Finite Group Databasebeta

← #37584109

Subgroups — #37584109

Order 24448.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259354C211yesmaximal, characteristic, complemented
2C2×#20259342C211yesmaximal, characteristic, complemented
3#27133332C211yesmaximal, characteristic, complemented
4#27133323C211yesmaximal, characteristic, complemented
5#27133049C211yesmaximal, characteristic, complemented
6#27132915C211yesmaximal, characteristic, complemented
7#27132894C211yesmaximal, characteristic, complemented
8#27132887C211yesmaximal, characteristic, complemented
9#27132862C211yesmaximal, characteristic, complemented
10#27132811C211yesmaximal, characteristic, complemented
11#27132810C211yesmaximal, characteristic, complemented
12#27132800C211yesmaximal, characteristic, complemented
13#27132767C211yesmaximal, characteristic, complemented
14#27132731C211yesmaximal, characteristic, complemented
15#452×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690287C2211yescharacteristic
18C2×Q8×D191C2211yescharacteristic
19#20259354—24no—
20#20259342—24no—
21C2×#15690277C2211yescharacteristic
22C2×#15690276C2211yescharacteristic
23C2×#15690275C2211yescharacteristic
24C2×#15690274C2211yescharacteristic
25C2×#15690274C2211yescharacteristic
26C2×D1528C2211yescharacteristic
27C2×#15690272C2211yescharacteristic
28C2×#15690270C2211yescharacteristic
29C22×C191⋊C8C2211yescharacteristic
30#20259318—21no—
31#20259318C2211yescharacteristic
32#20259311C2211yescharacteristic
33#20259308—21no—
34#20259300—21no—
35#20259297C2211yescharacteristic
36#20259296—21no—
37#20259296C2211yescharacteristic
38#20259295—21no—
39#20259293—21no—
40#20259293C2211yescharacteristic
41Q8×Dic191C2211yescharacteristic
42#20259287C2211yescharacteristic
43#20259284—21no—
44C4×C22⋊D191—21no—
45C4×C22⋊D191C2211yescharacteristic
46#20259253C2211yescharacteristic
47Dic191⋊C8C2211yescharacteristic
48#20259247C2211yescharacteristic
49#20259246C2211yescharacteristic
50#20259244C2211yescharacteristic
51#20259242C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53#20259232C2211yescharacteristic
54C2×#41×C191C2211yescharacteristic
55C2×Q16×C191C2211yescharacteristic
56C2×SD16×C191C2211yescharacteristic
57C22⋊Q8×C191C2211yescharacteristic
58#103×C191C2211yescharacteristic
59#102×C191C2211yescharacteristic
60C22⋊C8×C191C2211yescharacteristic
61#15690288—22no—
62#15690288—41no—
63#15690287—22no—
64#15690287—41no—
65C2×Dic382—21no—
66C2×Dic382C2311yescharacteristic
67Q8×D191—22no—
68Q8×D191—41no—
69C2×D764—21no—
70C2×D764—21no—
71C2×D764C2311yescharacteristic
72C2×D764C2311yescharacteristic
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78C2×C22⋊D191D411yescharacteristic
79C2×C4×D191—21no—
80C2×C4×D191—21no—
81C2×C4×D191—21no—
82C2×C4×D191C2311yescharacteristic
83#15690277D414yescomplemented
84#15690276—22no—
85#15690275—22no—
86#15690274—22no—
87#15690274—22no—
88D1528—41no—
89#15690272—22no—
90#15690270—22no—
91C2×C191⋊C8—22no—
92C2×C191⋊C8C2311yescharacteristic
93C2×C191⋊C8C2311yescharacteristic
94C4⋊Dic191—21no—
95C4⋊Dic191C2311yescharacteristic
96Dic191⋊C4—21no—
97Dic191⋊C4D411yescharacteristic
98C22⋊Dic191—21no—
99C22⋊Dic191D411yescharacteristic
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4D411yescharacteristic
105C4×Dic191—21no—
106C4×Dic191—21no—
