Finite Group Databasebeta

← #3316×C227 #44397272

Subgroups — #3316×C227 #44397272

Order 29056.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#554×C227C211yesmaximal, characteristic, complemented
2#548×C227C211yesmaximal, characteristic
3#489×C227C211yesmaximal, characteristic, complemented
4#483×C227C211yesmaximal, characteristic, complemented
5#481×C227C211yesmaximal, characteristic, complemented
6#475×C227C211yesmaximal, characteristic, complemented
7#470×C227C211yesmaximal, characteristic, complemented
8#465×C227C211yesmaximal, characteristic, complemented
9#464×C227C211yesmaximal, characteristic, complemented
10#452×C227C211yesmaximal, characteristic, complemented
11#451×C227C211yesmaximal, characteristic, complemented
12#441×C227C211yesmaximal, characteristic, complemented
13#435×C227C211yesmaximal, characteristic, complemented
14#417×C227C211yesmaximal, characteristic, complemented
15C2×#103×C227C211yesmaximal, characteristic, complemented
16C2×#41×C227C2211yescharacteristic
17C22×Q8×C227C2211yescharacteristic
18C2×Q16×C227C2211yescharacteristic
19C2×Q16×C227C2211yescharacteristic
20C2×SD16×C227C2211yescharacteristic
21C2×M16×C227C2211yescharacteristic
22C22×C8×C227C2211yescharacteristic
23C4⋊Q8×C227C2211yescharacteristic
24#125×C227—21no—
25#124×C227—21no—
26#124×C227C2211yescharacteristic
27C22⋊Q8×C227—21no—
28C22⋊Q8×C227—21no—
29C22⋊Q8×C227—21no—
30C22⋊Q8×C227C2211yescharacteristic
31C22⋊Q8×C227C2211yescharacteristic
32C22⋊Q8×C227C2211yescharacteristic
33#121×C227—21no—
34#121×C227C2211yescharacteristic
35C4×Q8×C227C2211yescharacteristic
36C4×Q8×C227C2211yescharacteristic
37C4×D4×C227—21no—
38C4×D4×C227—21no—
39C4×D4×C227C2211yescharacteristic
40C2×C4⋊C4×C227C2211yescharacteristic
41C2×C4⋊C4×C227C2211yescharacteristic
42#106×C227C2211yescharacteristic
43#106×C227C2211yescharacteristic
44#106×C227C2211yescharacteristic
45C4⋊C8×C227C2211yescharacteristic
46#103×C227C2211yescharacteristic
47#103×C227C2211yescharacteristic
48#103×C227C2211yescharacteristic
49#103×C227C2211yescharacteristic
50#103×C227C2211yescharacteristic
51#103×C227C2211yescharacteristic
52#103×C227C2211yescharacteristic
53#102×C227C2211yescharacteristic
54#102×C227C2211yescharacteristic
55#102×C227C2211yescharacteristic
56C22⋊C8×C227C2211yescharacteristic
57C22⋊C8×C227C2211yescharacteristic
58#97×C227C2211yescharacteristic
59#41×C227—22no—
60#41×C227—41no—
61C2×Q8×C227—21no—
62C2×Q8×C227—21no—
63C2×Q8×C227—22no—
64C2×Q8×C227—41no—
65C2×Q8×C227C2311yescharacteristic
66C2×Q8×C227C2311yescharacteristic
67C2×Q8×C227C2311yescharacteristic
68C2×D4×C227—21no—
69C2×D4×C227—21no—
70C2×D4×C227D411yescharacteristic
71C2×D4×C227C2311yescharacteristic
72C22×C4×C227—21no—
73C22×C4×C227—21no—
74C22×C4×C227—21no—
75C22×C4×C227—21no—
76C22×C4×C227C2311yescharacteristic
77C22×C4×C227C2311yescharacteristic
78Q16×C227—22no—
79Q16×C227—41no—
80SD16×C227—41no—
81M16×C227—41no—
82C2×C8×C227—41no—
83C2×C8×C227C2311yescharacteristic
84C2×C8×C227C2311yescharacteristic
85C2×C8×C227C2311yescharacteristic
86C2×C8×C227C2311yescharacteristic
87C4⋊C4×C227—21no—
88C4⋊C4×C227—21no—
89C4⋊C4×C227—21no—
90C4⋊C4×C227—21no—
91C4⋊C4×C227—21no—
92C4⋊C4×C227—21no—
93C4⋊C4×C227—21no—
94C4⋊C4×C227—21no—
95C4⋊C4×C227D411yescharacteristic
96C4⋊C4×C227C2311yescharacteristic
97C4⋊C4×C227C2311yescharacteristic
98C4⋊C4×C227C2311yescharacteristic
99C4⋊C4×C227C2311yescharacteristic
100C22⋊C4×C227—21no—
101C22⋊C4×C227—21no—
102C22⋊C4×C227—21no—
