Finite Group Databasebeta

← #3267×C233 #44449299

Subgroups — #3267×C233 #44449299

Order 29824.

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×AGL(1,ℤ/8)×C233C211yesmaximal, characteristic, complemented
2C2×#135×C233C211yesmaximal, characteristic, complemented
3#547×C233C211yesmaximal, characteristic, complemented
4#546×C233C211yesmaximal, characteristic, complemented
5#497×C233C211yesmaximal, characteristic, complemented
6#470×C233C211yesmaximal, characteristic, complemented
7#465×C233C211yesmaximal, characteristic, complemented
8#463×C233C211yesmaximal, characteristic, complemented
9#461×C233C211yesmaximal, characteristic, complemented
10#452×C233C211yesmaximal, characteristic, complemented
11#452×C233C211yesmaximal, characteristic, complemented
12#450×C233C211yesmaximal, characteristic, complemented
13#449×C233C211yesmaximal, characteristic, complemented
14#442×C233C211yesmaximal, characteristic, complemented
15#435×C233C211yesmaximal, characteristic, complemented
16C2×#41×C233C2211yescharacteristic
17C2×#41×C233C2211yescharacteristic
18C2×#41×C233C2211yescharacteristic, complemented
19C22×D4×C233C2211yescharacteristic
20AGL(1,ℤ/8)×C233—24no—
21#135×C233—24no—
22C2×Q16×C233C2211yescharacteristic, complemented
23C2×Q16×C233C2211yescharacteristic, complemented
24C2×SD16×C233C2211yescharacteristic
25C2×SD16×C233C2211yescharacteristic
26C2×SD16×C233C2211yescharacteristic
27C2×SD16×C233C2211yescharacteristic
28C2×SD16×C233C2211yescharacteristic, complemented
29C2×D8×C233C2211yescharacteristic
30C2×D8×C233C2211yescharacteristic
31C2×M16×C233C2211yescharacteristic
32C22×C8×C233C2211yescharacteristic, complemented
33C4⋊Q8×C233C2211yescharacteristic, complemented
34#124×C233C2211yescharacteristic
35#123×C233—21no—
36#123×C233—21no—
37#123×C233—21no—
38C22⋊Q8×C233C2211yescharacteristic
39C22⋊Q8×C233C2211yescharacteristic
40C22⋊Q8×C233C2211yescharacteristic, complemented
41#121×C233—21no—
42#121×C233—21no—
43#121×C233C2211yescharacteristic
44#120×C233—21no—
45C4×D4×C233C2211yescharacteristic
46C4×D4×C233C2211yescharacteristic
47C4×D4×C233C2211yescharacteristic, complemented
48C2×C4⋊C4×C233—21no—
49C2×C22⋊C4×C233—21no—
50#106×C233C2211yescharacteristic, complemented
51C4⋊C8×C233C2211yescharacteristic, complemented
52#103×C233C2211yescharacteristic, complemented
53#103×C233C2211yescharacteristic, complemented
54#103×C233C2211yescharacteristic, complemented
55#102×C233C2211yescharacteristic
56#102×C233C2211yescharacteristic
57#102×C233C2211yescharacteristic, complemented
58C22⋊C8×C233C2211yescharacteristic
59C22⋊C8×C233C2211yescharacteristic, complemented
60#97×C233C2211yescharacteristic, complemented
61C24×C233—21no—
62#41×C233—22no—
63#41×C233—22no—
64#41×C233—22no—
65#41×C233—41no—
66#41×C233—41no—
67#41×C233—41no—
68C2×Q8×C233C2311yescharacteristic
69C2×Q8×C233C2311yescharacteristic
70C2×Q8×C233C2311yescharacteristic
71C2×D4×C233—22no—
72C2×D4×C233—21no—
73C2×D4×C233—21no—
74C2×D4×C233—21no—
75C2×D4×C233—21no—
76C2×D4×C233—21no—
77C2×D4×C233—41no—
78C2×D4×C233D411yescharacteristic
79C2×D4×C233C2311yescharacteristic
80C2×D4×C233C2311yescharacteristic
81C2×D4×C233C2311yescharacteristic
82C22×C4×C233—21no—
