Finite Group Databasebeta

← #37583402

Subgroups — #37583402

Order 24448.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133241C211yesmaximal, characteristic
2#27133004C211yesmaximal, characteristic, complemented
3#27132996C211yesmaximal, characteristic, complemented
4#27132991C211yesmaximal, characteristic, complemented
5#27132987C211yesmaximal, characteristic, complemented
6#27132919C211yesmaximal, characteristic, complemented
7#27132911C211yesmaximal, characteristic, complemented
8#27132902C211yesmaximal, characteristic, complemented
9#27132894C211yesmaximal, characteristic, complemented
10#27132823C211yesmaximal, characteristic, complemented
11#27132823C211yesmaximal, characteristic, complemented
12#27132819C211yesmaximal, characteristic, complemented
13#27132802C211yesmaximal, characteristic, complemented
14#27132630C211yesmaximal, characteristic, complemented
15#536×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×#15690277C2211yescharacteristic
18C2×#15690277C2211yescharacteristic
19C2×#15690274C2211yescharacteristic
20C2×#15690274C2211yescharacteristic
21C2×#15690271C2211yescharacteristic
22C22×C191⋊C8C2211yescharacteristic
23#20259301C2211yescharacteristic
24#20259301C2211yescharacteristic
25#20259298—21no—
26#20259296C2211yescharacteristic
27#20259296C2211yescharacteristic
28#20259294—21no—
29#20259294—21no—
30#20259293—21no—
31C2×C4⋊Dic191C2211yescharacteristic
32#20259255C2211yescharacteristic
33#20259255C2211yescharacteristic
34#20259253C2211yescharacteristic
35#20259253C2211yescharacteristic
36#20259246C2211yescharacteristic
37#20259246C2211yescharacteristic
38#20259246C2211yescharacteristic
39#20259246C2211yescharacteristic
40#20259245C2211yescharacteristic
41#20259245C2211yescharacteristic
42#20259242C2211yescharacteristic
43#20259242C2211yescharacteristic
44C22⋊(C191⋊C8)C2211yescharacteristic
45C22⋊(C191⋊C8)C2211yescharacteristic
46C22⋊(C191⋊C8)C2211yescharacteristic
47C22⋊(C191⋊C8)C2211yescharacteristic
48C2×#41×C191C2211yescharacteristic
49C22×Q8×C191C2211yescharacteristic
50C4⋊Q8×C191—21no—
51#124×C191—21no—
52#123×C191—21no—
53#123×C191—21no—
54C22⋊Q8×C191C2211yescharacteristic
55C22⋊Q8×C191C2211yescharacteristic
56C22⋊Q8×C191C2211yescharacteristic
57C22⋊Q8×C191C2211yescharacteristic
58#117×C191C2211yescharacteristic
59#15690287—41no—
60#15690287—41no—
61C2×Dic382C2311yescharacteristic
62C2×D764C2311yescharacteristic
63C2×C22⋊D191—21no—
64C2×C22⋊D191—21no—
65C2×C22⋊D191—21no—
66C22×Dic191—21no—
67C2×C4×D191—21no—
68#15690277—41no—
69#15690277—41no—
70#15690274—41no—
71#15690274—41no—
72#15690271—41no—
73C2×C191⋊C8—41no—
74C2×C191⋊C8C2311yescharacteristic
75C2×C191⋊C8C2311yescharacteristic
76C2×C191⋊C8C2311yescharacteristic
77C2×C191⋊C8C2311yescharacteristic
78C4⋊Dic191—21no—
79C4⋊Dic191C2311yescharacteristic
80C4⋊Dic191C2311yescharacteristic
81Dic191⋊C4—21no—
82Dic191⋊C4—21no—
83C22⋊Dic191—21no—
84C22⋊Dic191—21no—
85(C2×D191)⋊C4—21no—
86(C2×D191)⋊C4—21no—
87#41×C191—41no—
88#41×C191—41no—
89C2×Q8×C191—21no—
90C2×Q8×C191—41no—
91C2×Q8×C191—41no—
92C2×Q8×C191D411yescharacteristic
93C2×Q8×C191C2311yescharacteristic
94C2×Q8×C191C2311yescharacteristic