103C22⋊C4×C227—21no—
104C22⋊C4×C227—21no—
105C22⋊C4×C227—21no—
106C22⋊C4×C227D411yescharacteristic
107C22⋊C4×C227D411yescharacteristic
108C42×C227—21no—
109C42×C227—21no—
110C42×C227C2311yescharacteristic
111C23×C227—21no—
112C23×C227C2×D411yescharacteristic
113C23×C227C2×D411yescharacteristic
114Q8×C227—21no—
115Q8×C227—21no—
116Q8×C227—21no—
117Q8×C227—21no—
118Q8×C227—21no—
119Q8×C227—42no—
120Q8×C227—81no—
121Q8×C227#4112yes—
122D4×C227—22no—
123D4×C227—21no—
124D4×C227—21no—
125D4×C227—42no—
126D4×C227—42no—
127C2×C4×C227—21no—
128C2×C4×C227—21no—
129C2×C4×C227—21no—
130C2×C4×C227—21no—
131C2×C4×C227—21no—
132C2×C4×C227—21no—
133C2×C4×C227—21no—
134C2×C4×C227—21no—
135C2×C4×C227—21no—
136C2×C4×C227—21no—
137C2×C4×C227—21no—
138C2×C4×C227—42no—
139C2×C4×C227—42no—
140C2×C4×C227—81no—
141C2×C4×C227—81no—
142C2×C4×C227C2×D411yescharacteristic
143C2×C4×C227C2×D411yescharacteristic
144C2×C4×C227C2×D411yescharacteristic
145C2×C4×C227C2×D411yescharacteristic
146C2×C4×C227C2411yescharacteristic
147C8×C227—21no—
148C8×C227—21no—
149C8×C227—21no—
150C8×C227—21no—
151C22×C227—21no—
152C22×C227—21no—
153C22×C227—21no—
154C22×C227—21no—
155C22×C227—21no—
156C22×C227—41no—
157C22×C227—41no—
158C22×C227—41no—
159C22×C227#13712yes—
160C22×C227C22×D411yescharacteristic
161C4×C227—21no—
162C4×C227—21no—
163C4×C227—22no—
164C4×C227—21no—
165C4×C227—21no—
166C4×C227—41no—
167C4×C227—41no—
168C4×C227—41no—
169C4×C227—41no—
170C4×C227—41no—
171C4×C227—41no—
172C4×C227C2×#4111yescharacteristic
173C4×C227H(5,𝔽2)11yescharacteristic
174C2×C227—22no—
175C2×C227—41no—
176C2×C227—81no—
177C2×C227#54611yescharacteristic
178C2×C227C2×#13711yescharacteristic
179C2×C227#57711yescharacteristic
180C227#331611yescharacteristic, direct product factor, Sylow
181#3316C22711yesmaximal, characteristic, direct product factor, Sylow
182#554C2×C22711yescharacteristic, complemented
183#548C2×C22711yescharacteristic
184#489C2×C22711yescharacteristic, complemented
185#483C2×C22711yescharacteristic, complemented
186#481C2×C22711yescharacteristic, complemented
187#475C2×C22711yescharacteristic, complemented
188#470C2×C22711yescharacteristic, complemented
189#465C2×C22711yescharacteristic, complemented
190#464C2×C22711yescharacteristic, complemented
191#452C2×C22711yescharacteristic, complemented
192#451C2×C22711yescharacteristic, complemented
193#441C2×C22711yescharacteristic, complemented
194#435C2×C22711yescharacteristic, complemented
195#417C2×C22711yescharacteristic, complemented
196C2×#103C2×C22711yescharacteristic, complemented
197C2×#41C22×C22711yescharacteristic
198C22×Q8C22×C22711yescharacteristic
199C2×Q16C22×C22711yescharacteristic
200C2×Q16C22×C22711yescharacteristic
201C2×SD16C22×C22711yescharacteristic
202C2×M16C22×C22711yescharacteristic
203C22×C8C22×C22711yescharacteristic
204C4⋊Q8C22×C22711yescharacteristic
205#125—21no—
206#124—21no—
207#124C22×C22711yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8C22×C22711yescharacteristic
212C22⋊Q8C22×C22711yescharacteristic
213C22⋊Q8C22×C22711yescharacteristic
214#121—21no—
215#121C22×C22711yescharacteristic
216C4×Q8C22×C22711yescharacteristic
217C4×Q8C22×C22711yescharacteristic
218C4×D4—21no—
219C4×D4—21no—
220C4×D4C22×C22711yescharacteristic
221C2×C4⋊C4C22×C22711yescharacteristic
222C2×C4⋊C4C22×C22711yescharacteristic
223#106C22×C22711yescharacteristic
224#106C22×C22711yescharacteristic
225#106C22×C22711yescharacteristic
226C4⋊C8C22×C22711yescharacteristic
