Finite Group Databasebeta

← #37584099

Subgroups — #37584099

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×#20259345C211yesmaximal, characteristic, complemented
2#27133332C211yesmaximal, characteristic, complemented
3#27133324C211yesmaximal, characteristic, complemented
4#27133049C211yesmaximal, characteristic, complemented
5#27132916C211yesmaximal, characteristic, complemented
6#27132889C211yesmaximal, characteristic, complemented
7#27132887C211yesmaximal, characteristic, complemented
8#27132861C211yesmaximal, characteristic, complemented
9#27132824C211yesmaximal, characteristic, complemented
10#27132808C211yesmaximal, characteristic, complemented
11#27132807C211yesmaximal, characteristic, complemented
12#27132802C211yesmaximal, characteristic, complemented
13#27132770C211yesmaximal, characteristic, complemented
14#27132732C211yesmaximal, characteristic, complemented
15C2×#137×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690288C2211yescharacteristic
18#20259345—24no—
19C2×#15690274C2211yescharacteristic
20C2×#15690274C2211yescharacteristic
21C2×#15690274C2211yescharacteristic
22C2×D1528C2211yescharacteristic
23C2×#15690272C2211yescharacteristic
24C2×C8×D191C2211yescharacteristic
25#20259318—21no—
26#20259318C2211yescharacteristic
27#20259311—21no—
28#20259311C2211yescharacteristic
29#20259298—21no—
30#20259298C2211yescharacteristic
31#20259296—21no—
32#20259296—21no—
33#20259296C2211yescharacteristic
34#20259296C2211yescharacteristic
35#20259293—21no—
36#20259293C2211yescharacteristic
37Q8×Dic191C2211yescharacteristic
38Q8×Dic191C2211yescharacteristic
39C4×C22⋊D191—21no—
40C4×C22⋊D191—21no—
41C4×C22⋊D191C2211yescharacteristic
42#20259254C2211yescharacteristic
43Dic191⋊C8C2211yescharacteristic
44#20259246C2211yescharacteristic
45#20259246C2211yescharacteristic
46#20259246C2211yescharacteristic
47#20259244C2211yescharacteristic
48#20259244C2211yescharacteristic
49#20259243C2211yescharacteristic
50C22⋊(C191⋊C8)C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic
52#20259232C2211yescharacteristic
53C2×#41×C191C2211yescharacteristic
54C22×Q8×C191C2211yescharacteristic
55#137×C191—24no—
56C2×Q16×C191C2211yescharacteristic
57C2×Q16×C191C2211yescharacteristic
58C2×SD16×C191C2211yescharacteristic
59C2×SD16×C191C2211yescharacteristic
60C2×M16×C191C2211yescharacteristic
61#15690288—22no—
62#15690288—22no—
63#15690288—41no—
64#15690288—41no—
65C2×D764—21no—
66C2×D764—21no—
67C2×D764C2311yescharacteristic
68C2×D764C2311yescharacteristic
69C2×C22⋊D191—21no—
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191D411yescharacteristic
75C2×C4×D191—21no—
76C2×C4×D191—21no—
77C2×C4×D191C2311yescharacteristic
78#15690274—22no—
79#15690274—22no—
80#15690274—41no—
81D1528—22no—
82#15690272—41no—
83C2×C191⋊C8C2311yescharacteristic
84C2×C191⋊C8C2311yescharacteristic
85C8×D191—22no—
86C4⋊Dic191—21no—
87C4⋊Dic191—21no—
88C4⋊Dic191C2311yescharacteristic
89C4⋊Dic191C2311yescharacteristic
90Dic191⋊C4—21no—
91Dic191⋊C4D411yescharacteristic
92C22⋊Dic191—21no—
93C22⋊Dic191D411yescharacteristic
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96(C2×D191)⋊C4—21no—
97(C2×D191)⋊C4—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4D411yescharacteristic
100C4×Dic191—21no—
101C4×Dic191—21no—
102C4×Dic191C2311yescharacteristic
103#41×C191—22no—
104#41×C191—41no—
105C2×Q8×C191—21no—
106C2×Q8×C191—22no—
107C2×Q8×C191—41no—
108C2×Q8×C191C2311yescharacteristic
109C2×Q8×C191C2311yescharacteristic
110C2×Q8×C191C2311yescharacteristic
111C2×D4×C191—21no—
112C2×D4×C191C2311yescharacteristic
113C22×C4×C191—21no—
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116Q16×C191—22no—
117Q16×C191D414yescomplemented
118SD16×C191—22no—
119SD16×C191—22no—
120M16×C191—22no—
121C2×C8×C191C2311yescharacteristic
122C2×C8×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—21no—
125C22×D191C2×D411yescharacteristic
126D764—21no—
127D764—21no—
128D764—21no—
129D764—21no—
130D764—42no—
