Finite Group Databasebeta

← #37583321

Subgroups — #37583321

Order 24448.

462 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
1C22⋊Q8×D191C211yesmaximal, characteristic, complemented
2#121×D191C211yesmaximal, characteristic
3#27133000C211yesmaximal, characteristic, complemented
4#27132997C211yesmaximal, characteristic, complemented
5#27132978C211yesmaximal, characteristic, complemented
6#27132975C211yesmaximal, characteristic, complemented
7#27132911C211yesmaximal, characteristic, complemented
8#27132909C211yesmaximal, characteristic, complemented
9#27132907C211yesmaximal, characteristic, complemented
10#27132815C211yesmaximal, characteristic, complemented
11#27132812C211yesmaximal, characteristic, complemented
12#27132803C211yesmaximal, characteristic, complemented
13#27132800C211yesmaximal, characteristic, complemented
14#27132626C211yesmaximal, characteristic, complemented
15#468×C191C211yesmaximal, characteristic, complemented
16C2×Q8×D191C2211yescharacteristic, complemented
17C2×D4×D191—21no—
18C2×D4×D191C2211yescharacteristic
19C22×C4×D191C2211yescharacteristic
20C2×#15690276C2211yescharacteristic, complemented
21C2×#15690275C2211yescharacteristic, complemented
22C2×#15690274C2211yescharacteristic, complemented
23C2×#15690271C2211yescharacteristic, complemented
24#20259302C2211yescharacteristic
25#20259301C2211yescharacteristic, complemented
26#20259300—21no—
27#20259299—21no—
28#20259298C2211yescharacteristic
29#20259297C2211yescharacteristic
30#20259296C2211yescharacteristic
31#20259295—21no—
32#20259294—21no—
33#20259293C2211yescharacteristic
34C4⋊C4×D191—21no—
35C4⋊C4×D191C2211yescharacteristic
36C4⋊C4×D191C2211yescharacteristic, complemented
37C22⋊C4×D191—21no—
38C22⋊C4×D191—21no—
39#20259256C2211yescharacteristic, complemented
40#20259255C2211yescharacteristic, complemented
41#20259255C2211yescharacteristic, complemented
42#20259247C2211yescharacteristic, complemented
43#20259246C2211yescharacteristic, complemented
44#20259245C2211yescharacteristic, complemented
45#20259244C2211yescharacteristic, complemented
46#20259243C2211yescharacteristic, complemented
47#20259242C2211yescharacteristic, complemented
48(C2×D191)⋊C8C2211yescharacteristic
49(C2×D191)⋊C8C2211yescharacteristic
50(C2×D191)⋊C8C2211yescharacteristic, complemented
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52C2×SD16×C191C2211yescharacteristic, complemented
53C2×M16×C191C2211yescharacteristic, complemented
54C22⋊Q8×C191C2211yescharacteristic, complemented
55#121×C191C2211yescharacteristic
56#107×C191C2211yescharacteristic, complemented
57#103×C191C2211yescharacteristic, complemented
58#102×C191C2211yescharacteristic, complemented
59C23×D191—21no—
60C23×D191D411yescharacteristic
61C2×Dic382—21no—
62C2×Dic382C2311yescharacteristic
63Q8×D191—41no—
64Q8×D191—41no—
65C2×D764—21no—
66C2×D764C2311yescharacteristic
67C2×C22⋊D191—21no—
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70D4×D191—41no—
71D4×D191—41no—
72D4×D191—44no—
73C22×Dic191D411yescharacteristic
74C2×C4×D191—21no—
75C2×C4×D191—21no—
76C2×C4×D191—21no—
77C2×C4×D191—41no—
78C2×C4×D191—41no—
79C2×C4×D191D411yescharacteristic
80C2×C4×D191D411yescharacteristic
81C2×C4×D191C2311yescharacteristic
82C2×C4×D191C2311yescharacteristic
83#15690276—41no—
84#15690275—41no—
85#15690274—41no—
86#15690271—41no—
87C2×C191⋊C8C2311yescharacteristic
88C2×C191⋊C8C2311yescharacteristic
89C4⋊Dic191—21no—
90C4⋊Dic191C2311yescharacteristic
91C4⋊Dic191C2311yescharacteristic
92Dic191⋊C4—21no—
93Dic191⋊C4—21no—
94Dic191⋊C4—21no—
95Dic191⋊C4—21no—
96C22⋊Dic191—21no—
97C22⋊Dic191—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4—21no—
102C2×Q8×C191C2311yescharacteristic
