Finite Group Databasebeta

← #3266×C227 #44397222

Subgroups — #3266×C227 #44397222

Order 29056.

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×#137×C227C211yesmaximal, characteristic, complemented
2C22×SD16×C227C211yesmaximal, characteristic, complemented
3D4×Q8×C227C211yesmaximal, characteristic, complemented
4#546×C227C211yesmaximal, characteristic
5#497×C227C211yesmaximal, characteristic, complemented
6#468×C227C211yesmaximal, characteristic, complemented
7#467×C227C211yesmaximal, characteristic, complemented
8#464×C227C211yesmaximal, characteristic, complemented
9#461×C227C211yesmaximal, characteristic, complemented
10#452×C227C211yesmaximal, characteristic, complemented
11#451×C227C211yesmaximal, characteristic, complemented
12#450×C227C211yesmaximal, characteristic, complemented
13#448×C227C211yesmaximal, characteristic, complemented
14#440×C227C211yesmaximal, characteristic, complemented
15#435×C227C211yesmaximal, characteristic, complemented
16C2×#41×C227C2211yescharacteristic
17C22×Q8×C227C2211yescharacteristic
18C22×Q8×C227C2211yescharacteristic, complemented
19C22×D4×C227C2211yescharacteristic
20#137×C227—24no—
21C2×Q16×C227C2211yescharacteristic, complemented
22C2×Q16×C227C2211yescharacteristic, complemented
23C2×Q16×C227C2211yescharacteristic, complemented
24C2×SD16×C227—24no—
25C2×SD16×C227C2211yescharacteristic
26C2×SD16×C227C2211yescharacteristic
27C2×SD16×C227C2211yescharacteristic
28C2×SD16×C227C2211yescharacteristic
29C2×SD16×C227C2211yescharacteristic
30C2×SD16×C227C2211yescharacteristic
31C2×M16×C227C2211yescharacteristic, complemented
32C22×C8×C227C2211yescharacteristic
33C4⋊Q8×C227—21no—
34C4⋊Q8×C227C2211yescharacteristic, complemented
35#124×C227C2211yescharacteristic
36#123×C227—21no—
37C22⋊Q8×C227—21no—
38C22⋊Q8×C227—21no—
39C22⋊Q8×C227C2211yescharacteristic
40C22⋊Q8×C227C2211yescharacteristic
41C22⋊Q8×C227C2211yescharacteristic, complemented
42#121×C227—21no—
43#121×C227C2211yescharacteristic
44#120×C227—21no—
45C4×Q8×C227C2211yescharacteristic, complemented
46C4×D4×C227—21no—
47C4×D4×C227C2211yescharacteristic
48C4×D4×C227C2211yescharacteristic
49C2×C22⋊C4×C227—21no—
50#107×C227C2211yescharacteristic, complemented
51C4⋊C8×C227C2211yescharacteristic, complemented
52#103×C227C2211yescharacteristic, complemented
53#103×C227C2211yescharacteristic, complemented
54#103×C227C2211yescharacteristic, complemented
55#103×C227C2211yescharacteristic, complemented
56#102×C227C2211yescharacteristic
57#102×C227C2211yescharacteristic
58C22⋊C8×C227C2211yescharacteristic
59C22⋊C8×C227C2211yescharacteristic, complemented
60#97×C227C2211yescharacteristic, complemented
61C24×C227—21no—
62#41×C227—22no—
63#41×C227—41no—
64C2×Q8×C227—21no—
65C2×Q8×C227—21no—
66C2×Q8×C227—22no—
67C2×Q8×C227—22no—
68C2×Q8×C227—41no—
69C2×Q8×C227—41no—
70C2×Q8×C227C2311yescharacteristic
71C2×Q8×C227C2311yescharacteristic
72C2×Q8×C227C2311yescharacteristic
73C2×Q8×C227C2311yescharacteristic
74C2×D4×C227—22no—
75C2×D4×C227—21no—
76C2×D4×C227—21no—
77C2×D4×C227—21no—
78C2×D4×C227—41no—
79C2×D4×C227D411yescharacteristic
80C2×D4×C227C2311yescharacteristic
81C2×D4×C227C2311yescharacteristic
82C22×C4×C227—21no—
83C22×C4×C227—21no—