83C22×C4×C233—21no—
84C22×C4×C233—21no—
85C22×C4×C233—21no—
86C22×C4×C233—21no—
87C22×C4×C233C2311yescharacteristic
88C22×C4×C233C2311yescharacteristic
89Q16×C233—22no—
90Q16×C233—41no—
91SD16×C233—22no—
92SD16×C233—22no—
93SD16×C233—22no—
94SD16×C233—22no—
95SD16×C233—41no—
96D8×C233—22no—
97D8×C233D414yescomplemented
98M16×C233—22no—
99C2×C8×C233—22no—
100C2×C8×C233C2311yescharacteristic
101C2×C8×C233C2311yescharacteristic
102C2×C8×C233C2311yescharacteristic
103C2×C8×C233C2311yescharacteristic
104C4⋊C4×C233—21no—
105C4⋊C4×C233—21no—
106C4⋊C4×C233—21no—
107C4⋊C4×C233—21no—
108C4⋊C4×C233—21no—
109C4⋊C4×C233D411yescharacteristic
110C4⋊C4×C233C2311yescharacteristic
111C4⋊C4×C233C2311yescharacteristic
112C22⋊C4×C233—21no—
113C22⋊C4×C233—21no—
114C22⋊C4×C233—21no—
115C22⋊C4×C233—21no—
116C22⋊C4×C233—21no—
117C22⋊C4×C233—21no—
118C22⋊C4×C233—21no—
119C22⋊C4×C233—21no—
120C22⋊C4×C233D411yescharacteristic
121C22⋊C4×C233D411yescharacteristic
122C42×C233C2311yescharacteristic
123C23×C233—21no—
124C23×C233—21no—
125C23×C233—21no—
126C23×C233—42no—
127C23×C233—42no—
128C23×C233—81no—
129C23×C233C2×D411yescharacteristic
130C23×C233C2×D411yescharacteristic
131Q8×C233—21no—
132Q8×C233—21no—
133Q8×C233—21no—
134Q8×C233—21no—
135Q8×C233—21no—
136Q8×C233—21no—
137D4×C233—21no—
138D4×C233—21no—
139D4×C233—21no—
140D4×C233—21no—
141D4×C233—41no—
142D4×C233—42no—
143D4×C233—42no—
144D4×C233—42no—
145D4×C233—42no—
146D4×C233—81no—
147D4×C233C2×D412yes—
148D4×C233C2×D412yes—
149C2×C4×C233—21no—
150C2×C4×C233—21no—
151C2×C4×C233—21no—
152C2×C4×C233—21no—
153C2×C4×C233—21no—
154C2×C4×C233—21no—
155C2×C4×C233—21no—
156C2×C4×C233—21no—
157C2×C4×C233—21no—
158C2×C4×C233—42no—
159C2×C4×C233—42no—
160C2×C4×C233—42no—
161C2×C4×C233—42no—
162C2×C4×C233—81no—
163C2×C4×C233C2×D411yescharacteristic
164C2×C4×C233C2×D411yescharacteristic
165C2×C4×C233C2×D411yescharacteristic
166C2×C4×C233C2×D411yescharacteristic
167C2×C4×C233C2411yescharacteristic
168C8×C233—21no—
169C8×C233—21no—
170C8×C233—21no—
171C8×C233C2×D412yes—
172C22×C233—21no—
173C22×C233—21no—
174C22×C233—22no—
175C22×C233—22no—
176C22×C233—21no—
177C22×C233—21no—
178C22×C233—21no—
179C22×C233—21no—
180C22×C233—41no—
181C22×C233—41no—
182C22×C233—41no—
183C22×C233—41no—
184C22×C233—41no—
185C22×C233—41no—
186C22×C233—41no—
187C22×C233—84no—
188C22×C233C22×D411yescharacteristic
189C4×C233—21no—
190C4×C233—21no—
191C4×C233—21no—
192C4×C233—21no—
193C4×C233—41no—
194C4×C233—41no—
195C4×C233—41no—
196C4×C233—41no—
197C4×C233—41no—
198C4×C233C22×D411yescharacteristic
199C4×C233H(5,𝔽2)11yescharacteristic
200C2×C233—41no—
201C2×C233—41no—
202C2×C233—42no—
203C2×C233—42no—
204C2×C233—81no—
205C2×C233D4211yescharacteristic
206C2×C233#57511yescharacteristic
207C2×C233#57711yescharacteristic
208C233#326711yescharacteristic, direct product factor, Sylow
209#3267C23311yesmaximal, characteristic, direct product factor, Sylow
210C2×AGL(1,ℤ/8)C2×C23311yescharacteristic, complemented
211C2×#135C2×C23311yescharacteristic, complemented
212#547C2×C23311yescharacteristic, complemented
213#546C2×C23311yescharacteristic, complemented
214#497C2×C23311yescharacteristic, complemented
