Finite Group Databasebeta

← #37584084

Subgroups — #37584084

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×Q16×D191C211yesmaximal, characteristic, complemented
2Q8×C22⋊D191C211yesmaximal, characteristic, complemented
3#27133320C211yesmaximal, characteristic
4#27133054C211yesmaximal, characteristic, complemented
5#27132921C211yesmaximal, characteristic, complemented
6#27132889C211yesmaximal, characteristic, complemented
7#27132883C211yesmaximal, characteristic, complemented
8#27132871C211yesmaximal, characteristic, complemented
9#27132825C211yesmaximal, characteristic, complemented
10#27132819C211yesmaximal, characteristic, complemented
11#27132817C211yesmaximal, characteristic, complemented
12#27132800C211yesmaximal, characteristic, complemented
13#27132770C211yesmaximal, characteristic, complemented
14#27132732C211yesmaximal, characteristic, complemented
15C2×#137×C191C211yesmaximal, characteristic, complemented
16C2×Q8×D191C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic
18C2×Dic764C2211yescharacteristic
19C2×#15690277C2211yescharacteristic
20C2×#15690277C2211yescharacteristic
21C2×#15690277C2211yescharacteristic
22Q16×D191—24no—
23C2×#15690275C2211yescharacteristic
24C2×C8×D191C2211yescharacteristic
25#20259320—21no—
26#20259320C2211yescharacteristic
27#20259309—21no—
28#20259309C2211yescharacteristic
29#20259301—21no—
30#20259301—21no—
31#20259301C2211yescharacteristic
32#20259301C2211yescharacteristic
33#20259298—21no—
34#20259298C2211yescharacteristic
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#20259247C2211yescharacteristic
45#20259247C2211yescharacteristic
46#20259246C2211yescharacteristic
47#20259246C2211yescharacteristic
48#20259246C2211yescharacteristic
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
61C2×Dic382—21no—
62C2×Dic382—21no—
63C2×Dic382C2311yescharacteristic
64C2×Dic382C2311yescharacteristic
65Q8×D191—22no—
66Q8×D191—22no—
67Q8×D191—41no—
68Q8×D191—41no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191D411yescharacteristic
71C2×C4×D191—21no—
72C2×C4×D191—21no—
73C2×C4×D191C2311yescharacteristic
74Dic764—22no—
75#15690277—22no—
76#15690277—22no—
77#15690277—41no—
78#15690275—41no—
79C2×C191⋊C8C2311yescharacteristic
80C2×C191⋊C8C2311yescharacteristic
81C8×D191—22no—
82C4⋊Dic191—21no—
83C4⋊Dic191—21no—
84C4⋊Dic191C2311yescharacteristic
85C4⋊Dic191C2311yescharacteristic
86Dic191⋊C4—21no—
87Dic191⋊C4—21no—
88Dic191⋊C4—21no—
89Dic191⋊C4—21no—
90Dic191⋊C4—21no—
91Dic191⋊C4D411yescharacteristic
92C22⋊Dic191—21no—
93C22⋊Dic191—21no—
94C22⋊Dic191—21no—
95C22⋊Dic191—21no—
96C22⋊Dic191—21no—
97C22⋊Dic191D411yescharacteristic
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×D191C2×D411yescharacteristic
124Dic382—21no—
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128Dic382—42no—
129Dic382—42no—
130C22⋊D191—22no—
131C22⋊D191—42no—
132C2×Dic191—21no—
133C2×Dic191—21no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191C2×D411yescharacteristic
137C2×Dic191C2×D411yescharacteristic
138C2×Dic191C2×D411yescharacteristic
139C4×D191—21no—
140C4×D191—21no—
141C4×D191—42no—
142C4×D191—42no—
143C191⋊C8—21no—
144C191⋊C8—21no—
145C23×C191—21no—
146C23×C191C2×D411yescharacteristic
147Q8×C191—21no—
148Q8×C191—21no—
149Q8×C191—21no—
150Q8×C191—21no—
151Q8×C191—42no—
152Q8×C191C2×D412yes—
153Q8×C191C2×D412yes—
154D4×C191—21no—
155D4×C191—21no—
156D4×C191—42no—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—42no—
163C2×C4×C191—42no—
164C2×C4×C191C2×D411yescharacteristic
165C2×C4×C191C2411yescharacteristic
166C8×C191—21no—
167C8×C191C2×D412yes—
168C22×C191—21no—
169C22×C191—21no—
170C22×C191—21no—
171C22×C191—41no—
172C22×C191—41no—
173C22×C191—41no—
174C22×C191C22×D411yescharacteristic
175C2×D191—21no—
176C2×D191—21no—
177C2×D191#13712yescomplemented
178C4×C191—21no—
179C4×C191—22no—
180C4×C191—22no—
181C4×C191—41no—
182C4×C191C22×D411yescharacteristic
183C4×C191H(5,𝔽2)11yescharacteristic
