Finite Group Databasebeta

← #37584039

Subgroups — #37584039

Order 24448.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259341C211yesmaximal, characteristic, complemented
2#27133312C211yesmaximal, characteristic, complemented
3#27133307C211yesmaximal, characteristic, complemented
4#27133054C211yesmaximal, characteristic, complemented
5#27132921C211yesmaximal, characteristic, complemented
6#27132889C211yesmaximal, characteristic, complemented
7#27132876C211yesmaximal, characteristic, complemented
8#27132866C211yesmaximal, characteristic, complemented
9#27132825C211yesmaximal, characteristic, complemented
10#27132822C211yesmaximal, characteristic, complemented
11#27132814C211yesmaximal, characteristic, complemented
12#27132805C211yesmaximal, characteristic, complemented
13#27132775C211yesmaximal, characteristic, complemented
14#27132732C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×#15690286C2211yescharacteristic, complemented
18#20259341—24no—
19C2×Dic764C2211yescharacteristic, complemented
20C2×#15690277C2211yescharacteristic, complemented
21C2×#15690275C2211yescharacteristic
22C2×#15690275C2211yescharacteristic
23C2×#15690275C2211yescharacteristic, complemented
24C2×C8×D191C2211yescharacteristic, complemented
25#20259320C2211yescharacteristic, complemented
26#20259309C2211yescharacteristic
27#20259305—21no—
28#20259303—21no—
29#20259303—21no—
30#20259301C2211yescharacteristic
31#20259301C2211yescharacteristic
32#20259298C2211yescharacteristic, complemented
33#20259294—21no—
34#20259294—21no—
35#20259293C2211yescharacteristic
36#20259291—21no—
37D4×Dic191C2211yescharacteristic
38D4×Dic191C2211yescharacteristic, complemented
39C4×C22⋊D191C2211yescharacteristic
40C2×Dic191⋊C4—21no—
41C2×C22⋊Dic191—21no—
42#20259254C2211yescharacteristic, complemented
43Dic191⋊C8C2211yescharacteristic, complemented
44#20259247C2211yescharacteristic, complemented
45#20259247C2211yescharacteristic, complemented
46#20259246C2211yescharacteristic, complemented
47#20259243C2211yescharacteristic
48#20259243C2211yescharacteristic
49#20259243C2211yescharacteristic, complemented
50C22⋊(C191⋊C8)C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52#20259232C2211yescharacteristic, complemented
53C2×#41×C191C2211yescharacteristic
54C22×D4×C191C2211yescharacteristic
55AGL(1,ℤ/8)×C191—24no—
56C2×SD16×C191C2211yescharacteristic
57C2×SD16×C191C2211yescharacteristic
58C2×D8×C191C2211yescharacteristic
59C2×D8×C191C2211yescharacteristic
60C2×M16×C191C2211yescharacteristic
61#15690286—22no—
62#15690286—22no—
63#15690286—41no—
64#15690286—41no—
65C2×Dic382C2311yescharacteristic
66C2×Dic382C2311yescharacteristic
67C2×C22⋊D191—21no—
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191D411yescharacteristic
71C22×Dic191—21no—
72C22×Dic191—21no—
73C22×Dic191—21no—
74C22×Dic191—21no—
75C2×C4×D191C2311yescharacteristic
76Dic764—22no—
77#15690277—41no—
78#15690275—22no—
79#15690275—22no—
80#15690275—41no—
81C2×C191⋊C8C2311yescharacteristic
82C2×C191⋊C8C2311yescharacteristic
83C8×D191—22no—
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⋊Dic191—21no—
98C22⋊Dic191—21no—
99C22⋊Dic191D411yescharacteristic
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4D411yescharacteristic
102C4×Dic191C2311yescharacteristic
103C24×C191—21no—
104#41×C191—22no—
105#41×C191—41no—
106C2×Q8×C191C2311yescharacteristic
107C2×D4×C191—22no—
108C2×D4×C191—21no—
109C2×D4×C191—21no—
110C2×D4×C191—41no—
