Finite Group Databasebeta

← #37584104

Subgroups — #37584104

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×#20259351C211yesmaximal, characteristic, complemented
2C2×#20259341C211yesmaximal, characteristic, complemented
3#27133333C211yesmaximal, characteristic, complemented
4#27133321C211yesmaximal, characteristic, complemented
5#27132913C211yesmaximal, characteristic, complemented
6#27132889C211yesmaximal, characteristic, complemented
7#27132882C211yesmaximal, characteristic, complemented
8#27132860C211yesmaximal, characteristic, complemented
9#27132822C211yesmaximal, characteristic, complemented
10#27132805C211yesmaximal, characteristic, complemented
11#27132805C211yesmaximal, characteristic, complemented
12#27132800C211yesmaximal, characteristic, complemented
13#27132771C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#496×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18C2×#15690286C2211yescharacteristic
19C2×Q8×D191C2211yescharacteristic
20#20259351—24no—
21#20259341—24no—
22C2×Dic764C2211yescharacteristic
23C2×#15690277C2211yescharacteristic
24C2×#15690275C2211yescharacteristic
25C2×#15690275C2211yescharacteristic
26C2×#15690275C2211yescharacteristic
27C2×#15690270C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic
29#20259318—21no—
30#20259308—21no—
31#20259299—21no—
32#20259298C2211yescharacteristic
33#20259295—21no—
34#20259294—21no—
35#20259294—21no—
36#20259293C2211yescharacteristic
37#20259293C2211yescharacteristic
38#20259293C2211yescharacteristic
39C4×Dic382C2211yescharacteristic
40C4×Dic382C2211yescharacteristic
41C4×D764C2211yescharacteristic
42D4×Dic191—21no—
43(C22⋊D191)⋊C4—21no—
44#20259254C2211yescharacteristic
45C4⋊(C191⋊C8)C2211yescharacteristic
46#20259247C2211yescharacteristic
47#20259247C2211yescharacteristic
48#20259246C2211yescharacteristic
49#20259243C2211yescharacteristic
50#20259243C2211yescharacteristic
51#20259243C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53(C2×D191)⋊C8C2211yescharacteristic
54C2×SD16×C191C2211yescharacteristic
55C2×SD16×C191C2211yescharacteristic
56C2×D8×C191C2211yescharacteristic
57C2×D8×C191C2211yescharacteristic
58#127×C191C2211yescharacteristic
59#124×C191C2211yescharacteristic
60#97×C191C2211yescharacteristic
61#15690286—22no—
62#15690286—22no—
63#15690286—22no—
64#15690286—41no—
65#15690286—41no—
66#15690286—41no—
67C2×Dic382—21no—
68C2×Dic382C2311yescharacteristic
69C2×Dic382C2311yescharacteristic
70Q8×D191—22no—
71Q8×D191—41no—
72C2×D764D411yescharacteristic
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C2×C22⋊D191—21no—
78C2×C22⋊D191—21no—
79C22×Dic191—21no—
80C22×Dic191—21no—
81C22×Dic191—21no—
82C2×C4×D191—21no—
83C2×C4×D191C2311yescharacteristic
84C2×C4×D191C2311yescharacteristic
85Dic764D414yescomplemented
86#15690277—22no—
87#15690275—22no—
88#15690275—22no—
89#15690275—22no—
90#15690270—22no—
91C2×C191⋊C8C2311yescharacteristic
92C2×C191⋊C8C2311yescharacteristic
93C8×D191—22no—
94C4⋊Dic191D411yescharacteristic
95C4⋊Dic191C2311yescharacteristic
96C4⋊Dic191C2311yescharacteristic
97Dic191⋊C4—21no—
98Dic191⋊C4—21no—
99Dic191⋊C4—21no—
100C22⋊Dic191—21no—
101C22⋊Dic191—21no—
102C22⋊Dic191—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4D411yescharacteristic
