Finite Group Databasebeta

← #37584030

Subgroups — #37584030

Order 24448.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259351C211yesmaximal, characteristic, complemented
2Q8×C22⋊D191C211yesmaximal, characteristic, complemented
3#27133307C211yesmaximal, characteristic
4#27133054C211yesmaximal, characteristic, complemented
5#27132906C211yesmaximal, characteristic, complemented
6#27132904C211yesmaximal, characteristic, complemented
7#27132883C211yesmaximal, characteristic, complemented
8#27132866C211yesmaximal, characteristic, complemented
9#27132822C211yesmaximal, characteristic, complemented
10#27132819C211yesmaximal, characteristic, complemented
11#27132814C211yesmaximal, characteristic, complemented
12#27132800C211yesmaximal, characteristic, complemented
13#27132764C211yesmaximal, characteristic, complemented
14#27132734C211yesmaximal, characteristic, complemented
15C22×SD16×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×Q8×D191C2211yescharacteristic, complemented
18#20259351—24no—
19C2×Dic764C2211yescharacteristic, complemented
20C2×#15690277C2211yescharacteristic, complemented
21C2×#15690277C2211yescharacteristic, complemented
22C2×#15690275C2211yescharacteristic
23C2×#15690275C2211yescharacteristic
24C2×#15690270C2211yescharacteristic, complemented
25#20259320—21no—
26#20259320C2211yescharacteristic, complemented
27#20259309C2211yescharacteristic
28#20259303—21no—
29#20259301—21no—
30#20259301C2211yescharacteristic
31#20259301C2211yescharacteristic
32#20259298—21no—
33#20259298C2211yescharacteristic, complemented
34#20259294—21no—
35#20259293C2211yescharacteristic
36#20259291—21no—
37Q8×Dic191C2211yescharacteristic, complemented
38D4×Dic191C2211yescharacteristic
39C4×C22⋊D191—21no—
40C4×C22⋊D191C2211yescharacteristic
41C2×C22⋊Dic191—21no—
42#20259256C2211yescharacteristic, complemented
43Dic191⋊C8C2211yescharacteristic, complemented
44#20259247C2211yescharacteristic, complemented
45#20259247C2211yescharacteristic, complemented
46#20259246C2211yescharacteristic, complemented
47#20259246C2211yescharacteristic, complemented
48#20259243C2211yescharacteristic
49#20259243C2211yescharacteristic
50C22⋊(C191⋊C8)C2211yescharacteristic
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52#20259232C2211yescharacteristic, complemented
53C22×Q8×C191C2211yescharacteristic
54C22×D4×C191C2211yescharacteristic
55C2×SD16×C191—24no—
56C2×SD16×C191C2211yescharacteristic
57C2×SD16×C191C2211yescharacteristic
58C2×SD16×C191C2211yescharacteristic
59C2×SD16×C191C2211yescharacteristic
60C22×C8×C191C2211yescharacteristic
61#15690286—22no—
62#15690286—41no—
63C2×Dic382—21no—
64C2×Dic382C2311yescharacteristic
65C2×Dic382C2311yescharacteristic
66Q8×D191—22no—
67Q8×D191—41no—
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191D411yescharacteristic
71C22×Dic191—21no—
72C22×Dic191—21no—
73C2×C4×D191—21no—
74C2×C4×D191C2311yescharacteristic
75Dic764—22no—
76#15690277—22no—
77#15690277—41no—
78#15690275—22no—
79#15690275—41no—
80#15690270—22no—
81C2×C191⋊C8C2311yescharacteristic
82C2×C191⋊C8C2311yescharacteristic
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⋊Dic191—21no—
98C22⋊Dic191D411yescharacteristic
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4D411yescharacteristic
101C4×Dic191—21no—
102C4×Dic191C2311yescharacteristic
103C24×C191—21no—
104C2×Q8×C191—21no—
105C2×Q8×C191—22no—
106C2×Q8×C191—41no—
107C2×Q8×C191C2311yescharacteristic
108C2×Q8×C191C2311yescharacteristic
109C2×D4×C191—22no—
110C2×D4×C191—21no—
