Finite Group Databasebeta

← #37584089

Subgroups — #37584089

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×#20259351C211yesmaximal, characteristic, complemented
2C2×Q16×D191C211yesmaximal, characteristic, complemented
3#27133328C211yesmaximal, characteristic, complemented
4#27133321C211yesmaximal, characteristic
5#27132918C211yesmaximal, characteristic, complemented
6#27132889C211yesmaximal, characteristic, complemented
7#27132882C211yesmaximal, characteristic, complemented
8#27132870C211yesmaximal, characteristic, complemented
9#27132822C211yesmaximal, characteristic, complemented
10#27132817C211yesmaximal, characteristic, complemented
11#27132817C211yesmaximal, characteristic, complemented
12#27132800C211yesmaximal, characteristic, complemented
13#27132771C211yesmaximal, characteristic, complemented
14#27132736C211yesmaximal, characteristic, complemented
15#497×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C2×Q8×D191C2211yescharacteristic
19C2×Q8×D191C2211yescharacteristic
20#20259351—24no—
21C2×Dic764C2211yescharacteristic
22C2×#15690277C2211yescharacteristic
23C2×#15690277C2211yescharacteristic
24C2×#15690277C2211yescharacteristic
25Q16×D191—24no—
26C2×#15690275C2211yescharacteristic
27C2×#15690270C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic
29#20259321—21no—
30#20259308—21no—
31#20259300—21no—
32#20259300—21no—
33#20259299—21no—
34#20259298C2211yescharacteristic
35#20259298C2211yescharacteristic
36#20259298C2211yescharacteristic
37#20259295—21no—
38#20259293C2211yescharacteristic
39C4×Dic382C2211yescharacteristic
40C4×Dic382C2211yescharacteristic
41C4×D764C2211yescharacteristic
42#20259284—21no—
43(C22⋊D191)⋊C4—21no—
44#20259254C2211yescharacteristic
45C4⋊(C191⋊C8)C2211yescharacteristic
46#20259247C2211yescharacteristic
47#20259247C2211yescharacteristic
48#20259246C2211yescharacteristic
49#20259246C2211yescharacteristic
50#20259246C2211yescharacteristic
51#20259243C2211yescharacteristic
52(C2×D191)⋊C8C2211yescharacteristic
53(C2×D191)⋊C8C2211yescharacteristic
54C2×Q16×C191C2211yescharacteristic
55C2×Q16×C191C2211yescharacteristic
56C2×SD16×C191C2211yescharacteristic
57C2×SD16×C191C2211yescharacteristic
58C4⋊Q8×C191C2211yescharacteristic
59#124×C191C2211yescharacteristic
60#97×C191C2211yescharacteristic
61#15690286—22no—
62#15690286—41no—
63C2×Dic382—21no—
64C2×Dic382—21no—
65C2×Dic382—21no—
66C2×Dic382C2311yescharacteristic
67C2×Dic382C2311yescharacteristic
68Q8×D191—22no—
69Q8×D191—22no—
70Q8×D191—22no—
71Q8×D191—41no—
72Q8×D191—41no—
73Q8×D191—41no—
74C2×D764D411yescharacteristic
75C2×C22⋊D191—21no—
76C2×C22⋊D191—21no—
77C22×Dic191—21no—
78C2×C4×D191—21no—
79C2×C4×D191—21no—
80C2×C4×D191—21no—
81C2×C4×D191C2311yescharacteristic
82C2×C4×D191C2311yescharacteristic
83Dic764D414yescomplemented
84#15690277—22no—
85#15690277—22no—
86#15690277—22no—
87#15690275—22no—
88#15690270—22no—
89C2×C191⋊C8C2311yescharacteristic
90C2×C191⋊C8C2311yescharacteristic
91C8×D191—22no—
92C4⋊Dic191D411yescharacteristic
93C4⋊Dic191C2311yescharacteristic
94C4⋊Dic191C2311yescharacteristic
95Dic191⋊C4—21no—
