Finite Group Databasebeta

← #37584091

Subgroups — #37584091

Order 24448.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259353C211yesmaximal, characteristic, complemented
2C22×#15690277C211yesmaximal, characteristic, complemented
3#27133322C211yesmaximal, characteristic
4#27133055C211yesmaximal, characteristic, complemented
5#27132919C211yesmaximal, characteristic, complemented
6#27132894C211yesmaximal, characteristic, complemented
7#27132885C211yesmaximal, characteristic, complemented
8#27132870C211yesmaximal, characteristic, complemented
9#27132823C211yesmaximal, characteristic, complemented
10#27132819C211yesmaximal, characteristic, complemented
11#27132818C211yesmaximal, characteristic, complemented
12#27132802C211yesmaximal, characteristic, complemented
13#27132768C211yesmaximal, characteristic, complemented
14#27132735C211yesmaximal, characteristic, complemented
15D4×Q8×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C22×Dic382C2211yescharacteristic
18#20259353—24no—
19C2×#15690277—24no—
20C2×#15690277C2211yescharacteristic
21C2×#15690277C2211yescharacteristic
22C2×#15690277C2211yescharacteristic
23C2×#15690277C2211yescharacteristic
24C2×#15690277C2211yescharacteristic
25C2×#15690277C2211yescharacteristic
26C2×#15690274C2211yescharacteristic
27C2×#15690274C2211yescharacteristic
28C2×#15690274C2211yescharacteristic
29C2×#15690271C2211yescharacteristic
30C22×C191⋊C8C2211yescharacteristic
31#20259310C2211yescharacteristic
32#20259308—21no—
33#20259301C2211yescharacteristic
34#20259300—21no—
35#20259296C2211yescharacteristic
36#20259294—21no—
37C4×Dic382C2211yescharacteristic
38(C22⋊D191)⋊C4—21no—
39#20259253C2211yescharacteristic
40C4⋊(C191⋊C8)C2211yescharacteristic
41#20259246C2211yescharacteristic
42#20259246C2211yescharacteristic
43#20259246C2211yescharacteristic
44#20259245C2211yescharacteristic
45#20259245C2211yescharacteristic
46#20259242C2211yescharacteristic
47C22⋊(C191⋊C8)C2211yescharacteristic
48C22⋊(C191⋊C8)C2211yescharacteristic
49#20259233C2211yescharacteristic
50C22×Q8×C191C2211yescharacteristic
51C22×Q8×C191C2211yescharacteristic
52C4⋊Q8×C191—21no—
53C4⋊Q8×C191C2211yescharacteristic
54C22⋊Q8×C191—21no—
55C22⋊Q8×C191—21no—
56C22⋊Q8×C191C2211yescharacteristic
57C22⋊Q8×C191C2211yescharacteristic
58C4×Q8×C191C2211yescharacteristic
59C4×D4×C191—21no—
60C4×D4×C191C2211yescharacteristic
61#15690287—22no—
62#15690287—41no—
63C2×Dic382—22no—
64C2×Dic382—21no—
65C2×Dic382—41no—
66C2×Dic382C2311yescharacteristic
67C2×Dic382C2311yescharacteristic
68C2×D764C2311yescharacteristic
69C2×C22⋊D191—21no—
70C2×C22⋊D191—21no—
71C22×Dic191—21no—
72C2×C4×D191—21no—
73#15690277—22no—
74#15690277—22no—
75#15690277—22no—
76#15690277—22no—
77#15690277—41no—
78#15690277D414yescomplemented
79#15690274—22no—
80#15690274—22no—
81#15690274—41no—
82#15690271—22no—
83C2×C191⋊C8—22no—
84C2×C191⋊C8C2311yescharacteristic
85C2×C191⋊C8C2311yescharacteristic
86C2×C191⋊C8C2311yescharacteristic
87C2×C191⋊C8C2311yescharacteristic
88C4⋊Dic191C2311yescharacteristic
89Dic191⋊C4—21no—
90Dic191⋊C4—21no—
91C22⋊Dic191—21no—
92(C2×D191)⋊C4—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95C4×Dic191—21no—
96C2×Q8×C191—21no—
97C2×Q8×C191—21no—
98C2×Q8×C191—22no—
99C2×Q8×C191—22no—
100C2×Q8×C191—41no—
101C2×Q8×C191—41no—
102C2×Q8×C191C2311yescharacteristic