184Dic191—21no—
185Dic191—21no—
186Dic191—21no—
187Dic191—41no—
188Dic191—41no—
189Dic191—41no—
190Dic191—41no—
191C2×C191—41no—
192C2×C191—81no—
193C2×C191D4211yescharacteristic
194C2×C191C2×#13711yescharacteristic
195C2×C191#57811yescharacteristic
196D191—22no—
197C191#327011yescharacteristic, complemented, Sylow
198#3270—1911nomaximal, Sylow
199C2×#137D19111yescharacteristic, complemented
200C22×Q16—1911no—
201D4×Q8—1911no—
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×#41C2×D19111yescharacteristic
215C22×Q8—1911no—
216C22×Q8—1911no—
217C22×Q8C2×D19111yescharacteristic
218#137—24no—
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×M16C2×D19111yescharacteristic
230C22×C8—1911no—
231C4⋊Q8—3821no—
232C4⋊Q8—1911no—
233#124—3821no—
234#124—1911no—
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#97—1911no—
259#41—22no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—3821no—
265C2×Q8—3821no—
266C2×Q8—3822no—
267C2×Q8—3822no—
268C2×Q8—1911no—
269C2×Q8—1911no—
270C2×Q8—7641no—
271C2×Q8—7641no—
272C2×Q8C22×D19111yescharacteristic
273C2×Q8C22×D19111yescharacteristic
274C2×Q8C22×D19111yescharacteristic
275C2×D4—21no—
276C2×D4—3821no—
277C2×D4—1911no—
278C2×D4C22×D19111yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—3821no—
282C22×C4—3821no—
283C22×C4—1911no—
284C22×C4C22×D19111yescharacteristic
285Q16—22no—
286Q16—3822no—
287Q16—3822no—
288Q16—3822no—
289Q16—7641no—
290Q16C22⋊D19114yescomplemented
291SD16—22no—
292SD16—22no—
293SD16—7641no—
294M16—22no—
295C2×C8—3822no—
296C2×C8—1911no—
297C2×C8—1911no—
298C2×C8C22×D19111yescharacteristic
299C2×C8C22×D19111yescharacteristic
300C4⋊C4—3821no—
301C4⋊C4—3821no—
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—3821no—
311C22⋊C4—3821no—
312C22⋊C4—3821no—
313C22⋊C4—3821no—
314C22⋊C4—3821no—
315C22⋊C4—3821no—
316C22⋊C4—1911no—
317C22⋊C4—1911no—
318C42—3821no—
319C42—3821no—
320C42—1911no—
321C23—21no—
322C23—1911no—
323C23D4×D19111yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—42no—
329Q8—3821no—
330Q8—3821no—
331Q8—3821no—
332Q8—3821no—
333Q8—7642no—
334Q8—7642no—
335Q8C2×C22⋊D19112yes—
336Q8C2×C22⋊D19112yes—
337D4—21no—
338D4—21no—
339D4—42no—
340D4—3822no—
341D4—7642no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—3821no—
350C2×C4—3821no—
351C2×C4—3821no—
352C2×C4—3821no—
353C2×C4—3821no—
354C2×C4—3821no—
355C2×C4—1911no—
356C2×C4—1911no—
357C2×C4—1911no—
358C2×C4—7642no—
359C2×C4—7642no—
360C2×C4D4×D19111yescharacteristic
361C2×C4C23×D19111yescharacteristic
362C8—21no—
363C8—3821no—
364C8—3821no—
365C8C2×C22⋊D19112yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—3821no—
373C22—3821no—
374C22—1912no—
375C22C2×D4×D19111yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—3821no—
381C4—3821no—
382C4—3821no—
383C4—7641no—
384C4—7641no—
385C4—7641no—
386C4—7641no—
387C4C22×C22⋊D19111yescharacteristic
388C4#2025936711yescharacteristic
389C2—41no—
390C2—81no—
391C2—3822no—
392C2D4×C22⋊D19111yescharacteristic
393C2#137×D19111yescharacteristic
394C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#137×C191C2
socleC22×C191C22×D4
prime radical#27133320C2
hypercenterC2×#137D191
hypocenterC191#3270
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×Q16×D191C2semidirect
2. Q8×C22⋊D191C2semidirect
4. #27133054C2semidirect
5. #27132921C2semidirect
6. #27132889C2semidirect
7. #27132883C2semidirect
8. #27132871C2semidirect
9. #27132825C2semidirect
10. #27132819C2semidirect
11. #27132817C2semidirect
12. #27132800C2semidirect
13. #27132770C2semidirect
14. #27132732C2semidirect
15. C2×#137×C191C2semidirect
117. Q16×C191D4semidirect
177. C2×D191#137semidirect
197. C191#3270semidirect
199. C2×#137D191semidirect
290. Q16C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1912
Q8×C22⋊D1912
#271333202
#271330542
#271329212
#271328892
#271328832
#271328712
#271328252
#271328192
#271328172
#271328002
#271327702
#271327322
C2×#137×C1912
#3270191