Finite Group Databasebeta

← #15684922

Subgroups — #15684922

Order 2944.

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×#10625082C211yesmaximal, characteristic, complemented
2#13093066C211yesmaximal, characteristic, complemented
3#13093055C211yesmaximal, characteristic, complemented
4#13092810C211yesmaximal, characteristic, complemented
5#13092672C211yesmaximal, characteristic, complemented
6#13092646C211yesmaximal, characteristic, complemented
7#13092634C211yesmaximal, characteristic, complemented
8#13092620C211yesmaximal, characteristic, complemented
9#13092579C211yesmaximal, characteristic, complemented
10#13092576C211yesmaximal, characteristic, complemented
11#13092566C211yesmaximal, characteristic, complemented
12#13092562C211yesmaximal, characteristic, complemented
13#13092529C211yesmaximal, characteristic, complemented
14#13092488C211yesmaximal, characteristic, complemented
15C2×C23×#135C211yesmaximal, characteristic, complemented
16C2×#41×C23C2211yescharacteristic
17C2×#41×C23C2211yescharacteristic, complemented
18C2×#67290C2211yescharacteristic
19C2×D4×D23C2211yescharacteristic
20C23×#135—24no—
21C2×Q16×C23C2211yescharacteristic, complemented
22C2×SD16×C23C2211yescharacteristic
23C2×SD16×C23C2211yescharacteristic, complemented
24C2×D8×C23C2211yescharacteristic
25C22×C8×C23C2211yescharacteristic, complemented
26#10625118C2211yescharacteristic
27#10625117C2211yescharacteristic
28#10625117C2211yescharacteristic
29#10625115C2211yescharacteristic, complemented
30C2×#67275C2211yescharacteristic, complemented
31C2×#67274C2211yescharacteristic
32#10625110—21no—
33#10625110—21no—
34#10625108—21no—
35#10625106—21no—
36D4×Dic23C2211yescharacteristic
37D4×Dic23C2211yescharacteristic, complemented
38C2×#67273C2211yescharacteristic
39C2×#67272C2211yescharacteristic
40#10625101C2211yescharacteristic
41#10625100—21no—
42#10625100—21no—
43C4×C22⋊D23C2211yescharacteristic
44C2×(C2×D23)⋊C4—21no—
45#10625095C2211yescharacteristic, complemented
46C2×Dic23⋊C4—21no—
47#10625082—24no—
48C2×#67263C2211yescharacteristic
49C2×#67262C2211yescharacteristic
50#10625017C2211yescharacteristic, complemented
51#10625017C2211yescharacteristic, complemented
52#10625014C2211yescharacteristic
53#10625014C2211yescharacteristic, complemented
54C22⋊(C23⋊C8)C2211yescharacteristic, complemented
55#10625003C2211yescharacteristic
56(C2×D23)⋊C8C2211yescharacteristic
57#10624999C2211yescharacteristic, complemented
58#10624998C2211yescharacteristic, complemented
59#10624997C2211yescharacteristic, complemented
60Dic23⋊C8C2211yescharacteristic, complemented
61C23×D23—21no—
62#41×C23—22no—
63#41×C23—22no—
64#41×C23—41no—
65#41×C23—41no—
66C2×Q8×C23C2311yescharacteristic
67C2×Q8×C23C2311yescharacteristic
68C2×D4×C23—21no—
69C2×D4×C23—21no—
70C2×D4×C23C2311yescharacteristic
71C2×D4×C23C2311yescharacteristic
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23C2311yescharacteristic
75C2×C22⋊D23—21no—
76C2×C22⋊D23—21no—
77C2×C22⋊D23—21no—
78C2×C22⋊D23D411yescharacteristic
79C22×Dic23—21no—
80C22×Dic23—21no—
81C22×Dic23—21no—
82#67290—22no—
83#67290—41no—
84D4×D23—22no—
85D4×D23—41no—
86C2×D92C2311yescharacteristic
87C2×C4×D23C2311yescharacteristic
88C2×Dic46C2311yescharacteristic
89Q16×C23—22no—
90SD16×C23—22no—
91SD16×C23—22no—
92D8×C23D414yescomplemented
93C2×C8×C23—22no—
94C2×C8×C23C2311yescharacteristic
95C2×C8×C23C2311yescharacteristic
96C22⋊Dic23—21no—
97C22⋊Dic23—21no—
98C22⋊Dic23—21no—
99C22⋊Dic23D411yescharacteristic
100#67275—41no—
101#67274—41no—
102#67273—22no—
103#67272—22no—