103C2×Q8×C191C2311yescharacteristic
104C2×Q8×C191C2311yescharacteristic
105C2×D4×C191D411yescharacteristic
106C22×C4×C191—21no—
107C22×C4×C191—21no—
108C22×C4×C191C2311yescharacteristic
109C22×C4×C191C2311yescharacteristic
110C4⋊C4×C191—21no—
111C4⋊C4×C191—21no—
112C4⋊C4×C191—21no—
113C4⋊C4×C191—21no—
114C4⋊C4×C191—21no—
115C4⋊C4×C191D411yescharacteristic
116C4⋊C4×C191C2311yescharacteristic
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C22⋊C4×C191D411yescharacteristic
120C22⋊C4×C191D411yescharacteristic
121C42×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191—21no—
124Dic382—21no—
125Dic382—21no—
126Dic382—21no—
127Dic382—42no—
128Dic382C2×D412yes—
129D764—21no—
130D764—21no—
131C22⋊D191—42no—
132C22⋊D191—42no—
133C2×Dic191—21no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191—42no—
137C4×D191—42no—
138C191⋊C8—21no—
139C191⋊C8—21no—
140C191⋊C8—21no—
141C191⋊C8C2×D412yes—
142C23×C191C2×D411yescharacteristic
143C23×C191C2×D411yescharacteristic
144Q8×C191—21no—
145Q8×C191—21no—
146Q8×C191—21no—
147Q8×C191—21no—
148Q8×C191—21no—
149Q8×C191—42no—
150Q8×C191—42no—
151Q8×C191C2×D412yes—
152D4×C191—22no—
153C2×C4×C191—21no—
154C2×C4×C191—21no—
155C2×C4×C191—21no—
156C2×C4×C191—21no—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—42no—
162C2×C4×C191—42no—
163C2×C4×C191C2×D411yescharacteristic
164C2×C4×C191C2×D411yescharacteristic
165C2×C4×C191C2×D411yescharacteristic
166C2×C4×C191C2×D411yescharacteristic
167C2×C4×C191C2411yescharacteristic
168C22×C191—21no—
169C22×C191—21no—
170C22×C191—21no—
171C22×C191—21no—
172C22×C191—21no—
173C22×C191#13712yescomplemented
174C22×C191C22×D411yescharacteristic
175C2×D191—41no—
176C2×D191—41no—
177C2×D191—41no—
178C4×C191—21no—
179C4×C191—21no—
180C4×C191—22no—
181C4×C191—21no—
182C4×C191—21no—
183C4×C191—41no—
184C4×C191—41no—
185C4×C191—41no—
186C4×C191C22×D411yescharacteristic
187C4×C191H(5,𝔽2)11yescharacteristic
188Dic191—22no—
189Dic191—41no—
190Dic191—41no—
191C2×C191—22no—
192C2×C191—41no—
193C2×C191D4211yescharacteristic
194C2×C191C2×#13711yescharacteristic
195C2×C191#57811yescharacteristic
196D191—81no—
197C191#327011yescharacteristic, complemented, Sylow
198#3270—1911nomaximal, Sylow
199C2×#137—1911no—
200C22×Q16—1911no—
201D4×Q8D19111yescharacteristic, complemented
202#548—1911no—
203#497—1911no—
204#470—1911no—
205#465—1911no—
206#464—1911no—
207#462—1911no—
208#452—1911no—
209#451—1911no—
210#451—1911no—
211#448—1911no—
212#441—1911no—
213#435—1911no—
214C2×#41—1911no—
215C22×Q8—1911no—
216C22×Q8C2×D19111yescharacteristic
217C22×Q8C2×D19111yescharacteristic
218#137—3824no—
219C2×Q16—3824no—
220C2×Q16—1911no—
221C2×Q16—1911no—
222C2×Q16—1911no—
223C2×Q16—1911no—
224C2×Q16—1911no—
225C2×Q16—1911no—
226C2×SD16—1911no—
227C2×SD16—1911no—
228C2×SD16—1911no—
229C2×M16—1911no—
230C22×C8—1911no—
231C4⋊Q8—21no—
232C4⋊Q8C2×D19111yescharacteristic
233#124—3821no—
234#124—1911no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—3821no—
238C22⋊Q8—1911no—
239C22⋊Q8C2×D19111yescharacteristic
240C22⋊Q8C2×D19111yescharacteristic
241#121—3821no—
242#121—1911no—
243C4×Q8—1911no—
244C4×Q8C2×D19111yescharacteristic
245C4×D4—21no—
246C4×D4—3821no—
247C4×D4C2×D19111yescharacteristic
248#106—1911no—
249C4⋊C8—1911no—
250#103—1911no—
251#103—1911no—
252#103—1911no—