227#103C22×C22711yescharacteristic
228#103C22×C22711yescharacteristic
229#103C22×C22711yescharacteristic
230#103C22×C22711yescharacteristic
231#103C22×C22711yescharacteristic
232#103C22×C22711yescharacteristic
233#103C22×C22711yescharacteristic
234#102C22×C22711yescharacteristic
235#102C22×C22711yescharacteristic
236#102C22×C22711yescharacteristic
237C22⋊C8C22×C22711yescharacteristic
238C22⋊C8C22×C22711yescharacteristic
239#97C22×C22711yescharacteristic
240#41—22no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8C23×C22711yescharacteristic
247C2×Q8C23×C22711yescharacteristic
248C2×Q8C23×C22711yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4D4×C22711yescharacteristic
252C2×D4C23×C22711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4C23×C22711yescharacteristic
258C22×C4C23×C22711yescharacteristic
259Q16—22no—
260Q16—41no—
261SD16—41no—
262M16—41no—
263C2×C8—41no—
264C2×C8C23×C22711yescharacteristic
265C2×C8C23×C22711yescharacteristic
266C2×C8C23×C22711yescharacteristic
267C2×C8C23×C22711yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4D4×C22711yescharacteristic
277C4⋊C4C23×C22711yescharacteristic
278C4⋊C4C23×C22711yescharacteristic
279C4⋊C4C23×C22711yescharacteristic
280C4⋊C4C23×C22711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4D4×C22711yescharacteristic
288C22⋊C4D4×C22711yescharacteristic
289C42—21no—
290C42—21no—
291C42C23×C22711yescharacteristic
292C23—21no—
293C23C2×D4×C22711yescharacteristic
294C23C2×D4×C22711yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—81no—
302Q8#41×C22712yes—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—42no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4C2×D4×C22711yescharacteristic
324C2×C4C2×D4×C22711yescharacteristic
325C2×C4C2×D4×C22711yescharacteristic
326C2×C4C2×D4×C22711yescharacteristic
327C2×C4C24×C22711yescharacteristic
328C8—21no—
329C8—21no—
330C8—21no—
331C8—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#137×C22712yes—
341C22C22×D4×C22711yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4C2×#41×C22711yescharacteristic
354C4H(5,𝔽2)×C22711yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2#546×C22711yescharacteristic
359C2C2×#137×C22711yescharacteristic
360C2#577×C22711yescharacteristic
subgroupG/H
center, socleC22×C227C22×D4
commutator, Frattini, cosocleC2×C4C24×C227
prime radical#548×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. #554×C227C2semidirect
3. #489×C227C2semidirect
4. #483×C227C2semidirect
5. #481×C227C2semidirect
6. #475×C227C2semidirect
7. #470×C227C2semidirect
8. #465×C227C2semidirect
9. #464×C227C2semidirect
10. #452×C227C2semidirect
11. #451×C227C2semidirect
12. #441×C227C2semidirect
13. #435×C227C2semidirect
14. #417×C227C2semidirect
15. C2×#103×C227C2semidirect
180. C227#3316direct (N × H)
181. #3316C227direct (N × H)
182. #554C2×C227semidirect
184. #489C2×C227semidirect
185. #483C2×C227semidirect
186. #481C2×C227semidirect
187. #475C2×C227semidirect
188. #470C2×C227semidirect
189. #465C2×C227semidirect
190. #464C2×C227semidirect
191. #452C2×C227semidirect
192. #451C2×C227semidirect
193. #441C2×C227semidirect
194. #435C2×C227semidirect
195. #417C2×C227semidirect
196. C2×#103C2×C227semidirect

Maximal subgroups

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

subgroupindex
#554×C2272
#548×C2272
#489×C2272
#483×C2272
#481×C2272
#475×C2272
#470×C2272
#465×C2272
#464×C2272
#452×C2272
#451×C2272
#441×C2272
#435×C2272
#417×C2272
C2×#103×C2272
#3316227