84C22×C4×C227—21no—
85C22×C4×C227—21no—
86C22×C4×C227C2311yescharacteristic
87C22×C4×C227C2311yescharacteristic
88Q16×C227—22no—
89Q16×C227—22no—
90Q16×C227—41no—
91SD16×C227—22no—
92SD16×C227—22no—
93SD16×C227—22no—
94SD16×C227—22no—
95SD16×C227—41no—
96SD16×C227D414yescomplemented
97M16×C227—22no—
98C2×C8×C227—22no—
99C2×C8×C227C2311yescharacteristic
100C2×C8×C227C2311yescharacteristic
101C2×C8×C227C2311yescharacteristic
102C2×C8×C227C2311yescharacteristic
103C4⋊C4×C227—21no—
104C4⋊C4×C227—21no—
105C4⋊C4×C227—21no—
106C4⋊C4×C227—21no—
107C4⋊C4×C227—21no—
108C4⋊C4×C227—21no—
109C4⋊C4×C227D411yescharacteristic
110C4⋊C4×C227C2311yescharacteristic
111C4⋊C4×C227C2311yescharacteristic
112C22⋊C4×C227—21no—
113C22⋊C4×C227—21no—
114C22⋊C4×C227—21no—
115C22⋊C4×C227—21no—
116C22⋊C4×C227—21no—
117C22⋊C4×C227—21no—
118C22⋊C4×C227—21no—
119C22⋊C4×C227D411yescharacteristic
120C22⋊C4×C227D411yescharacteristic
121C42×C227—21no—
122C42×C227C2311yescharacteristic
123C23×C227—21no—
124C23×C227—21no—
125C23×C227—42no—
126C23×C227—42no—
127C23×C227—81no—
128C23×C227C2×D411yescharacteristic
129C23×C227C2×D411yescharacteristic
130Q8×C227—21no—
131Q8×C227—21no—
132Q8×C227—21no—
133Q8×C227—21no—
134Q8×C227—21no—
135Q8×C227—21no—
136Q8×C227—21no—
137Q8×C227—42no—
138Q8×C227—42no—
139Q8×C227C2×D412yes—
140D4×C227—22no—
141D4×C227—21no—
142D4×C227—21no—
143D4×C227—21no—
144D4×C227—42no—
145D4×C227—42no—
146D4×C227—81no—
147D4×C227C2×D412yes—
148C2×C4×C227—21no—
149C2×C4×C227—21no—
150C2×C4×C227—21no—
151C2×C4×C227—21no—
152C2×C4×C227—21no—
153C2×C4×C227—21no—
154C2×C4×C227—21no—
155C2×C4×C227—21no—
156C2×C4×C227—21no—
157C2×C4×C227—21no—
158C2×C4×C227—42no—
159C2×C4×C227—42no—
160C2×C4×C227—42no—
161C2×C4×C227—81no—
162C2×C4×C227C2×D411yescharacteristic
163C2×C4×C227C2×D411yescharacteristic
164C2×C4×C227C2×D411yescharacteristic
165C2×C4×C227C2×D411yescharacteristic
166C2×C4×C227C2411yescharacteristic
167C8×C227—21no—
168C8×C227—21no—
169C8×C227—21no—
170C8×C227C2×D412yes—
171C22×C227—21no—
172C22×C227—21no—
173C22×C227—21no—
174C22×C227—22no—
175C22×C227—21no—
176C22×C227—21no—
177C22×C227—41no—
178C22×C227—41no—
179C22×C227—41no—
180C22×C227—41no—
181C22×C227—41no—
182C22×C227—84no—
183C22×C227#13712yescomplemented
184C22×C227C22×D411yescharacteristic
185C4×C227—21no—
186C4×C227—21no—
187C4×C227—22no—
188C4×C227—21no—
189C4×C227—21no—
190C4×C227—41no—
191C4×C227—41no—
192C4×C227—41no—
193C4×C227—41no—
194C4×C227—41no—
195C4×C227C22×D411yescharacteristic
196C4×C227H(5,𝔽2)11yescharacteristic
197C2×C227—22no—
198C2×C227—41no—
199C2×C227—42no—
200C2×C227—81no—
201C2×C227D4211yescharacteristic
202C2×C227C2×#13711yescharacteristic
203C2×C227#57711yescharacteristic
204C227#326611yescharacteristic, direct product factor, Sylow
205#3266C22711yesmaximal, characteristic, direct product factor, Sylow
206C2×#137C2×C22711yescharacteristic, complemented
207C22×SD16C2×C22711yescharacteristic, complemented
208D4×Q8C2×C22711yescharacteristic, complemented
209#546C2×C22711yescharacteristic
210#497C2×C22711yescharacteristic, complemented
211#468C2×C22711yescharacteristic, complemented
212#467C2×C22711yescharacteristic, complemented