107C4×Dic191C2311yescharacteristic
108#41×C191—22no—
109#41×C191—41no—
110C2×Q8×C191C2311yescharacteristic
111C2×Q8×C191C2311yescharacteristic
112C2×D4×C191—21no—
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116Q16×C191—22no—
117SD16×C191—41no—
118C2×C8×C191C2311yescharacteristic
119C2×C8×C191C2311yescharacteristic
120C4⋊C4×C191—21no—
121C4⋊C4×C191C2311yescharacteristic
122C22⋊C4×C191—21no—
123C22×D191—21no—
124C22×D191—21no—
125C22×D191C2×D411yescharacteristic
126Dic382—21no—
127Dic382—42no—
128Dic382C2×D412yes—
129D764—21no—
130D764—21no—
131D764—21no—
132D764—21no—
133D764—42no—
134D764—42no—
135C22⋊D191—41no—
136C22⋊D191—42no—
137C22⋊D191—42no—
138C22⋊D191—81no—
139C22⋊D191—81no—
140C22⋊D191—81no—
141C2×Dic191—21no—
142C2×Dic191—21no—
143C2×Dic191C2×D411yescharacteristic
144C2×Dic191C2×D411yescharacteristic
145C2×Dic191C2×D411yescharacteristic
146C4×D191—21no—
147C4×D191—21no—
148C4×D191—42no—
149C4×D191—42no—
150C4×D191—42no—
151C191⋊C8—21no—
152C191⋊C8C2×D412yes—
153C23×C191—21no—
154C23×C191C2×D411yescharacteristic
155Q8×C191—21no—
156Q8×C191—21no—
157Q8×C191—21no—
158Q8×C191C2×D412yes—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191—42no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191—42no—
168C2×C4×C191C2×D411yescharacteristic
169C2×C4×C191C2411yescharacteristic
170C8×C191—21no—
171C8×C191—21no—
172C22×C191—21no—
173C22×C191—21no—
174C22×C191—21no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191C22×D411yescharacteristic
179C2×D191—21no—
180C2×D191—21no—
181C2×D191—21no—
182C2×D191—41no—
183C2×D191—41no—
184C2×D191—41no—
185C2×D191—41no—
186C2×D191—41no—
187C2×D191—41no—
188C4×C191—21no—
189C4×C191—22no—
190C4×C191—41no—
191C4×C191—41no—
192C4×C191C22×D411yescharacteristic
193C4×C191H(5,𝔽2)11yescharacteristic
194Dic191—21no—
195Dic191—22no—
196Dic191—21no—
197Dic191—21no—
198Dic191—41no—
199C2×C191—41no—
200C2×C191—81no—
201C2×C191D4211yescharacteristic
202C2×C191#57511yescharacteristic
203C2×C191#57711yescharacteristic
204D191—41no—
205D191—81no—
206D191—81no—
207C191#327111yescharacteristic, complemented, Sylow
208#3271—1911nomaximal, Sylow
209C2×#137—1911no—
210C2×#135—1911no—
211#550—1911no—
212#548—1911no—
213#496—1911no—
214#469—1911no—
215#465—1911no—
216#464—1911no—
217#460—1911no—
218#452D19111yescharacteristic, complemented
219#449—1911no—
220#449—1911no—
221#448—1911no—
222#441—1911no—
223#435—1911no—
224C2×#41—1911no—
225C2×#41—1911no—
226C2×#41C2×D19111yescharacteristic
227C22×Q8—1911no—
228#137—3824no—
229#135—3824no—
230C2×Q16—1911no—
231C2×Q16C2×D19111yescharacteristic
232C2×SD16—1911no—
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16—1911no—
236C2×SD16C2×D19111yescharacteristic
237C2×D8—1911no—
238C2×D8—1911no—
239C2×M16—1911no—
240C22×C8—1911no—
241#127—3821no—
242#127—1911no—
243#124—3821no—
244#124—1911no—
245C22⋊Q8—3821no—
246C22⋊Q8C2×D19111yescharacteristic
247#121—3821no—
248#121—3821no—
249#121—3821no—
250#121—1911no—
251#121—1911no—
252#121—1911no—
253C4×Q8—1911no—
254C4×Q8—1911no—
255C4×D4—3821no—
256C4×D4—3821no—
257C4×D4—1911no—
258#106—1911no—
259C4⋊C8—1911no—
260#103—1911no—
261#103—1911no—
262#103C2×D19111yescharacteristic
263#102—1911no—
264#102—1911no—
265#102C2×D19111yescharacteristic
266C22⋊C8—1911no—
267C22⋊C8C2×D19111yescharacteristic
268#97—1911no—
269#41—22no—