131D764—42no—
132C22⋊D191—41no—
133C22⋊D191—42no—
134C22⋊D191—42no—
135C22⋊D191—81no—
136C22⋊D191—81no—
137C22⋊D191—81no—
138C2×Dic191—21no—
139C2×Dic191—21no—
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C2×Dic191C2×D411yescharacteristic
143C4×D191—21no—
144C4×D191—21no—
145C4×D191—42no—
146C4×D191—42no—
147C191⋊C8—21no—
148C191⋊C8—21no—
149C23×C191—21no—
150C23×C191C2×D411yescharacteristic
151Q8×C191—21no—
152Q8×C191—21no—
153Q8×C191—21no—
154Q8×C191—21no—
155Q8×C191—42no—
156Q8×C191C2×D412yes—
157Q8×C191C2×D412yes—
158D4×C191—21no—
159D4×C191—21no—
160D4×C191—42no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—42no—
167C2×C4×C191—42no—
168C2×C4×C191C2×D411yescharacteristic
169C2×C4×C191C2411yescharacteristic
170C8×C191—21no—
171C8×C191C2×D412yes—
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—22no—
191C4×C191—41no—
192C4×C191C22×D411yescharacteristic
193C4×C191H(5,𝔽2)11yescharacteristic
194Dic191—21no—
195Dic191—21no—
196Dic191—21no—
197Dic191—41no—
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×#137D19111yescharacteristic, complemented
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#452—1911no—
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×Q8C2×D19111yescharacteristic
228#137—24no—
229#135—3824no—
230C2×Q16C2×D19111yescharacteristic
231C2×Q16C2×D19111yescharacteristic
232C2×SD16—1911no—
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16C2×D19111yescharacteristic
236C2×SD16C2×D19111yescharacteristic
237C2×D8—1911no—
238C2×D8—1911no—
239C2×M16C2×D19111yescharacteristic
240C22×C8—1911no—
241#127—3821no—
242#127—1911no—
243#124—3821no—
244#124—1911no—
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#97—1911no—
269#41—22no—
270#41—41no—
271#41—3822no—
272#41—3822no—
273#41—7641no—
274#41—7641no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D19111yescharacteristic
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—21no—
295C22×C4—3821no—
296C22×C4—3821no—
297C22×C4—1911no—
298C22×C4C22×D19111yescharacteristic
299Q16—22no—
300Q16C22⋊D19114yescomplemented
301SD16—22no—
302SD16—22no—
303SD16—3822no—
304SD16—3822no—
305SD16—7641no—
306D8—3822no—
307D8—7641no—
308M16—22no—
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—3821no—
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—21no—
340Q8—42no—
341Q8C2×C22⋊D19112yes—
342Q8C2×C22⋊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—42no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—3821no—
368C2×C4—3821no—
369C2×C4—1911no—
370C2×C4—1911no—
371C2×C4—1911no—
372C2×C4—7642no—
373C2×C4—7642no—
374C2×C4D4×D19111yescharacteristic
375C2×C4C23×D19111yescharacteristic
376C8—21no—
377C8—3821no—
378C8—3821no—
379C8C2×C22⋊D19112yes—
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—22no—
399C4—41no—
400C4—3821no—
401C4—3821no—
402C4—3821no—
403C4—7641no—
404C4—7641no—
405C4C22×C22⋊D19111yescharacteristic
406C4#2025936711yescharacteristic
407C2—41no—
408C2—81no—
409C2—15281no—
410C2—15281no—
411C2—7641no—
412C2D4×C22⋊D19111yescharacteristic
413C2#2713346211yescharacteristic
414C2#2713347211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#137×C191C2
socleC22×C191C22×D4
hypercenterC2×#137D191
hypocenterC191#3271
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#20259345C2semidirect
2. #27133332C2semidirect
3. #27133324C2semidirect
4. #27133049C2semidirect
5. #27132916C2semidirect
6. #27132889C2semidirect
7. #27132887C2semidirect
8. #27132861C2semidirect
9. #27132824C2semidirect
10. #27132808C2semidirect
11. #27132807C2semidirect
12. #27132802C2semidirect
13. #27132770C2semidirect
14. #27132732C2semidirect
15. C2×#137×C191C2semidirect
117. Q16×C191D4semidirect
207. C191#3271semidirect
209. C2×#137D191semidirect
300. Q16C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593452
#271333322
#271333242
#271330492
#271329162
#271328892
#271328872
#271328612
#271328242
#271328082
#271328072
#271328022
#271327702
#271327322
C2×#137×C1912
#3271191