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116SD16×C191—22no—
117SD16×C191—22no—
118D8×C191—22no—
119D8×C191D414yescomplemented
120M16×C191—22no—
121C2×C8×C191C2311yescharacteristic
122C2×C8×C191C2311yescharacteristic
123C22×D191C2×D411yescharacteristic
124Dic382—21no—
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128C22⋊D191—41no—
129C22⋊D191—42no—
130C22⋊D191—42no—
131C22⋊D191—81no—
132C2×Dic191—21no—
133C2×Dic191—21no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191—42no—
137C2×Dic191—42no—
138C2×Dic191—42no—
139C2×Dic191—81no—
140C2×Dic191C2×D411yescharacteristic
141C2×Dic191C2×D411yescharacteristic
142C2×Dic191C2×D411yescharacteristic
143C4×D191—21no—
144C4×D191—21no—
145C191⋊C8—21no—
146C191⋊C8—21no—
147C23×C191—21no—
148C23×C191—21no—
149C23×C191—21no—
150C23×C191—42no—
151C23×C191—42no—
152C23×C191—81no—
153C23×C191C2×D411yescharacteristic
154Q8×C191—21no—
155Q8×C191—21no—
156D4×C191—21no—
157D4×C191—21no—
158D4×C191—21no—
159D4×C191—21no—
160D4×C191—42no—
161D4×C191—42no—
162D4×C191C2×D412yes—
163D4×C191C2×D412yes—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191—42no—
168C2×C4×C191C2×D411yescharacteristic
169C2×C4×C191C2411yescharacteristic
170C8×C191—21no—
171C8×C191C2×D412yes—
172C22×C191—21no—
173C22×C191—22no—
174C22×C191—22no—
175C22×C191—21no—
176C22×C191—21no—
177C22×C191—41no—
178C22×C191—41no—
179C22×C191—41no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191—84no—
185C22×C191C22×D411yescharacteristic
186C2×D191—21no—
187C2×D191—21no—
188C2×D191—21no—
189C4×C191—21no—
190C4×C191—41no—
191C4×C191C22×D411yescharacteristic
192C4×C191H(5,𝔽2)11yescharacteristic
193Dic191—21no—
194Dic191—21no—
195Dic191—21no—
196Dic191—41no—
197Dic191—41no—
198Dic191—41no—
199Dic191—41no—
200C2×C191—41no—
201C2×C191—42no—
202C2×C191—42no—
203C2×C191—81no—
204C2×C191D4211yescharacteristic
205C2×C191#57511yescharacteristic
206C2×C191#57711yescharacteristic
207D191—41no—
208C191#326711yescharacteristic, complemented, Sylow
209#3267—1911nomaximal, Sylow
210C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
211C2×#135—1911no—
212#547—1911no—
213#546—1911no—
214#497—1911no—
215#470—1911no—
216#465—1911no—
217#463—1911no—
218#461—1911no—
219#452—1911no—
220#452—1911no—
221#450—1911no—
222#449—1911no—
223#442—1911no—
224#435—1911no—
225C2×#41—1911no—
226C2×#41—1911no—
227C2×#41C2×D19111yescharacteristic
228C22×D4C2×D19111yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—3824no—
231C2×Q16—1911no—
232C2×Q16—1911no—
233C2×SD16—1911no—
234C2×SD16—1911no—
235C2×SD16—1911no—
236C2×SD16C2×D19111yescharacteristic
237C2×SD16C2×D19111yescharacteristic
238C2×D8C2×D19111yescharacteristic
239C2×D8C2×D19111yescharacteristic
240C2×M16C2×D19111yescharacteristic
241C22×C8—1911no—
242C4⋊Q8—1911no—
243#124—1911no—
244#123—3821no—
245#123—3821no—
246#123—3821no—
247C22⋊Q8—1911no—
248C22⋊Q8—1911no—
249C22⋊Q8—1911no—
250#121—3821no—
251#121—3821no—
252#121—1911no—
253#120—3821no—
254C4×D4—1911no—
255C4×D4—1911no—
256C4×D4—1911no—
257C2×C4⋊C4—3821no—
258C2×C22⋊C4—3821no—
259#106—1911no—
260C4⋊C8—1911no—
261#103—1911no—
262#103—1911no—
263#103—1911no—
264#102—1911no—
265#102—1911no—
266#102—1911no—
267C22⋊C8—1911no—
268C22⋊C8—1911no—
269#97—1911no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—3822no—
274#41—3822no—
275#41—7641no—
276#41—7641no—
277C2×Q8—1911no—
278C2×Q8—1911no—
279C2×Q8C22×D19111yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—3821no—
285C2×D4—3821no—
