Finite Group Databasebeta

← #61303

Subgroups — #61303

Order 256.

920 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3480C211yesmaximal, characteristic, complemented
2Q8×#41C211yesmaximal, characteristic, complemented
3#3365C213yesmaximal, complemented
4#3364C213yesmaximal, complemented
5#3363C211yesmaximal, characteristic, complemented
6#3362C211yesmaximal, characteristic, complemented
7#3361C213yesmaximal, complemented
8#3360C213yesmaximal, complemented
9#3359C211yesmaximal, characteristic, complemented
10#3358C211yesmaximal, characteristic, complemented
11#3131C213yesmaximal, complemented
12#3076C213yesmaximal, complemented
13#3035C213yesmaximal, complemented
14C2×C8×Q8C211yesmaximal, characteristic, complemented
15C4×#135C213yesmaximal, complemented
16C2×#135C2213yes—
17#559C2213yes—
18Q82C2211yescharacteristic
19#557C2211yescharacteristic
20#557C2213yes—
21#556C2213yes—
22#554—23no—
23#554C2213yes—
24#553—23no—
25#550C2211yescharacteristic
26D4×Q8—21no—
27D4×Q8C2211yescharacteristic
28#548—21no—
29#533—23no—
30#533C2213yes—
31#532C2213yes—
32#529—23no—
33C4×#41—21no—
34C4×#41C2213yes—
35C4×#41C2213yes—
36C2×C4×Q8—21no—
37C2×C4×Q8C2211yescharacteristic
38#488C2216yes—
39#488C2216yes—
40#487C2213yes—
41#487C2213yes—
42#486C2213yes—
43#486C2213yes—
44#479C2216yes—
45#478C2216yes—
46#477C2213yes—
47#476C2213yes—
48#475C2213yes—
49#474C2213yes—
50#464C2216yes—
51#463C2216yes—
52#462C2213yes—
53#461C2213yes—
54#460C2213yes—
55#459C2213yes—
56C8×Q8C2211yescharacteristic
57C8×Q8C2211yescharacteristic
58C8×Q8C2213yes—
59C8×Q8C2213yes—
60C4×Q16C2213yes—
61C4×Q16C2213yes—
62C4×SD16C2213yes—
63C4×SD16C2213yes—
64C4×SD16C2213yes—
65C4×SD16C2213yes—
66C4×D8C2213yes—
67C4×D8C2213yes—
68#427C2213yes—
69C2×C4⋊C8C2213yes—
70#416C2216yes—
71#416C2213yes—
72#416C2211yescharacteristic
73C2×C4×C8C2213yes—
74C2×#41—21no—
75C2×#41C2313yes—
76C2×#41C2311yescharacteristic
77C22×Q8—21no—
78C22×Q8D411yescharacteristic
79#135—212no—
80C2×Q16C2313yes—
81C2×SD16C2313yes—
82C2×SD16C2313yes—
83C2×D8C2313yes—
84C22×C8C2313yes—
85C22×C8C2311yescharacteristic
86C4⋊Q8—23no—
87C4⋊Q8—23no—
88C4⋊Q8C2313yes—
89C4⋊Q8C2313yes—
90C4⋊Q8C2313yes—
91#127C2313yes—
92#126—26no—
93#125—23no—
94#125—26no—
95#125C2316yes—
96#124—23no—
97#124C2316yes—
98#123—26no—
99C22⋊Q8—23no—
100C22⋊Q8—23no—
101C22⋊Q8—23no—
102C22⋊Q8—23no—
103C22⋊Q8—23no—
104C22⋊Q8—26no—
105#121—23no—
106#121—23no—
107#121—23no—
108C4×Q8—21no—
109C4×Q8—21no—
110C4×Q8—23no—
111C4×Q8—23no—
112C4×Q8—23no—
113C4×Q8D411yescharacteristic
114C4×Q8D413yes—
115C4×Q8C2313yes—
116C4×Q8C2313yes—
117C4×Q8C2313yes—
118C4×Q8C2313yes—
119C4×Q8C2311yescharacteristic
120C4×Q8C2313yes—
121C4×D4—23no—
122C4×D4—23no—
123C4×D4—23no—
124C4×D4—23no—
125C4×D4—23no—
126C4×D4—23no—
127C4×D4—23no—
128C4×D4C2313yes—
129C4×D4C2313yes—
130C4×D4C2313yes—
131C4×D4C2313yes—
132#117—23no—
133#117—23no—
134#117—23no—