103C2×D4×C191—21no—
104C2×D4×C191C2311yescharacteristic
105C22×C4×C191C2311yescharacteristic
106SD16×C191—41no—
107M16×C191—41no—
108C2×C8×C191C2311yescharacteristic
109C2×C8×C191C2311yescharacteristic
110C4⋊C4×C191—21no—
111C4⋊C4×C191C2311yescharacteristic
112C4⋊C4×C191C2311yescharacteristic
113C22⋊C4×C191—21no—
114C22⋊C4×C191—21no—
115C22×D191—22no—
116C22×D191—22no—
117C22×D191—21no—
118C22×D191—41no—
119C22×D191—42no—
120C22×D191—42no—
121C22×D191—42no—
122C22×D191C2×D411yescharacteristic
123C22×D191C2×D411yescharacteristic
124Dic382—21no—
125Dic382—21no—
126Dic382—42no—
127D764—21no—
128D764—21no—
129D764—42no—
130C22⋊D191—42no—
131C22⋊D191—42no—
132C22⋊D191—42no—
133C2×Dic191—21no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191—41no—
137C2×Dic191C2×D411yescharacteristic
138C2×Dic191C2×D411yescharacteristic
139C4×D191—21no—
140C4×D191—21no—
141C4×D191—21no—
142C4×D191—21no—
143C4×D191—41no—
144C4×D191—41no—
145C4×D191—42no—
146C4×D191—42no—
147C4×D191—42no—
148C191⋊C8—21no—
149C191⋊C8—21no—
150C23×C191—21no—
151C23×C191C2×D411yescharacteristic
152Q8×C191—21no—
153Q8×C191—21no—
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×C191C2×D411yescharacteristic
163C2×C4×C191C2411yescharacteristic
164C8×C191—21no—
165C8×C191—21no—
166C22×C191—21no—
167C22×C191—21no—
168C22×C191—21no—
169C22×C191—41no—
170C22×C191—41no—
171C22×C191—41no—
172C22×C191C22×D411yescharacteristic
173C2×D191—21no—
174C2×D191—21no—
175C2×D191—21no—
176C2×D191—21no—
177C2×D191—44no—
178C2×D191—41no—
179C2×D191—41no—
180C2×D191—41no—
181C2×D191—84no—
182C2×D191AGL(1,ℤ/8)12yes—
183C2×D191#13712yes—
184C4×C191—21no—
185C4×C191—41no—
186C4×C191—41no—
187C4×C191—41no—
188C4×C191H(5,𝔽2)11yescharacteristic
189C4×C191H(5,𝔽2)11yescharacteristic
190Dic191—21no—
191Dic191—21no—
192Dic191—41no—
193Dic191—41no—
194Dic191—41no—
195C2×C191—41no—
196C2×C191—81no—
197C2×C191#53411yescharacteristic
198C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
199C2×C191C2×#13711yescharacteristic
200D191—22no—
201D191—22no—
202D191—81no—
203C191#319311yescharacteristic, complemented, Sylow
204#3193—1911nomaximal, Sylow
205C2×C22⋊Q8—1911no—
206C2×#121—1911no—
207#484—1911no—
208#484—1911no—
209#480—1911no—
210#480—1911no—
211#468—1911no—
212#468—1911no—
213#468—1911no—
214#468D19111yescharacteristic, complemented
215#450—1911no—
216#450—1911no—
217#448—1911no—
218#448—1911no—
219#407—1911no—
220C22×Q8—1911no—
221C22×D4—3821no—
222C22×D4—1911no—
223C23×C4—1911no—
224C2×SD16—1911no—
225C2×SD16—1911no—
226C2×SD16—1911no—
227C2×SD16C2×D19111yescharacteristic, complemented
228C2×M16—1911no—
229C2×M16C2×D19111yescharacteristic, complemented
230C22⋊Q8—3821no—
231C22⋊Q8—3821no—
232C22⋊Q8—1911no—
233C22⋊Q8—1911no—
234C22⋊Q8—1911no—
235C22⋊Q8C2×D19111yescharacteristic, complemented
236#121—3821no—
237#121—3821no—
238#121—1911no—
239#121—1911no—
240#121—1911no—
241#121C2×D19111yescharacteristic
242C2×C4⋊C4—3821no—
243C2×C4⋊C4—1911no—
244C2×C4⋊C4—1911no—
245C2×C22⋊C4—3821no—
246C2×C22⋊C4—3821no—
247#107—1911no—
248#107—1911no—
249#107—1911no—
250#107C2×D19111yescharacteristic, complemented
251#103—1911no—
252#103—1911no—
253#103—1911no—
254#103C2×D19111yescharacteristic, complemented
255#102—1911no—
256#102—1911no—
257#102—1911no—
258#102C2×D19111yescharacteristic, complemented
259C22⋊C8—1911no—
260C22⋊C8—1911no—
261C22⋊C8—1911no—
262C22⋊C8—1911no—
263C24—3821no—
264C24—1911no—
265C2×Q8—3821no—
266C2×Q8—1911no—
267C2×Q8—7641no—
268C2×Q8—7641no—
269C2×Q8C22×D19111yescharacteristic
270C2×D4—21no—
271C2×D4—3821no—
272C2×D4—3821no—
273C2×D4—3821no—
274C2×D4—3821no—
275C2×D4—1911no—
276C2×D4—7641no—
277C2×D4—7641no—
278C2×D4—7644no—
279C2×D4C22×D19111yescharacteristic