270#41—41no—
271#41—3822no—
272#41—3822no—
273#41—7641no—
274#41—7641no—
275C2×Q8—3821no—
276C2×Q8—3822no—
277C2×Q8—1911no—
278C2×Q8—7641no—
279C2×Q8C22×D19111yescharacteristic
280C2×Q8C22×D19111yescharacteristic
281C2×D4—21no—
282C2×D4—3821no—
283C2×D4—3821no—
284C2×D4—3821no—
285C2×D4—3821no—
286C2×D4—3821no—
287C2×D4—3821no—
288C2×D4—3821no—
289C2×D4—1911no—
290C2×D4—1911no—
291C2×D4—1911no—
292C2×D4C22×D19111yescharacteristic
293C22×C4—21no—
294C22×C4—3821no—
295C22×C4—3821no—
296C22×C4—3821no—
297C22×C4—1911no—
298C22×C4C22×D19111yescharacteristic
299Q16—22no—
300Q16—1914no—
301SD16—41no—
302SD16—3822no—
303SD16—3822no—
304SD16—3822no—
305SD16—3822no—
306D8—3822no—
307D8—7641no—
308M16—3822no—
309C2×C8—3822no—
310C2×C8—1911no—
311C2×C8—1911no—
312C2×C8C22×D19111yescharacteristic
313C2×C8C22×D19111yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—3821no—
316C4⋊C4—3821no—
317C4⋊C4—1911no—
318C4⋊C4—1911no—
319C4⋊C4C22×D19111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—3821no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—3821no—
325C22⋊C4—3821no—
326C22⋊C4—1911no—
327C22⋊C4—1911no—
328C42—3821no—
329C42—3821no—
330C42—1911no—
331C23—21no—
332C23—3821no—
333C23—3821no—
334C23—1911no—
335C23D4×D19111yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—3821no—
340Q8—1912no—
341Q8—7642no—
342Q8D4×D19112yes—
343D4—21no—
344D4—21no—
345D4—42no—
346D4—3821no—
347D4—3821no—
348D4—3821no—
349D4—3821no—
350D4—15281no—
351D4—15281no—
352D4—15281no—
353D4—7642no—
354D4—7642no—
355D4—7641no—
356D4—7642no—
357D4—7642no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—3821no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—3821no—
368C2×C4—1911no—
369C2×C4—1911no—
370C2×C4—1911no—
371C2×C4—7642no—
372C2×C4—7642no—
373C2×C4—7642no—
374C2×C4D4×D19111yescharacteristic
375C2×C4C23×D19111yescharacteristic
376C8—21no—
377C8—21no—
378C8—3821no—
379C8—1912no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—3821no—
387C22—3821no—
388C22—3821no—
389C22—7641no—
390C22—7641no—
391C22—7641no—
392C22—7641no—
393C22—7641no—
394C22—7641no—
395C22C2×D4×D19111yescharacteristic
396C4—21no—
397C4—22no—
398C4—41no—
399C4—41no—
400C4—3821no—
401C4—3821no—
402C4—3821no—
403C4—3822no—
404C4—7641no—
405C4C2×D4×D19111yescharacteristic
406C4#2025936711yescharacteristic
407C2—41no—
408C2—81no—
409C2—15281no—
410C2—15281no—
411C2—7641no—
412C2#2713329511yescharacteristic
413C2#2713346011yescharacteristic
414C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#452×C191C2
socleC22×C191C22×D4
hypercenter#452D191
hypocenterC191#3271
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#20259354C2semidirect
2. C2×#20259342C2semidirect
3. #27133332C2semidirect
4. #27133323C2semidirect
5. #27133049C2semidirect
6. #27132915C2semidirect
7. #27132894C2semidirect
8. #27132887C2semidirect
9. #27132862C2semidirect
10. #27132811C2semidirect
11. #27132810C2semidirect
12. #27132800C2semidirect
13. #27132767C2semidirect
14. #27132731C2semidirect
15. #452×C191C2semidirect
83. #15690277D4semidirect
207. C191#3271semidirect
218. #452D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593542
C2×#202593422
#271333322
#271333232
#271330492
#271329152
#271328942
#271328872
#271328622
#271328112
#271328102
#271328002
#271327672
#271327312
#452×C1912
#3271191