286C2×D4—3821no—
287C2×D4—1911no—
288C2×D4C22×D19111yescharacteristic
289C2×D4C22×D19111yescharacteristic
290C2×D4C22×D19111yescharacteristic
291C22×C4—21no—
292C22×C4—3821no—
293C22×C4—3821no—
294C22×C4—3821no—
295C22×C4—3821no—
296C22×C4—1911no—
297C22×C4C22×D19111yescharacteristic
298Q16—3822no—
299Q16—7641no—
300SD16—22no—
301SD16—22no—
302SD16—3822no—
303SD16—3822no—
304SD16—7641no—
305D8—22no—
306D8C22⋊D19114yescomplemented
307M16—22no—
308C2×C8—3822no—
309C2×C8—1911no—
310C2×C8—1911no—
311C2×C8C22×D19111yescharacteristic
312C2×C8C22×D19111yescharacteristic
313C4⋊C4—3821no—
314C4⋊C4—3821no—
315C4⋊C4—3821no—
316C4⋊C4—3821no—
317C4⋊C4—3821no—
318C4⋊C4—1911no—
319C4⋊C4—1911no—
320C4⋊C4—1911no—
321C22⋊C4—3821no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—3821no—
325C22⋊C4—3821no—
326C22⋊C4—3821no—
327C22⋊C4—3821no—
328C22⋊C4—3821no—
329C22⋊C4—1911no—
330C22⋊C4—1911no—
331C42—1911no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1911no—
339C23D4×D19111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3821no—
343Q8—3821no—
344Q8—3821no—
345Q8—3821no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—15281no—
353D4—7641no—
354D4—7642no—
355D4—7642no—
356D4C2×C22⋊D19112yes—
357D4C2×C22⋊D19112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—3821no—
363C2×C4—3821no—
364C2×C4—3821no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—3821no—
368C2×C4—1911no—
369C2×C4—1911no—
370C2×C4—1911no—
371C2×C4—15281no—
372C2×C4—7642no—
373C2×C4—7642no—
374C2×C4—7642no—
375C2×C4D4×D19111yescharacteristic
376C2×C4C23×D19111yescharacteristic
377C8—21no—
378C8—3821no—
379C8—3821no—
380C8C2×C22⋊D19112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—3821no—
395C22—3821no—
396C22—3821no—
397C22C2×D4×D19111yescharacteristic
398C4—21no—
399C4—41no—
400C4—3821no—
401C4—3821no—
402C4—3821no—
403C4—7641no—
404C4—7641no—
405C4—7641no—
406C4—7641no—
407C4C22×C22⋊D19111yescharacteristic
408C4#2025936711yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—7641no—
414C2D4×C22⋊D19111yescharacteristic
415C2#2713345911yescharacteristic
416C2#2713347311yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×AGL(1,ℤ/8)×C191C2
socleC22×C191C22×D4
hypercenterC2×AGL(1,ℤ/8)D191
hypocenterC191#3267
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20259341C2semidirect
2. #27133312C2semidirect
3. #27133307C2semidirect
4. #27133054C2semidirect
5. #27132921C2semidirect
6. #27132889C2semidirect
7. #27132876C2semidirect
8. #27132866C2semidirect
9. #27132825C2semidirect
10. #27132822C2semidirect
11. #27132814C2semidirect
12. #27132805C2semidirect
13. #27132775C2semidirect
14. #27132732C2semidirect
15. C2×AGL(1,ℤ/8)×C191C2semidirect
17. C2×#15690286C22semidirect
19. C2×Dic764C22semidirect
20. C2×#15690277C22semidirect
23. C2×#15690275C22semidirect
24. C2×C8×D191C22semidirect
25. #20259320C22semidirect
32. #20259298C22semidirect
38. D4×Dic191C22semidirect
42. #20259254C22semidirect
43. Dic191⋊C8C22semidirect
44. #20259247C22semidirect
45. #20259247C22semidirect
46. #20259246C22semidirect
49. #20259243C22semidirect
51. (C2×D191)⋊C8C22semidirect
52. #20259232C22semidirect
119. D8×C191D4semidirect
208. C191#3267semidirect
210. C2×AGL(1,ℤ/8)D191semidirect
306. D8C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593412
#271333122
#271333072
#271330542
#271329212
#271328892
#271328762
#271328662
#271328252
#271328222
#271328142
#271328052
#271327752
#271327322
C2×AGL(1,ℤ/8)×C1912
#3267191