253#103—1911no—
254#103—1911no—
255#102—1911no—
256C22⋊C8—1911no—
257C22⋊C8—1911no—
258#97—1911no—
259#41—3822no—
260#41—7641no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—3822no—
268C2×Q8—3821no—
269C2×Q8—1911no—
270C2×Q8—1911no—
271C2×Q8—7641no—
272C2×Q8C22×D19111yescharacteristic
273C2×Q8C22×D19111yescharacteristic
274C2×Q8C22×D19111yescharacteristic
275C2×D4—3821no—
276C2×D4—3821no—
277C2×D4—1911no—
278C2×D4C22⋊D19111yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3821no—
282C22×C4—3821no—
283C22×C4C22×D19111yescharacteristic
284C22×C4C22×D19111yescharacteristic
285Q16—3822no—
286Q16—3822no—
287Q16—3822no—
288Q16—3822no—
289Q16—1914no—
290Q16—7641no—
291SD16—3822no—
292SD16—3822no—
293SD16—7641no—
294M16—3822no—
295C2×C8—3822no—
296C2×C8—1911no—
297C2×C8—1911no—
298C2×C8—1911no—
299C2×C8—1911no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—3821no—
306C4⋊C4—3821no—
307C4⋊C4—1911no—
308C4⋊C4C22⋊D19111yescharacteristic
309C4⋊C4C22×D19111yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3821no—
313C22⋊C4—3821no—
314C22⋊C4—3821no—
315C22⋊C4—3821no—
316C22⋊C4C22⋊D19111yescharacteristic
317C22⋊C4C22⋊D19111yescharacteristic
318C42—21no—
319C42—3821no—
320C42C22×D19111yescharacteristic
321C23—3821no—
322C23C2×C22⋊D19111yescharacteristic
323C23C2×C22⋊D19111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—42no—
331Q8—3821no—
332Q8—3821no—
333Q8—3821no—
334Q8—1912no—
335Q8—7642no—
336Q8D4×D19112yes—
337D4—22no—
338D4—3821no—
339D4—3821no—
340D4—7642no—
341D4—7642no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—3821no—
353C2×C4—3821no—
354C2×C4—3821no—
355C2×C4—7642no—
356C2×C4—7642no—
357C2×C4C2×C22⋊D19111yescharacteristic
358C2×C4C2×C22⋊D19111yescharacteristic
359C2×C4C2×C22⋊D19111yescharacteristic
360C2×C4C2×C22⋊D19111yescharacteristic
361C2×C4C23×D19111yescharacteristic
362C8—3821no—
363C8—3821no—
364C8—3821no—
365C8—1912no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—7641no—
372C22—7641no—
373C22—7641no—
374C22#2025935312yescomplemented
375C22C22×C22⋊D19111yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3822no—
385C4—7641no—
386C4—7641no—
387C4C2×D4×D19111yescharacteristic
388C4#2025936711yescharacteristic
389C2—22no—
390C2—41no—
391C2—15281no—
392C2D4×C22⋊D19111yescharacteristic
393C2C2×#2025935311yescharacteristic
394C2#2713347811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingD4×Q8×C191C2
socleC22×C191C22×D4
prime radical#27133322C2
hypercenterD4×Q8D191
hypocenterC191#3270
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20259353C2semidirect
2. C22×#15690277C2semidirect
4. #27133055C2semidirect
5. #27132919C2semidirect
6. #27132894C2semidirect
7. #27132885C2semidirect
8. #27132870C2semidirect
9. #27132823C2semidirect
10. #27132819C2semidirect
11. #27132818C2semidirect
12. #27132802C2semidirect
13. #27132768C2semidirect
14. #27132735C2semidirect
15. D4×Q8×C191C2semidirect
78. #15690277D4semidirect
173. C22×C191#137semidirect
197. C191#3270semidirect
201. D4×Q8D191semidirect
374. C22#20259353semidirect

Maximal subgroups

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

subgroupindex
C2×#202593532
C22×#156902772
#271333222
#271330552
#271329192
#271328942
#271328852
#271328702
#271328232
#271328192
#271328182
#271328022
#271327682
#271327352
D4×Q8×C1912
#3270191