105(C2×D191)⋊C4D411yescharacteristic
106C4×Dic191—21no—
107C4×Dic191—21no—
108C2×Q8×C191C2311yescharacteristic
109C2×D4×C191—21no—
110C2×D4×C191—21no—
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114SD16×C191—22no—
115SD16×C191—41no—
116D8×C191—22no—
117D8×C191—41no—
118C2×C8×C191C2311yescharacteristic
119C2×C8×C191C2311yescharacteristic
120C22⋊C4×C191—21no—
121C22⋊C4×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191C2×D411yescharacteristic
124C22×D191C2×D411yescharacteristic
125Dic382—21no—
126Dic382—21no—
127Dic382—42no—
128Dic382C2×D412yes—
129Dic382C2×D412yes—
130D764—41no—
131C22⋊D191—42no—
132C22⋊D191—42no—
133C22⋊D191—42no—
134C22⋊D191—42no—
135C22⋊D191—81no—
136C22⋊D191—81no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—21no—
140C2×Dic191—21no—
141C2×Dic191—42no—
142C2×Dic191—42no—
143C2×Dic191—42no—
144C2×Dic191C2×D411yescharacteristic
145C2×Dic191C2×D411yescharacteristic
146C4×D191—21no—
147C4×D191—21no—
148C4×D191—21no—
149C4×D191—21no—
150C4×D191—42no—
151C191⋊C8—21no—
152C191⋊C8—21no—
153C23×C191—21no—
154C23×C191—21no—
155C23×C191—21no—
156Q8×C191—21no—
157Q8×C191—21no—
158D4×C191—21no—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191—21no—
162D4×C191—21no—
163D4×C191—21no—
164D4×C191—42no—
165D4×C191—81no—
166C2×C4×C191—21no—
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2×D411yescharacteristic
169C2×C4×C191C2411yescharacteristic
170C8×C191—21no—
171C8×C191C2×D412yes—
172C22×C191—41no—
173C22×C191—41no—
174C22×C191—41no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191—41no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191C22×D411yescharacteristic
182C2×D191—21no—
183C2×D191—21no—
184C2×D191—21no—
185C2×D191—21no—
186C2×D191—21no—
187C2×D191—21no—
188C4×C191—21no—
189C4×C191—21no—
190C4×C191—41no—
191C4×C191C22×D411yescharacteristic
192C4×C191H(5,𝔽2)11yescharacteristic
193Dic191—21no—
194Dic191—21no—
195Dic191—22no—
196Dic191—22no—
197Dic191—41no—
198Dic191—41no—
199C2×C191—81no—
200C2×C191—81no—
201C2×C191—81no—
202C2×C191D4211yescharacteristic
203C2×C191#57511yescharacteristic
204C2×C191#57711yescharacteristic
205D191—41no—
206D191—41no—
207C191#327111yescharacteristic, complemented, Sylow
208#3271—1911nomaximal, Sylow
209C2×#137—1911no—
210C2×#135—1911no—
211#550—1911no—
212#548—1911no—
213#496D19111yescharacteristic, complemented
214#469—1911no—
215#465—1911no—
216#464—1911no—
217#460—1911no—
218#452—1911no—
219#449—1911no—
220#449—1911no—
221#448—1911no—
222#441—1911no—
223#435—1911no—
224C2×#41—1911no—
225C2×#41—1911no—
226C2×#41—1911no—
227C22×Q8—1911no—
228#137—3824no—
229#135—3824no—
230C2×Q16—1911no—
231C2×Q16—1911no—
232C2×SD16—1911no—
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16C2×D19111yescharacteristic
236C2×SD16C2×D19111yescharacteristic
237C2×D8C2×D19111yescharacteristic
238C2×D8C2×D19111yescharacteristic
239C2×M16—1911no—
240C22×C8—1911no—
241#127—3821no—
242#127C2×D19111yescharacteristic
243#124—3821no—
244#124C2×D19111yescharacteristic
245C22⋊Q8—3821no—
246C22⋊Q8—1911no—
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#103—1911no—
263#102—1911no—
264#102—1911no—
265#102—1911no—
266C22⋊C8—1911no—
267C22⋊C8—1911no—
268#97C2×D19111yescharacteristic