213#464C2×C22711yescharacteristic, complemented
214#461C2×C22711yescharacteristic, complemented
215#452C2×C22711yescharacteristic, complemented
216#451C2×C22711yescharacteristic, complemented
217#450C2×C22711yescharacteristic, complemented
218#448C2×C22711yescharacteristic, complemented
219#440C2×C22711yescharacteristic, complemented
220#435C2×C22711yescharacteristic, complemented
221C2×#41C22×C22711yescharacteristic
222C22×Q8C22×C22711yescharacteristic
223C22×Q8C22×C22711yescharacteristic, complemented
224C22×D4C22×C22711yescharacteristic
225#137—24no—
226C2×Q16C22×C22711yescharacteristic, complemented
227C2×Q16C22×C22711yescharacteristic, complemented
228C2×Q16C22×C22711yescharacteristic, complemented
229C2×SD16—24no—
230C2×SD16C22×C22711yescharacteristic
231C2×SD16C22×C22711yescharacteristic
232C2×SD16C22×C22711yescharacteristic
233C2×SD16C22×C22711yescharacteristic
234C2×SD16C22×C22711yescharacteristic
235C2×SD16C22×C22711yescharacteristic
236C2×M16C22×C22711yescharacteristic, complemented
237C22×C8C22×C22711yescharacteristic
238C4⋊Q8—21no—
239C4⋊Q8C22×C22711yescharacteristic, complemented
240#124C22×C22711yescharacteristic
241#123—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8C22×C22711yescharacteristic
245C22⋊Q8C22×C22711yescharacteristic
246C22⋊Q8C22×C22711yescharacteristic, complemented
247#121—21no—
248#121C22×C22711yescharacteristic
249#120—21no—
250C4×Q8C22×C22711yescharacteristic, complemented
251C4×D4—21no—
252C4×D4C22×C22711yescharacteristic
253C4×D4C22×C22711yescharacteristic
254C2×C22⋊C4—21no—
255#107C22×C22711yescharacteristic, complemented
256C4⋊C8C22×C22711yescharacteristic, complemented
257#103C22×C22711yescharacteristic, complemented
258#103C22×C22711yescharacteristic, complemented
259#103C22×C22711yescharacteristic, complemented
260#103C22×C22711yescharacteristic, complemented
261#102C22×C22711yescharacteristic
262#102C22×C22711yescharacteristic
263C22⋊C8C22×C22711yescharacteristic
264C22⋊C8C22×C22711yescharacteristic, complemented
265#97C22×C22711yescharacteristic, complemented
266C24—21no—
267#41—22no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8C23×C22711yescharacteristic
276C2×Q8C23×C22711yescharacteristic
277C2×Q8C23×C22711yescharacteristic
278C2×Q8C23×C22711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4D4×C22711yescharacteristic
285C2×D4C23×C22711yescharacteristic
286C2×D4C23×C22711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4C23×C22711yescharacteristic
292C22×C4C23×C22711yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—41no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—22no—
300SD16—41no—
301SD16D4×C22714yescomplemented
302M16—22no—
303C2×C8—22no—
304C2×C8C23×C22711yescharacteristic
305C2×C8C23×C22711yescharacteristic
306C2×C8C23×C22711yescharacteristic
307C2×C8C23×C22711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4D4×C22711yescharacteristic
315C4⋊C4C23×C22711yescharacteristic
316C4⋊C4C23×C22711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4D4×C22711yescharacteristic
325C22⋊C4D4×C22711yescharacteristic
326C42—21no—
327C42C23×C22711yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×D4×C22711yescharacteristic