95C2×D4×C191D411yescharacteristic
96C2×D4×C191D411yescharacteristic
97C2×D4×C191D411yescharacteristic
98C22×C4×C191—21no—
99C22×C4×C191C2311yescharacteristic
100C22×C4×C191C2311yescharacteristic
101C22×C4×C191C2311yescharacteristic
102C4⋊C4×C191—21no—
103C4⋊C4×C191—21no—
104C4⋊C4×C191—21no—
105C4⋊C4×C191—21no—
106C4⋊C4×C191C2311yescharacteristic
107C4⋊C4×C191C2311yescharacteristic
108C22⋊C4×C191—21no—
109C22⋊C4×C191—21no—
110C22⋊C4×C191—21no—
111C22⋊C4×C191—21no—
112C22⋊C4×C191—21no—
113C42×C191—21no—
114C22×D191—21no—
115Dic382—21no—
116Dic382—21no—
117D764—21no—
118D764—21no—
119C22⋊D191—42no—
120C22⋊D191—81no—
121C22⋊D191—81no—
122C2×Dic191—21no—
123C2×Dic191—21no—
124C2×Dic191—21no—
125C2×Dic191—81no—
126C4×D191—42no—
127C191⋊C8—21no—
128C191⋊C8—21no—
129C191⋊C8—21no—
130C191⋊C8—21no—
131C23×C191C2×D411yescharacteristic
132C23×C191C2×D411yescharacteristic
133C23×C191C2×D411yescharacteristic
134Q8×C191—21no—
135Q8×C191—21no—
136Q8×C191—21no—
137Q8×C191—21no—
138Q8×C191—41no—
139Q8×C191—42no—
140D4×C191—41no—
141D4×C191—41no—
142D4×C191—41no—
143C2×C4×C191—21no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191—21no—
147C2×C4×C191—21no—
148C2×C4×C191—21no—
149C2×C4×C191—21no—
150C2×C4×C191—21no—
151C2×C4×C191—21no—
152C2×C4×C191—21no—
153C2×C4×C191—42no—
154C2×C4×C191C2×D411yescharacteristic
155C2×C4×C191C2×D411yescharacteristic
156C2×C4×C191C2×D411yescharacteristic
157C2×C4×C191C2411yescharacteristic
158C22×C191—21no—
159C22×C191—21no—
160C22×C191—21no—
161C22×C191—21no—
162C22×C191—21no—
163C22×C191—21no—
164C22×C191—21no—
165C22×C191—21no—
166C22×C191—21no—
167C22×C191C22×D411yescharacteristic
168C2×D191—41no—
169C2×D191—41no—
170C2×D191—41no—
171C4×C191—21no—
172C4×C191—21no—
173C4×C191—21no—
174C4×C191—41no—
175C4×C191—41no—
176C4×C191—41no—
177C4×C191—41no—
178C4×C191H(5,𝔽2)11yescharacteristic
179C4×C191H(5,𝔽2)11yescharacteristic
180Dic191—41no—
181Dic191—41no—
182Dic191—41no—
183C2×C191—41no—
184C2×C191—41no—
185C2×C191—41no—
186C2×C191#53411yescharacteristic
187C2×C191#57711yescharacteristic
188C2×C191#57811yescharacteristic
189D191—81no—
190C191#320111yescharacteristic, complemented, Sylow
191#3201—1911nomaximal, Sylow
192#537—1911no—
193#536D19111yescharacteristic, complemented
194#485—1911no—
195#484—1911no—
196#483—1911no—
197#482—1911no—
198#470—1911no—
199#468—1911no—
200#467—1911no—
201#465—1911no—
202#452—1911no—
203#452—1911no—
204#451—1911no—
205#448—1911no—
206#408—1911no—
207C2×#41—1911no—
208C2×#41C2×D19111yescharacteristic
209C22×Q8C2×D19111yescharacteristic
210C2×Q16—1911no—
211C2×Q16—1911no—
212C2×SD16—1911no—
213C2×SD16—1911no—
214C2×M16—1911no—
215C22×C8—1911no—
216C4⋊Q8—21no—
217#124—21no—
218#123—21no—
219#123—21no—
220C22⋊Q8—3821no—
221C22⋊Q8—1911no—
222C22⋊Q8—1911no—
223C22⋊Q8C2×D19111yescharacteristic
224C22⋊Q8C2×D19111yescharacteristic
225C22⋊Q8C2×D19111yescharacteristic
226C22⋊Q8C2×D19111yescharacteristic
227#121—3821no—
228#121—3821no—
229#121—3821no—
230#121—1911no—
231#121—1911no—
232#117C2×D19111yescharacteristic
233C2×C4⋊C4—1911no—
234#107—1911no—
235#107—1911no—
236#106—1911no—
237#106—1911no—
238#103—1911no—
239#103—1911no—