280C22×C4—3821no—
281C22×C4—3821no—
282C22×C4—3821no—
283C22×C4—1911no—
284C22×C4—1911no—
285C22×C4—1911no—
286C22×C4—1911no—
287C22×C4—1911no—
288C22×C4—7641no—
289C22×C4—7641no—
290C22×C4C22×D19111yescharacteristic
291SD16—41no—
292SD16—7641no—
293SD16—7641no—
294SD16—7641no—
295M16—41no—
296M16—7641no—
297C2×C8—1911no—
298C2×C8—1911no—
299C2×C8C22×D19111yescharacteristic
300C2×C8C22×D19111yescharacteristic
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⋊C4C22×D19111yescharacteristic
310C4⋊C4C22×D19111yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—3821no—
314C22⋊C4—3821no—
315C22⋊C4—3821no—
316C22⋊C4—3821no—
317C22⋊C4—3821no—
318C22⋊C4—3821no—
319C23—21no—
320C23—3822no—
321C23—3822no—
322C23—3821no—
323C23—1911no—
324C23—1911no—
325C23—7641no—
326C23—7642no—
327C23—7642no—
328C23—7642no—
329C23D4×D19111yescharacteristic
330Q8—21no—
331Q8—21no—
332Q8—3821no—
333Q8—3821no—
334Q8—7642no—
335D4—21no—
336D4—21no—
337D4—42no—
338D4—3821no—
339D4—3821no—
340D4—7642no—
341D4—7642no—
342D4—7642no—
343D4—7642no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—3821no—
350C2×C4—3821no—
351C2×C4—3821no—
352C2×C4—3821no—
353C2×C4—3821no—
354C2×C4—3821no—
355C2×C4—3821no—
356C2×C4—1911no—
357C2×C4—1911no—
358C2×C4—7641no—
359C2×C4—7641no—
360C2×C4—7642no—
361C2×C4—7642no—
362C2×C4—7642no—
363C2×C4—7641no—
364C2×C4D4×D19111yescharacteristic
365C2×C4C23×D19111yescharacteristic
366C8—21no—
367C8—21no—
368C8—3821no—
369C8—3821no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—3821no—
377C22—3821no—
378C22—3821no—
379C22—3821no—
380C22—1912no—
381C22—1912no—
382C22—15284no—
383C22—7644no—
384C22—7641no—
385C22—7641no—
386C22—7641no—
387C22C2×D4×D19111yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—3821no—
393C4—3821no—
394C4—7641no—
395C4—7641no—
396C4—7641no—
397C4#2025936711yescharacteristic
398C4#2025936811yescharacteristic
399C2—41no—
400C2—81no—
401C2—3822no—
402C2—3822no—
403C2—15281no—
404C2#2713322111yescharacteristic
405C2AGL(1,ℤ/8)×D19111yescharacteristic
406C2#137×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#468×C191C2
socleC22×C191C22×D4
prime radical#121×D191C2
hypercenter#468D191
hypocenterC191#3193
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D191C2semidirect
3. #27133000C2semidirect
4. #27132997C2semidirect
5. #27132978C2semidirect
6. #27132975C2semidirect
7. #27132911C2semidirect
8. #27132909C2semidirect
9. #27132907C2semidirect
10. #27132815C2semidirect
11. #27132812C2semidirect
12. #27132803C2semidirect
13. #27132800C2semidirect
14. #27132626C2semidirect
15. #468×C191C2semidirect
16. C2×Q8×D191C22semidirect
20. C2×#15690276C22semidirect
21. C2×#15690275C22semidirect
22. C2×#15690274C22semidirect
23. C2×#15690271C22semidirect
25. #20259301C22semidirect
36. C4⋊C4×D191C22semidirect
39. #20259256C22semidirect
40. #20259255C22semidirect
41. #20259255C22semidirect
42. #20259247C22semidirect
43. #20259246C22semidirect
44. #20259245C22semidirect
45. #20259244C22semidirect
46. #20259243C22semidirect
47. #20259242C22semidirect
50. (C2×D191)⋊C8C22semidirect
51. (C2×D191)⋊C8C22semidirect
52. C2×SD16×C191C22semidirect
53. C2×M16×C191C22semidirect
54. C22⋊Q8×C191C22semidirect
56. #107×C191C22semidirect
57. #103×C191C22semidirect
58. #102×C191C22semidirect
203. C191#3193semidirect
214. #468D191semidirect
227. C2×SD16C2×D191semidirect
229. C2×M16C2×D191semidirect
235. C22⋊Q8C2×D191semidirect
250. #107C2×D191semidirect
254. #103C2×D191semidirect
258. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1912
#121×D1912
#271330002
#271329972
#271329782
#271329752
#271329112
#271329092
#271329072
#271328152
#271328122
#271328032
#271328002
#271326262
#468×C1912
#3193191