334C23C2×D4×C22711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—42no—
343Q8—42no—
344Q8C2×D4×C22712yes—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4C2×D4×C22712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4C2×D4×C22711yescharacteristic
368C2×C4C2×D4×C22711yescharacteristic
369C2×C4C2×D4×C22711yescharacteristic
370C2×C4C2×D4×C22711yescharacteristic
371C2×C4C24×C22711yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8C2×D4×C22712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—21no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#137×C22712yescomplemented
389C22C22×D4×C22711yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4C22×D4×C22711yescharacteristic
401C4H(5,𝔽2)×C22711yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D42×C22711yescharacteristic
407C2C2×#137×C22711yescharacteristic
408C2#577×C22711yescharacteristic
subgroupG/H
center, socleC22×C227C22×D4
commutator, Frattini, cosocleC2×C4C24×C227
prime radical#546×C227C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C227 ◁ C4×D4×C227 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#137×C227C2semidirect
2. C22×SD16×C227C2semidirect
3. D4×Q8×C227C2semidirect
5. #497×C227C2semidirect
6. #468×C227C2semidirect
7. #467×C227C2semidirect
8. #464×C227C2semidirect
9. #461×C227C2semidirect
10. #452×C227C2semidirect
11. #451×C227C2semidirect
12. #450×C227C2semidirect
13. #448×C227C2semidirect
14. #440×C227C2semidirect
15. #435×C227C2semidirect
18. C22×Q8×C227C22semidirect
21. C2×Q16×C227C22semidirect
22. C2×Q16×C227C22semidirect
23. C2×Q16×C227C22semidirect
31. C2×M16×C227C22semidirect
34. C4⋊Q8×C227C22semidirect
41. C22⋊Q8×C227C22semidirect
45. C4×Q8×C227C22semidirect
50. #107×C227C22semidirect
51. C4⋊C8×C227C22semidirect
52. #103×C227C22semidirect
53. #103×C227C22semidirect
54. #103×C227C22semidirect
55. #103×C227C22semidirect
59. C22⋊C8×C227C22semidirect
60. #97×C227C22semidirect
96. SD16×C227D4semidirect
183. C22×C227#137semidirect
204. C227#3266direct (N × H)
205. #3266C227direct (N × H)
206. C2×#137C2×C227semidirect
207. C22×SD16C2×C227semidirect
208. D4×Q8C2×C227semidirect
210. #497C2×C227semidirect
211. #468C2×C227semidirect
212. #467C2×C227semidirect
213. #464C2×C227semidirect
214. #461C2×C227semidirect
215. #452C2×C227semidirect
216. #451C2×C227semidirect
217. #450C2×C227semidirect
218. #448C2×C227semidirect
219. #440C2×C227semidirect
220. #435C2×C227semidirect
223. C22×Q8C22×C227semidirect
226. C2×Q16C22×C227semidirect
227. C2×Q16C22×C227semidirect
228. C2×Q16C22×C227semidirect
236. C2×M16C22×C227semidirect
239. C4⋊Q8C22×C227semidirect
246. C22⋊Q8C22×C227semidirect
250. C4×Q8C22×C227semidirect
255. #107C22×C227semidirect
256. C4⋊C8C22×C227semidirect
257. #103C22×C227semidirect
258. #103C22×C227semidirect
259. #103C22×C227semidirect
260. #103C22×C227semidirect
264. C22⋊C8C22×C227semidirect
265. #97C22×C227semidirect
301. SD16D4×C227semidirect
388. C22#137×C227semidirect

Maximal subgroups

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

subgroupindex
C2×#137×C2272
C22×SD16×C2272
D4×Q8×C2272
#546×C2272
#497×C2272
#468×C2272
#467×C2272
#464×C2272
#461×C2272
#452×C2272
#451×C2272
#450×C2272
#448×C2272
#440×C2272
#435×C2272
#3266227