240#103—1911no—
241#103—1911no—
242#103—1911no—
243#103—1911no—
244#102—1911no—
245#102—1911no—
246C22⋊C8—1911no—
247C22⋊C8—1911no—
248C22⋊C8—1911no—
249C22⋊C8—1911no—
250#41—41no—
251#41—41no—
252#41—7641no—
253#41—7641no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—1911no—
258C2×Q8C22⋊D19111yescharacteristic
259C2×Q8C22×D19111yescharacteristic
260C2×Q8C22×D19111yescharacteristic
261C2×D4—3821no—
262C2×D4—3821no—
263C2×D4—3821no—
264C2×D4—1911no—
265C2×D4C22⋊D19111yescharacteristic
266C2×D4C22⋊D19111yescharacteristic
267C2×D4C22⋊D19111yescharacteristic
268C22×C4—21no—
269C22×C4—3821no—
270C22×C4—3821no—
271C22×C4C22×D19111yescharacteristic
272C22×C4C22×D19111yescharacteristic
273C22×C4C22×D19111yescharacteristic
274Q16—7641no—
275Q16—7641no—
276SD16—7641no—
277SD16—7641no—
278M16—7641no—
279C2×C8—1911no—
280C2×C8—1911no—
281C2×C8—1911no—
282C2×C8—1911no—
283C2×C8—7641no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—3821no—
289C4⋊C4—3821no—
290C4⋊C4—3821no—
291C4⋊C4—1911no—
292C4⋊C4—1911no—
293C4⋊C4C22×D19111yescharacteristic
294C4⋊C4C22×D19111yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3821no—
301C22⋊C4—3821no—
302C22⋊C4—3821no—
303C22⋊C4—3821no—
304C42—21no—
305C23—3821no—
306C23C2×C22⋊D19111yescharacteristic
307C23C2×C22⋊D19111yescharacteristic
308C23C2×C22⋊D19111yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—41no—
314Q8—42no—
315Q8—3821no—
316Q8—3821no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—3821no—
321D4—3821no—
322D4—15281no—
323D4—15281no—
324D4—7642no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—3821no—
337C2×C4—3821no—
338C2×C4—3821no—
339C2×C4—15281no—
340C2×C4—7642no—
341C2×C4C2×C22⋊D19111yescharacteristic
342C2×C4C2×C22⋊D19111yescharacteristic
343C2×C4C2×C22⋊D19111yescharacteristic
344C2×C4C23×D19111yescharacteristic
345C8—3821no—
346C8—3821no—
347C8—3821no—
348C8—3821no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—7641no—
359C22—7641no—
360C22—7641no—
361C22C22×C22⋊D19111yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—7641no—
370C4—7641no—
371C4—7641no—
372C4#2025936711yescharacteristic
373C4#2025936711yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—15281no—
378C2#2713322211yescharacteristic
379C2#2713347111yescharacteristic
380C2#2713347811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#536×C191C2
socleC22×C191C22×D4
prime radical#27133241C2
hypercenter#536D191
hypocenterC191#3201
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #27133004C2semidirect
3. #27132996C2semidirect
4. #27132991C2semidirect
5. #27132987C2semidirect
6. #27132919C2semidirect
7. #27132911C2semidirect
8. #27132902C2semidirect
9. #27132894C2semidirect
10. #27132823C2semidirect
11. #27132823C2semidirect
12. #27132819C2semidirect
13. #27132802C2semidirect
14. #27132630C2semidirect
15. #536×C191C2semidirect
190. C191#3201semidirect
193. #536D191semidirect

Maximal subgroups

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

subgroupindex
#271332412
#271330042
#271329962
#271329912
#271329872
#271329192
#271329112
#271329022
#271328942
#271328232
#271328192
#271328022
#271326302
#536×C1912
#3201191