Finite Group Databasebeta

← #37583374

Subgroups — #37583374

Order 24448.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133241C211yesmaximal, characteristic, complemented
2#27133227C211yesmaximal, characteristic, complemented
3#27133005C211yesmaximal, characteristic, complemented
4#27132988C211yesmaximal, characteristic, complemented
5#27132982C211yesmaximal, characteristic, complemented
6#27132977C211yesmaximal, characteristic, complemented
7#27132916C211yesmaximal, characteristic, complemented
8#27132909C211yesmaximal, characteristic, complemented
9#27132899C211yesmaximal, characteristic, complemented
10#27132892C211yesmaximal, characteristic, complemented
11#27132816C211yesmaximal, characteristic, complemented
12#27132810C211yesmaximal, characteristic, complemented
13#27132810C211yesmaximal, characteristic, complemented
14#27132799C211yesmaximal, characteristic, complemented
15#408×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic, complemented
17C22×D764C2211yescharacteristic
18C2×#15690276C2211yescharacteristic
19C2×#15690276C2211yescharacteristic
20C2×D1528C2211yescharacteristic
21C2×D1528C2211yescharacteristic
22#20259318—21no—
23#20259311—21no—
24#20259301C2211yescharacteristic
25#20259301C2211yescharacteristic, complemented
26#20259298—21no—
27#20259296C2211yescharacteristic
28#20259296C2211yescharacteristic
29#20259296C2211yescharacteristic
30#20259296C2211yescharacteristic
31#20259296C2211yescharacteristic
32#20259296C2211yescharacteristic, complemented
33#20259294—21no—
34#20259294—21no—
35#20259293—21no—
36#20259290—21no—
37#20259290—21no—
38#20259281C2211yescharacteristic
39C2×C4⋊Dic191C2211yescharacteristic, complemented
40#20259256C2211yescharacteristic, complemented
41#20259256C2211yescharacteristic, complemented
42#20259254C2211yescharacteristic, complemented
43#20259254C2211yescharacteristic, complemented
44#20259247C2211yescharacteristic, complemented
45#20259247C2211yescharacteristic, complemented
46#20259244C2211yescharacteristic
47#20259244C2211yescharacteristic
48#20259244C2211yescharacteristic
49#20259244C2211yescharacteristic
50#20259244C2211yescharacteristic, complemented
51#20259244C2211yescharacteristic, complemented
52C2×#41×C191C2211yescharacteristic
53C2×M16×C191C2211yescharacteristic, complemented
54C22×C8×C191C2211yescharacteristic, complemented
55C22⋊C8×C191C2211yescharacteristic
56C22⋊C8×C191C2211yescharacteristic
57C22⋊C8×C191C2211yescharacteristic, complemented
58C22⋊C8×C191C2211yescharacteristic, complemented
59C23×D191—21no—
60#15690287—41no—
61#15690287—41no—
62C2×Dic382C2311yescharacteristic
63C2×D764—21no—
64C2×D764—41no—
65C2×D764—41no—
66C2×D764C2311yescharacteristic
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⋊D191—21no—
75C2×C22⋊D191—21no—
76C22×Dic191—21no—
77C2×C4×D191—21no—
78#15690276—41no—
79#15690276—41no—
80D1528—41no—
81D1528—41no—
82C4⋊Dic191—21no—
83C4⋊Dic191C2311yescharacteristic
84C4⋊Dic191C2311yescharacteristic
85C4⋊Dic191C2311yescharacteristic
86C4⋊Dic191C2311yescharacteristic
87Dic191⋊C4—21no—
88Dic191⋊C4—21no—
89C22⋊Dic191—21no—
90C22⋊Dic191—21no—
91C22⋊Dic191—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96(C2×D191)⋊C4—21no—
97(C2×D191)⋊C4—21no—
98C4×Dic191—21no—
99#41×C191—41no—
100#41×C191—41no—
101C2×Q8×C191D411yescharacteristic
102C2×D4×C191D411yescharacteristic
103C2×D4×C191D411yescharacteristic
104C2×D4×C191D411yescharacteristic
105C22×C4×C191C2311yescharacteristic
106C22×C4×C191C2311yescharacteristic
107C22×C4×C191C2311yescharacteristic
