—41no—
197C22×C73#46512yes—
198C22×C73C2×#12111yescharacteristic
199C2×D73—82no—
200C2×D73—82no—
201C2×D73—82no—
202C2×D73—164no—
203C4×C73—21no—
204C4×C73—22no—
205C4×C73—21no—
206C4×C73—41no—
207C4×C73—41no—
208C4×C73#46512yes—
209C4×C73#47212yes—
210C4×C73C22×SD1611yescharacteristic
211C4×C73C2×#13511yescharacteristic
212C4×C73#57512yes—
213Dic73—82no—
214#19274—731nomaximal, Sylow
215C2×C73—22no—
216C2×C73—42no—
217C2×C73#304011yescharacteristic
218C2×C73C2×#46511yescharacteristic
219C2×C73C2×#47211yescharacteristic
220D73—162no—
221C2×#486—731no—
222#3079—731no—
223#3038—731no—
224C8×#41D7311yescharacteristic, complemented
225#1678—734no—
226#1677—734no—
227#1566—731no—
228#1548—731no—
229#1545—731no—
230C73#1927411yescharacteristic, complemented, Sylow
231C2×AGL(1,ℤ/8)—1461no—
232C2×C4⋊Q8—731no—
233C2×#127—731no—
234C4×#41C2×D7311yescharacteristic
235#486—734no—
236#486—1461no—
237#486—1461no—
238#477—732no—
239#476—732no—
240#475—1461no—
241#474—1461no—
242#460—732no—
243#460—1461no—
244#459—732no—
245#459—1461no—
246C8×Q8C2×D7312yescomplemented
247C8×D4—22no—
248C8×D4C2×D7312yescomplemented
249#431—21no—
250#431C2×D7311yescharacteristic
251#428—1461no—
252#423—731no—
253#423—731no—
254#418—1461no—
255#417—1461no—
256C2×#102—1462no—
257C2×C4×C8—21no—
258C2×C4×C8C2×D7311yescharacteristic
259#334—734no—
260#333—734no—
261#331—734no—
262C2×#41D29211yescharacteristic
263C22×Q8—1461no—
264C22×D4—1461no—
265C22×D4—1461no—
266C22×D4—1461no—
267AGL(1,ℤ/8)—2924no—
268C2×SD16—1462no—
269C2×D8—1462no—
270C2×M16—1461no—
271C2×M16—1461no—
272C22×C8—22no—
273C22×C8C22⋊D7312yes—
274C4⋊Q8—732no—
275C4⋊Q8—1461no—
276C4⋊Q8—1461no—
277C4⋊Q8—1461no—
278#127—732no—
279#127—1461no—
280#127—1461no—
281#127—1461no—
282C4×Q8D29211yescharacteristic
283C4×Q8C22×D7311yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4D29211yescharacteristic
287C4×D4C22×D7311yescharacteristic
288#117—21no—
289#117D29211yescharacteristic
290C2×C4⋊C4—1461no—
291C2×C4⋊C4—1461no—
292C2×C42—21no—
293C2×C42C22×D7311yescharacteristic
294#107—1462no—
295#106—1462no—
296C4⋊C8—22no—
297C4⋊C8—732no—
298C4⋊C8—732no—
299C4⋊C8—1461no—
300C4⋊C8—1461no—
301C4⋊C8C22×D7312yes—
302#103—1462no—
303#102—1462no—
304#102—1464no—
305#102—1464no—
306#102—1462no—
307#102—1462no—
308C22⋊C8—21no—
309C22⋊C8—22no—
310C4×C8—22no—
311C4×C8—21no—
312C4×C8C22⋊D7311yescharacteristic
313C4×C8C22⋊D7311yescharacteristic
314C4×C8C22×D7312yes—
315C24—2921no—
316#41—22no—
317#41#1192401714yes—
318C2×Q8—2921no—
319C2×Q8—5842no—
320C2×Q8—1462no—
321C2×Q8C2×D29211yescharacteristic
322C2×D4—21no—
323C2×D4—2924no—
324C2×D4—2921no—
325C2×D4—2921no—
326C2×D4—2921no—
327C2×D4—2922no—
328C2×D4—2922no—
329C2×D4—5842no—
330C2×D4—1462no—
331C2×D4—1462no—
332C2×D4—1462no—
333C2×D4C2×D29211yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—2921no—
338C22×C4C2×C22⋊D7311yescharacteristic
339C22×C4C2×D29211yescharacteristic
340C22×C4#1192402811yescharacteristic
341SD16—2924no—
342D8—2924no—
343M16—2922no—
344M16—2922no—
345C2×C8—22no—
346C2×C8—42no—
347C2×C8—44no—
348C2×C8—1462no—
349C2×C8—1462no—
350C2×C8C2×C22⋊D7312yes—
351C2×C8C2×C22⋊D7312yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—2921no—
355C4⋊C4—2921no—
356C4⋊C4—1462no—
357C4⋊C4—1462no—
358C4⋊C4C2×D29211yescharacteristic
359C4⋊C4C2×D29211yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42C2×C22⋊D7311yescharacteristic
367C42C2×D29211yescharacteristic
368C42#1192402811yescharacteristic
369C42C23×D7311yescharacteristic
370C23—21no—
371C23—2922no—
372C23—5842no—
373C23—5842no—
374C23—5842no—
375C23#1455656511yescharacteristic
376Q8—21no—
377Q8—5842no—
378Q8—5842no—
379Q8C2×#1192401712yes—
380D4—24no—
381D4—21no—
382D4—2924no—
383D4—2922no—
384D4—2922no—
385D4—5842no—
386D4—5842no—
387D4—5842no—
388D4—5842no—
389D4C2×#1192401712yes—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—2922no—
399C2×C4—5842no—
400C2×C4#1455656511yescharacteristic
401C2×C4#1455656511yescharacteristic
402C2×C4#1455656511yescharacteristic
403C2×C4C2×#1192401712yes—
404C2×C4#1455661312yes—
405C2×C4C22×C22⋊D7311yescharacteristic
406C2×C4C22×D29211yescharacteristic
407C2×C4C2×#1192402811yescharacteristic
408C8—22no—
409C8—22no—
410C8—42no—
411C8—2922no—
412C8—2922no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—5842no—
419C22—5842no—
420C22—5842no—
421C22—11684no—
422C22#1798881612yes—
423C22C2×#1455656511yescharacteristic
424C4—21no—
425C4—22no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—5842no—
430C4#1798881612yes—
431C4#1798885712yes—
432C4C22×#1192401711yescharacteristic
433C4C2×#1455661311yescharacteristic
434C4#1798938512yes—
435C2—22no—
436C2—42no—
437C2—11682no—
438C2#2261942511yescharacteristic
439C2C2×#1798881611yescharacteristic
440C2C2×#1798885711yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC8×#41×C73C2
socleC22×C73C2×#121
prime radical#22619411C2
hypercenterC8×#41D73
hypocenterC73#19274
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. #22619746C2semidirect
4. #22609347C2semidirect
5. #22609340C2semidirect
6. #22608687C2semidirect
7. #22608611C2semidirect
8. #22608598C2semidirect
9. C8×#41×C73C2semidirect
16. #17988882C22semidirect
17. #17988877C22semidirect
30. #17988338C22semidirect
31. #17988335C22semidirect
34. C8×Q8×C73C22semidirect
36. C8×D4×C73C22semidirect
224. C8×#41D73semidirect
230. C73#19274semidirect
246. C8×Q8C2×D73semidirect
248. 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
#226197462
#226194112
#226093472
#226093402
#226086872
#226086112
#226085982
C8×#41×C732
#1927473