96Dic191⋊C4—21no—
97Dic191⋊C4—21no—
98Dic191⋊C4—21no—
99Dic191⋊C4—21no—
100C22⋊Dic191—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4D411yescharacteristic
105(C2×D191)⋊C4D411yescharacteristic
106C4×Dic191—21no—
107C4×Dic191—21no—
108C2×Q8×C191C2311yescharacteristic
109C2×Q8×C191C2311yescharacteristic
110C2×Q8×C191C2311yescharacteristic
111C2×D4×C191C2311yescharacteristic
112Q16×C191—22no—
113Q16×C191—41no—
114SD16×C191—22no—
115SD16×C191—41no—
116C2×C8×C191C2311yescharacteristic
117C2×C8×C191C2311yescharacteristic
118C4⋊C4×C191—21no—
119C4⋊C4×C191—21no—
120C22⋊C4×C191—21no—
121C22⋊C4×C191—21no—
122C42×C191C2311yescharacteristic
123C22×D191C2×D411yescharacteristic
124C22×D191C2×D411yescharacteristic
125Dic382—21no—
126Dic382—21no—
127Dic382—42no—
128Dic382—42no—
129Dic382—42no—
130Dic382C2×D412yes—
131Dic382C2×D412yes—
132D764—22no—
133C22⋊D191—42no—
134C22⋊D191—42no—
135C2×Dic191—21no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—42no—
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C4×D191—21no—
143C4×D191—21no—
144C4×D191—21no—
145C4×D191—21no—
146C4×D191—42no—
147C4×D191—42no—
148C4×D191—42no—
149C191⋊C8—21no—
150C191⋊C8—21no—
151C23×C191—21no—
152Q8×C191—21no—
153Q8×C191—21no—
154Q8×C191—21no—
155Q8×C191—21no—
156Q8×C191—21no—
157Q8×C191—21no—
158D4×C191—21no—
159D4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191C2×D411yescharacteristic
164C2×C4×C191C2×D411yescharacteristic
165C2×C4×C191C2411yescharacteristic
166C8×C191—21no—
167C8×C191C2×D412yes—
168C22×C191—41no—
169C22×C191—41no—
170C22×C191—41no—
171C22×C191C22×D411yescharacteristic
172C2×D191—21no—
173C2×D191—21no—
174C2×D191—21no—
175C2×D191—21no—
176C2×D191—21no—
177C2×D191#13712yescomplemented
178C4×C191—21no—
179C4×C191—21no—
180C4×C191—41no—
181C4×C191—41no—
182C4×C191—41no—
183C4×C191C22×D411yescharacteristic
184C4×C191H(5,𝔽2)11yescharacteristic
185Dic191—21no—
186Dic191—21no—
187Dic191—22no—
188Dic191—22no—
189Dic191—41no—
190Dic191—41no—
191C2×C191—81no—
192C2×C191D4211yescharacteristic
193C2×C191C2×#13711yescharacteristic
194C2×C191#57811yescharacteristic
195D191—22no—
196D191—41no—
197C191#327011yescharacteristic, complemented, Sylow
198#3270—1911nomaximal, Sylow
199C2×#137—1911no—
200C22×Q16—1911no—
201D4×Q8—1911no—
202#548—1911no—
203#497D19111yescharacteristic, complemented
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×Q8—1911no—
217C22×Q8—1911no—
218#137—3824no—
219C2×Q16—3824no—
220C2×Q16—1911no—
221C2×Q16—1911no—
222C2×Q16—1911no—
223C2×Q16—1911no—
224C2×Q16C2×D19111yescharacteristic
225C2×Q16C2×D19111yescharacteristic
226C2×SD16—1911no—
227C2×SD16C2×D19111yescharacteristic
228C2×SD16C2×D19111yescharacteristic
229C2×M16—1911no—
230C22×C8—1911no—
231C4⋊Q8—3821no—
232C4⋊Q8C2×D19111yescharacteristic
233#124—3821no—
234#124C2×D19111yescharacteristic
235C22⋊Q8—3821no—
236C22⋊Q8—3821no—
237C22⋊Q8—3821no—
238C22⋊Q8—1911no—
239C22⋊Q8—1911no—
240C22⋊Q8—1911no—
241#121—3821no—
242#121—1911no—
243C4×Q8—1911no—
244C4×Q8—1911no—
245C4×D4—3821no—
246C4×D4—3821no—
247C4×D4—1911no—
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#97C2×D19111yescharacteristic