108M16×C191—41no—
109C2×C8×C191—41no—
110C2×C8×C191C2311yescharacteristic
111C2×C8×C191C2311yescharacteristic
112C2×C8×C191C2311yescharacteristic
113C2×C8×C191C2311yescharacteristic
114C22×D191—21no—
115C22×D191—21no—
116C22×D191—21no—
117C22×D191—42no—
118C22×D191—81no—
119C22×D191—81no—
120Dic382—21no—
121Dic382—21no—
122D764—21no—
123D764—21no—
124D764—21no—
125D764—21no—
126D764—21no—
127D764—21no—
128D764—42no—
129C22⋊D191—42no—
130C22⋊D191—81no—
131C22⋊D191—81no—
132C22⋊D191—81no—
133C22⋊D191—81no—
134C22⋊D191—81no—
135C22⋊D191—81no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—21no—
140C2×Dic191—21no—
141C2×Dic191—81no—
142C4×D191—42no—
143C23×C191C2×D411yescharacteristic
144C23×C191C2×D411yescharacteristic
145C23×C191C2×D411yescharacteristic
146Q8×C191—41no—
147D4×C191—41no—
148D4×C191—41no—
149D4×C191—41no—
150C2×C4×C191—21no—
151C2×C4×C191—21no—
152C2×C4×C191—21no—
153C2×C4×C191—21no—
154C2×C4×C191—21no—
155C2×C4×C191—21no—
156C2×C4×C191C2×D411yescharacteristic
157C2×C4×C191C2×D411yescharacteristic
158C2×C4×C191C2×D411yescharacteristic
159C2×C4×C191C2411yescharacteristic
160C8×C191—21no—
161C8×C191—21no—
162C8×C191—21no—
163C8×C191—21no—
164C22×C191—21no—
165C22×C191—21no—
166C22×C191—21no—
167C22×C191—21no—
168C22×C191—21no—
169C22×C191—21no—
170C22×C191—21no—
171C22×C191—21no—
172C22×C191—21no—
173C22×C191C22×D411yescharacteristic
174C2×D191—41no—
175C2×D191—41no—
176C2×D191—41no—
177C2×D191—41no—
178C2×D191—41no—
179C2×D191—41no—
180C2×D191—41no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—84no—
184C4×C191—21no—
185C4×C191—21no—
186C4×C191—21no—
187C4×C191H(5,𝔽2)11yescharacteristic
188C4×C191H(5,𝔽2)11yescharacteristic
189Dic191—41no—
190Dic191—41no—
191Dic191—41no—
192Dic191—41no—
193Dic191—41no—
194C2×C191—41no—
195C2×C191—41no—
196C2×C191—41no—
197C2×C191#53411yescharacteristic
198C2×C191#57611yescharacteristic
199C2×C191#57711yescharacteristic
200D191—81no—
201D191—81no—
202D191—81no—
203C191#319811yescharacteristic, complemented, Sylow
204#3198—1911nomaximal, Sylow
205#537—1911no—
206#535—1911no—
207#485—1911no—
208#482—1911no—
209#481—1911no—
210#480—1911no—
211#469—1911no—
212#468—1911no—
213#466—1911no—
214#465—1911no—
215#450—1911no—
216#449—1911no—
217#449—1911no—
218#447—1911no—
219#408D19111yescharacteristic, complemented
220C2×#41—1911no—
221C2×#41C2×D19111yescharacteristic
222C22×D4—1911no—
223C2×SD16—1911no—
224C2×SD16—1911no—
225C2×D8—1911no—
226C2×D8—1911no—
227C2×M16C2×D19111yescharacteristic, complemented
228C22×C8C2×D19111yescharacteristic, complemented
229#127—3821no—
230#124—3821no—
231C22⋊Q8—3821no—
232C22⋊Q8—1911no—
233C22⋊Q8—1911no—
234#121—3821no—
235#121—3821no—
236#121—3821no—
237#121—1911no—
238#121—1911no—
239#121—1911no—
240#121—1911no—
241#121—1911no—
242#121—1911no—
243#120—3821no—
244#120—3821no—
245#117—1911no—
246C2×C4⋊C4—1911no—
247#107—1911no—
248#107—1911no—
249#106—1911no—
250#106—1911no—
251#103—1911no—
252#103—1911no—
253#102—1911no—
254#102—1911no—
255#102—1911no—
256#102—1911no—
257#102—1911no—
258#102—1911no—
259C22⋊C8C2×D19111yescharacteristic
260C22⋊C8C2×D19111yescharacteristic
261C22⋊C8C2×D19111yescharacteristic, complemented
262C22⋊C8C2×D19111yescharacteristic, complemented
263C24—3821no—
264#41—41no—
265#41—41no—
266#41—7641no—
267#41—7641no—
268C2×Q8—1911no—
269C2×Q8D76411yescharacteristic
270C2×D4—3821no—
271C2×D4—3821no—
272C2×D4—3821no—
273C2×D4—3821no—
274C2×D4—3821no—
275C2×D4—3821no—
276C2×D4—3821no—
277C2×D4—3821no—
278C2×D4—1911no—
279C2×D4—1911no—
280C2×D4—1911no—