104(C2×D23)⋊C4—21no—
105(C2×D23)⋊C4—21no—
106(C2×D23)⋊C4—21no—
107(C2×D23)⋊C4—21no—
108(C2×D23)⋊C4—21no—
109(C2×D23)⋊C4D411yescharacteristic
110C4⋊Dic23C2311yescharacteristic
111C4⋊Dic23C2311yescharacteristic
112Dic23⋊C4—21no—
113Dic23⋊C4—21no—
114Dic23⋊C4—21no—
115Dic23⋊C4—21no—
116Dic23⋊C4—21no—
117Dic23⋊C4D411yescharacteristic
118C4×Dic23C2311yescharacteristic
119C2×C23⋊C8C2311yescharacteristic
120C2×C23⋊C8C2311yescharacteristic
121#67263—22no—
122#67262—22no—
123C23×C23—21no—
124C23×C23—21no—
125C23×C23C2×D411yescharacteristic
126C22×D23—21no—
127C22×D23—42no—
128C22×D23—42no—
129C22×D23—81no—
130C22×D23C2×D411yescharacteristic
131Q8×C23—21no—
132Q8×C23—21no—
133Q8×C23—21no—
134Q8×C23—21no—
135D4×C23—21no—
136D4×C23—21no—
137D4×C23—42no—
138D4×C23—42no—
139D4×C23C2×D412yes—
140D4×C23C2×D412yes—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—42no—
146C2×C4×C23—42no—
147C2×C4×C23C2×D411yescharacteristic
148C2×C4×C23C2411yescharacteristic
149C22⋊D23—41no—
150C22⋊D23—42no—
151C22⋊D23—42no—
152C22⋊D23—81no—
153C2×Dic23—21no—
154C2×Dic23—21no—
155C2×Dic23—21no—
156C2×Dic23—42no—
157C2×Dic23—42no—
158C2×Dic23—81no—
159C2×Dic23C2×D411yescharacteristic
160C2×Dic23C2×D411yescharacteristic
161C2×Dic23C2×D411yescharacteristic
162D92—21no—
163D92—21no—
164C4×D23—21no—
165C4×D23—21no—
166Dic46—21no—
167Dic46—21no—
168C8×C23—21no—
169C8×C23C2×D412yes—
170C23⋊C8—21no—
171C23⋊C8—21no—
172#3267—231nomaximal, Sylow
173C22×C23—22no—
174C22×C23—22no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—41no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23C22×D411yescharacteristic
183C2×D23—21no—
184C2×D23—21no—
185C2×D23—21no—
186C2×D23—41no—
187C2×D23—41no—
188C2×D23—41no—
189C2×D23—84no—
190C4×C23—21no—
191C4×C23—41no—
192C4×C23—41no—
193C4×C23C22×D411yescharacteristic
194C4×C23H(5,𝔽2)11yescharacteristic
195Dic23—21no—
196Dic23—21no—
197Dic23—21no—
198Dic23—41no—
199Dic23—41no—
200Dic23—41no—
201C2×AGL(1,ℤ/8)—231no—
202C2×#135D2311yescharacteristic, complemented
203#547—231no—
204#546—231no—
205#497—231no—
206#470—231no—
207#465—231no—
208#463—231no—
209#461—231no—
210#452—231no—
211#452—231no—
212#450—231no—
213#449—231no—
214#442—231no—
215#435—231no—
216C2×C23—42no—
217C2×C23—41no—
218C2×C23—42no—
219C2×C23D4211yescharacteristic
220C2×C23#57511yescharacteristic
221C2×C23#57711yescharacteristic
222D23—41no—
223D23—81no—
224C2×#41—231no—
225C2×#41C2×D2311yescharacteristic
226C2×#41C2×D2311yescharacteristic, complemented
227C22×D4—231no—
228AGL(1,ℤ/8)—464no—
229#135—24no—
230C2×Q16—231no—
231C2×Q16C2×D2311yescharacteristic, complemented
232C2×SD16—231no—
233C2×SD16—231no—
234C2×SD16—231no—
235C2×SD16C2×D2311yescharacteristic
236C2×SD16C2×D2311yescharacteristic, complemented
237C2×D8—231no—
238C2×D8C2×D2311yescharacteristic
239C2×M16—231no—
240C22×C8C2×D2311yescharacteristic, complemented
241C4⋊Q8—231no—
242#124—231no—
243#123—461no—
244#123—461no—
245#123—461no—
246C22⋊Q8—231no—
247C22⋊Q8—231no—
248C22⋊Q8—231no—
249#121—231no—
250#121—461no—
251#121—461no—
252#120—461no—
253C4×D4—231no—
254C4×D4—231no—
255C4×D4—231no—
256C2×C4⋊C4—461no—
257C2×C22⋊C4—461no—
258#106—231no—
259C4⋊C8—231no—
260#103—231no—
261#103—231no—
262#103—231no—
263#102—231no—
264#102—231no—
265#102—231no—
266C22⋊C8—231no—
267C22⋊C8—231no—
268#97—231no—
269C23#326711yescharacteristic, complemented, Sylow
270C24—461no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—462no—
276#41—921no—
277C2×Q8—231no—
278C2×Q8C22×D2311yescharacteristic
279C2×Q8C22×D2311yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—231no—