215#470C2×C23311yescharacteristic, complemented
216#465C2×C23311yescharacteristic, complemented
217#463C2×C23311yescharacteristic, complemented
218#461C2×C23311yescharacteristic, complemented
219#452C2×C23311yescharacteristic, complemented
220#452C2×C23311yescharacteristic, complemented
221#450C2×C23311yescharacteristic, complemented
222#449C2×C23311yescharacteristic, complemented
223#442C2×C23311yescharacteristic, complemented
224#435C2×C23311yescharacteristic, complemented
225C2×#41C22×C23311yescharacteristic
226C2×#41C22×C23311yescharacteristic
227C2×#41C22×C23311yescharacteristic, complemented
228C22×D4C22×C23311yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—24no—
231C2×Q16C22×C23311yescharacteristic, complemented
232C2×Q16C22×C23311yescharacteristic, complemented
233C2×SD16C22×C23311yescharacteristic
234C2×SD16C22×C23311yescharacteristic
235C2×SD16C22×C23311yescharacteristic
236C2×SD16C22×C23311yescharacteristic
237C2×SD16C22×C23311yescharacteristic, complemented
238C2×D8C22×C23311yescharacteristic
239C2×D8C22×C23311yescharacteristic
240C2×M16C22×C23311yescharacteristic
241C22×C8C22×C23311yescharacteristic, complemented
242C4⋊Q8C22×C23311yescharacteristic, complemented
243#124C22×C23311yescharacteristic
244#123—21no—
245#123—21no—
246#123—21no—
247C22⋊Q8C22×C23311yescharacteristic
248C22⋊Q8C22×C23311yescharacteristic
249C22⋊Q8C22×C23311yescharacteristic, complemented
250#121—21no—
251#121—21no—
252#121C22×C23311yescharacteristic
253#120—21no—
254C4×D4C22×C23311yescharacteristic
255C4×D4C22×C23311yescharacteristic
256C4×D4C22×C23311yescharacteristic, complemented
257C2×C4⋊C4—21no—
258C2×C22⋊C4—21no—
259#106C22×C23311yescharacteristic, complemented
260C4⋊C8C22×C23311yescharacteristic, complemented
261#103C22×C23311yescharacteristic, complemented
262#103C22×C23311yescharacteristic, complemented
263#103C22×C23311yescharacteristic, complemented
264#102C22×C23311yescharacteristic
265#102C22×C23311yescharacteristic
266#102C22×C23311yescharacteristic, complemented
267C22⋊C8C22×C23311yescharacteristic
268C22⋊C8C22×C23311yescharacteristic, complemented
269#97C22×C23311yescharacteristic, complemented
270C24—21no—
271#41—22no—
272#41—22no—
273#41—22no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8C23×C23311yescharacteristic
278C2×Q8C23×C23311yescharacteristic
279C2×Q8C23×C23311yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4D4×C23311yescharacteristic
288C2×D4C23×C23311yescharacteristic
289C2×D4C23×C23311yescharacteristic
290C2×D4C23×C23311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4C23×C23311yescharacteristic
297C22×C4C23×C23311yescharacteristic
298Q16—22no—
299Q16—41no—
300SD16—22no—
301SD16—22no—
302SD16—22no—
303SD16—22no—
304SD16—41no—
305D8—22no—
306D8D4×C23314yescomplemented
307M16—22no—
308C2×C8—22no—
309C2×C8C23×C23311yescharacteristic
310C2×C8C23×C23311yescharacteristic
311C2×C8C23×C23311yescharacteristic
312C2×C8C23×C23311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4D4×C23311yescharacteristic
319C4⋊C4C23×C23311yescharacteristic
320C4⋊C4C23×C23311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4D4×C23311yescharacteristic
330C22⋊C4D4×C23311yescharacteristic