111C2×D4×C191—41no—
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116SD16×C191—22no—
117SD16×C191—22no—
118SD16×C191—22no—
119SD16×C191D414yescomplemented
120C2×C8×C191—22no—
121C2×C8×C191C2311yescharacteristic
122C2×C8×C191C2311yescharacteristic
123C22×D191C2×D411yescharacteristic
124Dic382—21no—
125Dic382—21no—
126Dic382—21no—
127Dic382—21no—
128Dic382—42no—
129C22⋊D191—22no—
130C22⋊D191—42no—
131C22⋊D191—81no—
132C2×Dic191—21no—
133C2×Dic191—21no—
134C2×Dic191—21no—
135C2×Dic191—21no—
136C2×Dic191—42no—
137C2×Dic191—81no—
138C2×Dic191C2×D411yescharacteristic
139C2×Dic191C2×D411yescharacteristic
140C2×Dic191C2×D411yescharacteristic
141C4×D191—21no—
142C4×D191—21no—
143C4×D191—42no—
144C191⋊C8—21no—
145C191⋊C8—21no—
146C23×C191—21no—
147C23×C191—21no—
148C23×C191—42no—
149C23×C191—42no—
150C23×C191—81no—
151C23×C191C2×D411yescharacteristic
152Q8×C191—21no—
153Q8×C191—21no—
154Q8×C191—21no—
155Q8×C191—42no—
156Q8×C191C2×D412yes—
157D4×C191—21no—
158D4×C191—21no—
159D4×C191—21no—
160D4×C191—42no—
161D4×C191C2×D412yes—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—42no—
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2411yescharacteristic
169C8×C191—21no—
170C8×C191C2×D412yes—
171C22×C191—21no—
172C22×C191—21no—
173C22×C191—22no—
174C22×C191—41no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191—41no—
179C22×C191—84no—
180C22×C191#13712yescomplemented
181C22×C191C22×D411yescharacteristic
182C2×D191—21no—
183C2×D191—21no—
184C2×D191—21no—
185C4×C191—21no—
186C4×C191—22no—
187C4×C191—41no—
188C4×C191C22×D411yescharacteristic
189C4×C191H(5,𝔽2)11yescharacteristic
190Dic191—21no—
191Dic191—21no—
192Dic191—21no—
193Dic191—41no—
194Dic191—41no—
195Dic191—41no—
196Dic191—41no—
197C2×C191—22no—
198C2×C191—42no—
199C2×C191—81no—
200C2×C191D4211yescharacteristic
201C2×C191C2×#13711yescharacteristic
202C2×C191#57711yescharacteristic
203D191—41no—
204C191#326611yescharacteristic, complemented, Sylow
205#3266—1911nomaximal, Sylow
206C2×#137—1911no—
207C22×SD16D19111yescharacteristic, complemented
208D4×Q8—1911no—
209#546—1911no—
210#497—1911no—
211#468—1911no—
212#467—1911no—
213#464—1911no—
214#461—1911no—
215#452—1911no—
216#451—1911no—
217#450—1911no—
218#448—1911no—
219#440—1911no—
220#435—1911no—
221C2×#41—1911no—
222C22×Q8—1911no—
223C22×Q8C2×D19111yescharacteristic
224C22×D4C2×D19111yescharacteristic
225#137—3824no—
226C2×Q16—1911no—
227C2×Q16—1911no—
228C2×Q16—1911no—
229C2×SD16—24no—
230C2×SD16—1911no—
231C2×SD16—1911no—
232C2×SD16C2×D19111yescharacteristic
233C2×SD16C2×D19111yescharacteristic
234C2×SD16C2×D19111yescharacteristic
235C2×SD16C2×D19111yescharacteristic
236C2×M16—1911no—
237C22×C8C2×D19111yescharacteristic
238C4⋊Q8—3821no—
239C4⋊Q8—1911no—
240#124—1911no—
241#123—3821no—
242C22⋊Q8—3821no—
243C22⋊Q8—3821no—
244C22⋊Q8—1911no—
245C22⋊Q8—1911no—
246C22⋊Q8—1911no—
247#121—3821no—
248#121—1911no—
249#120—3821no—
250C4×Q8—1911no—
251C4×D4—3821no—
252C4×D4—1911no—
253C4×D4—1911no—
254C2×C22⋊C4—3821no—
255#107—1911no—
256C4⋊C8—1911no—
257#103—1911no—
258#103—1911no—
259#103—1911no—
260#103—1911no—
261#102—1911no—
262#102—1911no—
263C22⋊C8—1911no—
264C22⋊C8—1911no—
265#97—1911no—
266C24—21no—
267#41—3822no—
268#41—7641no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3821no—
273C2×Q8—3822no—
274C2×Q8—1911no—
275C2×Q8—1911no—
276C2×Q8—7641no—
277C2×Q8C22×D19111yescharacteristic
278C2×Q8C22×D19111yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—3821no—
283C2×D4—3821no—
284C2×D4—1911no—