283C2×D4—231no—
284C2×D4—461no—
285C2×D4—462no—
286C2×D4—461no—
287C2×D4—461no—
288C2×D4—921no—
289C2×D4C22×D2311yescharacteristic
290C2×D4C22×D2311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—231no—
294C22×C4—461no—
295C22×C4—461no—
296C22×C4—461no—
297C22×C4C22×D2311yescharacteristic
298Q16—22no—
299Q16—921no—
300SD16—22no—
301SD16—22no—
302SD16—462no—
303SD16—462no—
304SD16—921no—
305D8—462no—
306D8C22⋊D2314yescomplemented
307M16—462no—
308C2×C8—22no—
309C2×C8—231no—
310C2×C8—231no—
311C2×C8C22×D2311yescharacteristic
312C2×C8C22×D2311yescharacteristic
313C4⋊C4—231no—
314C4⋊C4—231no—
315C4⋊C4—231no—
316C4⋊C4—461no—
317C4⋊C4—461no—
318C4⋊C4—461no—
319C4⋊C4—461no—
320C4⋊C4—461no—
321C22⋊C4—231no—
322C22⋊C4—231no—
323C22⋊C4—461no—
324C22⋊C4—461no—
325C22⋊C4—461no—
326C22⋊C4—461no—
327C22⋊C4—461no—
328C22⋊C4—461no—
329C22⋊C4—461no—
330C22⋊C4—461no—
331C42—231no—
332C23—21no—
333C23—21no—
334C23—231no—
335C23—461no—
336C23—922no—
337C23—922no—
338C23—1841no—
339C23D4×D2311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—461no—
345Q8—461no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—42no—
350D4—461no—
351D4—461no—
352D4—921no—
353D4—922no—
354D4—922no—
355D4—1841no—
356D4C2×C22⋊D2312yes—
357D4C2×C22⋊D2312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—231no—
365C2×C4—231no—
366C2×C4—231no—
367C2×C4—461no—
368C2×C4—461no—
369C2×C4—461no—
370C2×C4—461no—
371C2×C4—461no—
372C2×C4—922no—
373C2×C4—922no—
374C2×C4—1841no—
375C2×C4D4×D2311yescharacteristic
376C2×C4C23×D2311yescharacteristic
377C8—21no—
378C8—461no—
379C8—461no—
380C8C2×C22⋊D2312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—461no—
391C22—461no—
392C22—461no—
393C22—921no—
394C22—921no—
395C22—921no—
396C22—1844no—
397C22C2×D4×D2311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—461no—
402C4—461no—
403C4—461no—
404C4—921no—
405C4—921no—
406C4—921no—
407C4#1062515511yescharacteristic
408C4C22×C22⋊D2311yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—921no—
413C2—1841no—
414C2D4×C22⋊D2311yescharacteristic
415C2#1309321411yescharacteristic
416C2#1309322411yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC2×C23×#135C2
socleC22×C23C22×D4
hypercenterC2×#135D23
hypocenterC23#3267
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#10625082C2semidirect
2. #13093066C2semidirect
3. #13093055C2semidirect
4. #13092810C2semidirect
5. #13092672C2semidirect
6. #13092646C2semidirect
7. #13092634C2semidirect
8. #13092620C2semidirect
9. #13092579C2semidirect
10. #13092576C2semidirect
11. #13092566C2semidirect
12. #13092562C2semidirect
13. #13092529C2semidirect
14. #13092488C2semidirect
15. C2×C23×#135C2semidirect
17. C2×#41×C23C22semidirect
21. C2×Q16×C23C22semidirect
23. C2×SD16×C23C22semidirect
25. C22×C8×C23C22semidirect
29. #10625115C22semidirect
30. C2×#67275C22semidirect
37. D4×Dic23C22semidirect
45. #10625095C22semidirect
50. #10625017C22semidirect
51. #10625017C22semidirect
53. #10625014C22semidirect
54. C22⋊(C23⋊C8)C22semidirect
57. #10624999C22semidirect
58. #10624998C22semidirect
59. #10624997C22semidirect
60. Dic23⋊C8C22semidirect
92. D8×C23D4semidirect
202. C2×#135D23semidirect
226. C2×#41C2×D23semidirect
231. C2×Q16C2×D23semidirect
236. C2×SD16C2×D23semidirect
240. C22×C8C2×D23semidirect
269. C23#3267semidirect
306. D8C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
C2×#106250822
#130930662
#130930552
#130928102
#130926722
#130926462
#130926342
#130926202
#130925792
#130925762
#130925662
#130925622
#130925292
#130924882
C2×C23×#1352
#326723