259#41—3822no—
260#41—7641no—
261C2×Q8—3821no—
262C2×Q8—3821no—
263C2×Q8—3821no—
264C2×Q8—3822no—
265C2×Q8—3822no—
266C2×Q8—3822no—
267C2×Q8—1911no—
268C2×Q8—1911no—
269C2×Q8—7641no—
270C2×Q8—7641no—
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—3821no—
280C22×C4—3821no—
281C22×C4—3821no—
282C22×C4—3821no—
283C22×C4—1911no—
284C22×C4—1911no—
285Q16—22no—
286Q16—41no—
287Q16—3822no—
288Q16—3822no—
289Q16—3822no—
290Q16—1914no—
291SD16—22no—
292SD16—41no—
293SD16—3822no—
294M16—3822no—
295C2×C8—3822no—
296C2×C8—1911no—
297C2×C8—1911no—
298C2×C8C22×D19111yescharacteristic
299C2×C8C22×D19111yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—3821no—
303C4⋊C4—3821no—
304C4⋊C4—3821no—
305C4⋊C4—3821no—
306C4⋊C4—3821no—
307C4⋊C4—1911no—
308C4⋊C4—1911no—
309C4⋊C4—1911no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3821no—
313C22⋊C4—3821no—
314C22⋊C4—3821no—
315C22⋊C4—3821no—
316C22⋊C4—1911no—
317C22⋊C4—1911no—
318C42—3821no—
319C42—3821no—
320C42C22×D19111yescharacteristic
321C23—21no—
322C23—1911no—
323C23—1911no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—3821no—
331Q8—3821no—
332Q8—1912no—
333Q8—1912no—
334Q8—7642no—
335Q8—7642no—
336Q8—7642no—
337D4—21no—
338D4—21no—
339D4—3822no—
340D4—7642no—
341D4—7642no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—3821no—
346C2×C4—3821no—
347C2×C4—3821no—
348C2×C4—3821no—
349C2×C4—3821no—
350C2×C4—3821no—
351C2×C4—3821no—
352C2×C4—3821no—
353C2×C4—1911no—
354C2×C4—1911no—
355C2×C4—7642no—
356C2×C4—7642no—
357C2×C4—7642no—
358C2×C4—7642no—
359C2×C4D4×D19111yescharacteristic
360C2×C4D4×D19111yescharacteristic
361C2×C4C23×D19111yescharacteristic
362C8—21no—
363C8—3821no—
364C8—3821no—
365C8D4×D19112yes—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—3821no—
370C22—3821no—
371C22—3821no—
372C22—3821no—
373C22—3821no—
374C22—1912no—
375C22C2×D4×D19111yescharacteristic
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3821no—
382C4—3821no—
383C4—3822no—
384C4—3822no—
385C4—7641no—
386C4—7641no—
387C4C2×D4×D19111yescharacteristic
388C4#2025936711yescharacteristic
389C2—81no—
390C2—3822no—
391C2—7641no—
392C2#2713329811yescharacteristic
393C2#137×D19111yescharacteristic
394C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#497×C191C2
socleC22×C191C22×D4
prime radical#27133321C2
hypercenter#497D191
hypocenterC191#3270
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#20259351C2semidirect
2. C2×Q16×D191C2semidirect
3. #27133328C2semidirect
5. #27132918C2semidirect
6. #27132889C2semidirect
7. #27132882C2semidirect
8. #27132870C2semidirect
9. #27132822C2semidirect
10. #27132817C2semidirect
11. #27132817C2semidirect
12. #27132800C2semidirect
13. #27132771C2semidirect
14. #27132736C2semidirect
15. #497×C191C2semidirect
83. Dic764D4semidirect
177. C2×D191#137semidirect
197. C191#3270semidirect
203. #497D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593512
C2×Q16×D1912
#271333282
#271333212
#271329182
#271328892
#271328822
#271328702
#271328222
#271328172
#271328002
#271327712
#271327362
#497×C1912
#3270191