331C42C23×C23311yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23C2×D4×C23311yescharacteristic
339C23C2×D4×C23311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—41no—
351D4—42no—
352D4—42no—
353D4—42no—
354D4—42no—
355D4—81no—
356D4C2×D4×C23312yes—
357D4C2×D4×C23312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—81no—
372C2×C4C2×D4×C23311yescharacteristic
373C2×C4C2×D4×C23311yescharacteristic
374C2×C4C2×D4×C23311yescharacteristic
375C2×C4C2×D4×C23311yescharacteristic
376C2×C4C24×C23311yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8C2×D4×C23312yes—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—84no—
397C22C22×D4×C23311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4C22×D4×C23311yescharacteristic
408C4H(5,𝔽2)×C23311yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—42no—
413C2—81no—
414C2D42×C23311yescharacteristic
415C2#575×C23311yescharacteristic
416C2#577×C23311yescharacteristic
subgroupG/H
center, socleC22×C233C22×D4
commutator, Frattini, cosocleC2×C4C24×C233
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C233 ◁ C4×D4×C233 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×AGL(1,ℤ/8)×C233C2semidirect
2. C2×#135×C233C2semidirect
3. #547×C233C2semidirect
4. #546×C233C2semidirect
5. #497×C233C2semidirect
6. #470×C233C2semidirect
7. #465×C233C2semidirect
8. #463×C233C2semidirect
9. #461×C233C2semidirect
10. #452×C233C2semidirect
11. #452×C233C2semidirect
12. #450×C233C2semidirect
13. #449×C233C2semidirect
14. #442×C233C2semidirect
15. #435×C233C2semidirect
18. C2×#41×C233C22semidirect
22. C2×Q16×C233C22semidirect
23. C2×Q16×C233C22semidirect
28. C2×SD16×C233C22semidirect
32. C22×C8×C233C22semidirect
33. C4⋊Q8×C233C22semidirect
40. C22⋊Q8×C233C22semidirect
47. C4×D4×C233C22semidirect
50. #106×C233C22semidirect
51. C4⋊C8×C233C22semidirect
52. #103×C233C22semidirect
53. #103×C233C22semidirect
54. #103×C233C22semidirect
57. #102×C233C22semidirect
59. C22⋊C8×C233C22semidirect
60. #97×C233C22semidirect
97. D8×C233D4semidirect
208. C233#3267direct (N × H)
209. #3267C233direct (N × H)
210. C2×AGL(1,ℤ/8)C2×C233semidirect
211. C2×#135C2×C233semidirect
212. #547C2×C233semidirect
213. #546C2×C233semidirect
214. #497C2×C233semidirect
215. #470C2×C233semidirect
216. #465C2×C233semidirect
217. #463C2×C233semidirect
218. #461C2×C233semidirect
219. #452C2×C233semidirect
220. #452C2×C233semidirect
221. #450C2×C233semidirect
222. #449C2×C233semidirect
223. #442C2×C233semidirect
224. #435C2×C233semidirect
227. C2×#41C22×C233semidirect
231. C2×Q16C22×C233semidirect
232. C2×Q16C22×C233semidirect
237. C2×SD16C22×C233semidirect
241. C22×C8C22×C233semidirect
242. C4⋊Q8C22×C233semidirect
249. C22⋊Q8C22×C233semidirect
256. C4×D4C22×C233semidirect
259. #106C22×C233semidirect
260. C4⋊C8C22×C233semidirect
261. #103C22×C233semidirect
262. #103C22×C233semidirect
263. #103C22×C233semidirect
266. #102C22×C233semidirect
268. C22⋊C8C22×C233semidirect
269. #97C22×C233semidirect
306. D8D4×C233semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,ℤ/8)×C2332
C2×#135×C2332
#547×C2332
#546×C2332
#497×C2332
#470×C2332
#465×C2332
#463×C2332
#461×C2332
#452×C2332
#450×C2332
#449×C2332
#442×C2332
#435×C2332
#3267233