281C2×D4—7641no—
282C2×D4—7641no—
283C2×D4D76411yescharacteristic
284C2×D4D76411yescharacteristic
285C2×D4D76411yescharacteristic
286C22×C4—3821no—
287C22×C4—3821no—
288C22×C4C22×D19111yescharacteristic
289C22×C4C22×D19111yescharacteristic
290C22×C4C22×D19111yescharacteristic
291SD16—7641no—
292SD16—7641no—
293D8—7641no—
294D8—7641no—
295M16—41no—
296C2×C8—41no—
297C2×C8C22×D19111yescharacteristic
298C2×C8C22×D19111yescharacteristic
299C2×C8C22×D19111yescharacteristic
300C2×C8C22×D19111yescharacteristic
301C4⋊C4—3821no—
302C4⋊C4—3821no—
303C4⋊C4—3821no—
304C4⋊C4—1911no—
305C4⋊C4—1911no—
306C4⋊C4—1911no—
307C4⋊C4—1911no—
308C22⋊C4—3821no—
309C22⋊C4—3821no—
310C22⋊C4—3821no—
311C22⋊C4—3821no—
312C22⋊C4—3821no—
313C22⋊C4—3821no—
314C22⋊C4—3821no—
315C22⋊C4—3821no—
316C22⋊C4—3821no—
317C42—3821no—
318C23—3821no—
319C23—3821no—
320C23—3821no—
321C23—15281no—
322C23—15281no—
323C23—7642no—
324C23C2×D76411yescharacteristic
325C23C2×D76411yescharacteristic
326C23C2×D76411yescharacteristic
327Q8—41no—
328Q8—3821no—
329Q8—3821no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—3821no—
334D4—3821no—
335D4—3821no—
336D4—3821no—
337D4—3821no—
338D4—3821no—
339D4—15281no—
340D4—15281no—
341D4—15281no—
342D4—15281no—
343D4—15281no—
344D4—15281no—
345D4—7642no—
346D4—7642no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—3821no—
354C2×C4—3821no—
355C2×C4—3821no—
356C2×C4—3821no—
357C2×C4—3821no—
358C2×C4—15281no—
359C2×C4—7642no—
360C2×C4C2×D76411yescharacteristic
361C2×C4C2×D76411yescharacteristic
362C2×C4C2×D76411yescharacteristic
363C2×C4C23×D19111yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—15284no—
378C22—7641no—
379C22—7641no—
380C22—7641no—
381C22—7641no—
382C22—7641no—
383C22—7641no—
384C22—7641no—
385C22—7641no—
386C22—7641no—
387C22C22×D76411yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—7641no—
392C4—7641no—
393C4—7641no—
394C4—7641no—
395C4—7641no—
396C4#2025936711yescharacteristic
397C4#2025936711yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—15281no—
402C2—15281no—
403C2—15281no—
404C2#2713322511yescharacteristic
405C2#2713346811yescharacteristic
406C2#2713347511yescharacteristic
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#408×C191C2
socleC22×C191C22×D4
hypercenter#408D191
hypocenterC191#3198
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #27133241C2semidirect
2. #27133227C2semidirect
3. #27133005C2semidirect
4. #27132988C2semidirect
5. #27132982C2semidirect
6. #27132977C2semidirect
7. #27132916C2semidirect
8. #27132909C2semidirect
9. #27132899C2semidirect
10. #27132892C2semidirect
11. #27132816C2semidirect
12. #27132810C2semidirect
13. #27132810C2semidirect
14. #27132799C2semidirect
15. #408×C191C2semidirect
16. C2×#15690287C22semidirect
25. #20259301C22semidirect
32. #20259296C22semidirect
39. C2×C4⋊Dic191C22semidirect
40. #20259256C22semidirect
41. #20259256C22semidirect
42. #20259254C22semidirect
43. #20259254C22semidirect
44. #20259247C22semidirect
45. #20259247C22semidirect
50. #20259244C22semidirect
51. #20259244C22semidirect
53. C2×M16×C191C22semidirect
54. C22×C8×C191C22semidirect
57. C22⋊C8×C191C22semidirect
58. C22⋊C8×C191C22semidirect
203. C191#3198semidirect
219. #408D191semidirect
227. C2×M16C2×D191semidirect
228. C22×C8C2×D191semidirect
261. C22⋊C8C2×D191semidirect
262. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332412
#271332272
#271330052
#271329882
#271329822
#271329772
#271329162
#271329092
#271328992
#271328922
#271328162
#271328102
#271327992
#408×C1912
#3198191