269#41—3822no—
270#41—3822no—
271#41—3822no—
272#41—7641no—
273#41—7641no—
274#41—7641no—
275C2×Q8—3821no—
276C2×Q8—3822no—
277C2×Q8—1911no—
278C2×Q8—1911no—
279C2×Q8—7641no—
280C2×Q8C22×D19111yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3821no—
284C2×D4—3821no—
285C2×D4—3821no—
286C2×D4—3821no—
287C2×D4—3821no—
288C2×D4—3821no—
289C2×D4—1911no—
290C2×D4C22×D19111yescharacteristic
291C2×D4C22×D19111yescharacteristic
292C2×D4C22×D19111yescharacteristic
293C22×C4—3821no—
294C22×C4—3821no—
295C22×C4—3821no—
296C22×C4—3821no—
297C22×C4—1911no—
298C22×C4—1911no—
299Q16—3822no—
300Q16—1914no—
301SD16—22no—
302SD16—41no—
303SD16—3822no—
304SD16—3822no—
305SD16—3822no—
306D8—22no—
307D8—41no—
308M16—3822no—
309C2×C8—3822no—
310C2×C8—1911no—
311C2×C8—1911no—
312C2×C8C22×D19111yescharacteristic
313C2×C8C22×D19111yescharacteristic
314C4⋊C4—3821no—
315C4⋊C4—3821no—
316C4⋊C4—3821no—
317C4⋊C4—1911no—
318C4⋊C4—1911no—
319C4⋊C4—1911no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—3821no—
325C22⋊C4—3821no—
326C22⋊C4—1911no—
327C22⋊C4—1911no—
328C42—3821no—
329C42—3821no—
330C42C22×D19111yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23—1911no—
335C23—1911no—
336Q8—21no—
337Q8—21no—
338Q8—3821no—
339Q8—3821no—
340Q8—1912no—
341Q8—1912no—
342Q8—7642no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—81no—
351D4—15281no—
352D4—15281no—
353D4—7641no—
354D4—7642no—
355D4—7642no—
356D4—7642no—
357D4—7642no—
358C2×C4—21no—
359C2×C4—3821no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—3821no—
363C2×C4—3821no—
364C2×C4—3821no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—1911no—
368C2×C4—1911no—
369C2×C4—7642no—
370C2×C4—7642no—
371C2×C4—7642no—
372C2×C4—7642no—
373C2×C4D4×D19111yescharacteristic
374C2×C4D4×D19111yescharacteristic
375C2×C4C23×D19111yescharacteristic
376C8—21no—
377C8—3821no—
378C8—3821no—
379C8D4×D19112yes—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3821no—
390C22—3821no—
391C22—3821no—
392C22—3821no—
393C22—3821no—
394C22—3821no—
395C22C2×D4×D19111yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—3821no—
400C4—3821no—
401C4—3822no—
402C4—3822no—
403C4—7641no—
404C4—7641no—
405C4C2×D4×D19111yescharacteristic
406C4#2025936711yescharacteristic
407C2—81no—
408C2—81no—
409C2—81no—
410C2—7641no—
411C2—7641no—
412C2#2713329811yescharacteristic
413C2#2713345911yescharacteristic
414C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#496×C191C2
socleC22×C191C22×D4
hypercenter#496D191
hypocenterC191#3271
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#20259351C2semidirect
2. C2×#20259341C2semidirect
3. #27133333C2semidirect
4. #27133321C2semidirect
5. #27132913C2semidirect
6. #27132889C2semidirect
7. #27132882C2semidirect
8. #27132860C2semidirect
9. #27132822C2semidirect
10. #27132805C2semidirect
11. #27132805C2semidirect
12. #27132800C2semidirect
13. #27132771C2semidirect
14. #27132736C2semidirect
15. #496×C191C2semidirect
85. Dic764D4semidirect
207. C191#3271semidirect
213. #496D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593512
C2×#202593412
#271333332
#271333212
#271329132
#271328892
#271328822
#271328602
#271328222
#271328052
#271328002
#271327712
#271327362
#496×C1912
#3271191