285C2×D4C22×D19111yescharacteristic
286C2×D4C22×D19111yescharacteristic
287C22×C4—21no—
288C22×C4—3821no—
289C22×C4—3821no—
290C22×C4—3821no—
291C22×C4—1911no—
292C22×C4C22×D19111yescharacteristic
293Q16—3822no—
294Q16—3822no—
295Q16—7641no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—3822no—
300SD16—7641no—
301SD16C22⋊D19114yescomplemented
302M16—3822no—
303C2×C8—22no—
304C2×C8—1911no—
305C2×C8—1911no—
306C2×C8C22×D19111yescharacteristic
307C2×C8C22×D19111yescharacteristic
308C4⋊C4—3821no—
309C4⋊C4—3821no—
310C4⋊C4—3821no—
311C4⋊C4—3821no—
312C4⋊C4—3821no—
313C4⋊C4—3821no—
314C4⋊C4—1911no—
315C4⋊C4—1911no—
316C4⋊C4—1911no—
317C22⋊C4—3821no—
318C22⋊C4—3821no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4—3821no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—1911no—
325C22⋊C4—1911no—
326C42—3821no—
327C42—1911no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1911no—
334C23D4×D19111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—3821no—
340Q8—3821no—
341Q8—3821no—
342Q8—3821no—
343Q8—7642no—
344Q8C2×C22⋊D19112yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—3822no—
350D4—15281no—
351D4—7642no—
352D4C2×C22⋊D19112yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—3821no—
359C2×C4—3821no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—3821no—
363C2×C4—3821no—
364C2×C4—1911no—
365C2×C4—1911no—
366C2×C4—1911no—
367C2×C4—15281no—
368C2×C4—7642no—
369C2×C4—7642no—
370C2×C4D4×D19111yescharacteristic
371C2×C4C23×D19111yescharacteristic
372C8—21no—
373C8—3821no—
374C8—3821no—
375C8C2×C22⋊D19112yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—3821no—
386C22—3821no—
387C22—3821no—
388C22#2025935112yescomplemented
389C22C2×D4×D19111yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—3821no—
394C4—3821no—
395C4—3821no—
396C4—7641no—
397C4—7641no—
398C4—7641no—
399C4—7641no—
400C4C22×C22⋊D19111yescharacteristic
401C4#2025936711yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—7641no—
406C2D4×C22⋊D19111yescharacteristic
407C2C2×#2025935111yescharacteristic
408C2#2713346911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×SD16×C191C2
socleC22×C191C22×D4
prime radical#27133307C2
hypercenterC22×SD16D191
hypocenterC191#3266
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#20259351C2semidirect
2. Q8×C22⋊D191C2semidirect
4. #27133054C2semidirect
5. #27132906C2semidirect
6. #27132904C2semidirect
7. #27132883C2semidirect
8. #27132866C2semidirect
9. #27132822C2semidirect
10. #27132819C2semidirect
11. #27132814C2semidirect
12. #27132800C2semidirect
13. #27132764C2semidirect
14. #27132734C2semidirect
15. C22×SD16×C191C2semidirect
17. C2×Q8×D191C22semidirect
19. C2×Dic764C22semidirect
20. C2×#15690277C22semidirect
21. C2×#15690277C22semidirect
24. C2×#15690270C22semidirect
26. #20259320C22semidirect
33. #20259298C22semidirect
37. Q8×Dic191C22semidirect
42. #20259256C22semidirect
43. Dic191⋊C8C22semidirect
44. #20259247C22semidirect
45. #20259247C22semidirect
46. #20259246C22semidirect
47. #20259246C22semidirect
51. (C2×D191)⋊C8C22semidirect
52. #20259232C22semidirect
119. SD16×C191D4semidirect
180. C22×C191#137semidirect
204. C191#3266semidirect
207. C22×SD16D191semidirect
301. SD16C22⋊D191semidirect
388. C22#20259351semidirect

Maximal subgroups

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

subgroupindex
C2×#202593512
Q8×C22⋊D1912
#271333072
#271330542
#271329062
#271329042
#271328832
#271328662
#271328222
#271328192
#271328142
#271328002
#271327642
#271327342
C22×SD16×C1912
#3266191