135#117C2311yescharacteristic
136#117C2313yes—
137C2×C4⋊C4—23no—
138C2×C4⋊C4D413yes—
139C2×C42—23no—
140C2×C42—23no—
141C2×C42C2313yes—
142#107C2313yes—
143#107C2313yes—
144#106C2313yes—
145#106C2313yes—
146C4⋊C8C2313yes—
147C4⋊C8C2313yes—
148C4⋊C8C2316yes—
149#103C2311yescharacteristic
150#103C2311yescharacteristic
151#103C2313yes—
152#103C2313yes—
153#103C2316yes—
154#103C2316yes—
155#102C2311yescharacteristic
156#102C2311yescharacteristic
157#102C2313yes—
158#102C2313yes—
159#102C2316yes—
160#102C2316yes—
161C4×C8C2313yes—
162C4×C8C2313yes—
163C4×C8C2313yes—
164C4×C8C2313yes—
165#41—22no—
166#41—26no—
167#41—26no—
168#41—44no—
169#41#4114yescomplemented
170C2×Q8—24no—
171C2×Q8—23no—
172C2×Q8—23no—
173C2×Q8—44no—
174C2×Q8C2×D413yes—
175C2×Q8C2×D411yescharacteristic
176C2×Q8C2413yes—
177C2×Q8C2411yescharacteristic
178C2×D4—21no—
179C2×D4—23no—
180C2×D4—23no—
181C2×D4—23no—
182C2×D4C2413yes—
183C2×D4C2411yescharacteristic
184C22×C4—21no—
185C22×C4—23no—
186C22×C4—23no—
187C22×C4—23no—
188C22×C4—23no—
189C22×C4—23no—
190C22×C4C2×D413yes—
191C22×C4C2×D413yes—
192C22×C4C2411yescharacteristic
193Q16—26no—
194SD16—26no—
195SD16—26no—
196D8—26no—
197C2×C8—26no—
198C2×C8—22no—
199C2×C8C2411yescharacteristic
200C2×C8C2411yescharacteristic
201C2×C8C2413yes—
202C2×C8C2413yes—
203C4⋊C4—26no—
204C4⋊C4—23no—
205C4⋊C4—23no—
206C4⋊C4—23no—
207C4⋊C4—23no—
208C4⋊C4—23no—
209C4⋊C4—23no—
210C4⋊C4—23no—
211C4⋊C4—23no—
212C4⋊C4—26no—
213C4⋊C4C2×D413yes—
214C4⋊C4C2×D413yes—
215C4⋊C4C2×D416yes—
216C4⋊C4C2413yes—
217C4⋊C4C2413yes—
218C4⋊C4C2411yescharacteristic
219C4⋊C4C2411yescharacteristic
220C22⋊C4—21no—
221C22⋊C4—23no—
222C22⋊C4—26no—
223C22⋊C4—23no—
224C22⋊C4—23no—
225C22⋊C4—23no—
226C22⋊C4—23no—
227C22⋊C4—26no—
228C42—23no—
229C42—23no—
230C42—23no—
231C42—23no—
232C42—21no—
233C42—23no—
234C42C2×D413yes—
235C42C2×D413yes—
236C42C2413yes—
237C42C2413yes—
238C23—21no—
239C23—23no—
240C23C22×D411yescharacteristic
241Q8—26no—
242Q8—21no—
243Q8—23no—
244Q8—23no—
245Q8—46no—
246Q8—83no—
247Q8#13514yescomplemented
248Q8C2×#4112yes—
249D4—24no—
250D4—23no—
251D4—21no—
252D4—23no—
253D4—46no—
254D4—46no—
255D4—83no—
256D4C2×#4112yes—
257C2×C4—21no—
258C2×C4—24no—
259C2×C4—26no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—23no—
263C2×C4—23no—
264C2×C4—23no—
265C2×C4—43no—
266C2×C4—46no—
267C2×C4—46no—
268C2×C4—46no—
269C2×C4—83no—
270C2×C4—83no—
271C2×C4C22×D413yes—
272C2×C4C22×D413yes—
273C2×C4C22×D413yes—
274C2×C4C22×D413yes—
275C2×C4C22×D411yescharacteristic
276C2×C4C2×#4112yes—
277C2×C42−1+412yes—
278C2×C4C2511yescharacteristic
279C8—23no—
280C8—23no—
281C8—21no—
282C8—21no—
283C22—21no—
284C22—21no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22—43no—
289C22—43no—
290C22—43no—
291C22#54812yes—
292C22C23×D411yescharacteristic
293C4—23no—
294C4—23no—
295C4—22no—
296C4—23no—
297C4—41no—
298C4—43no—
299C4—43no—
300C4—41no—
301C4—43no—
302C4#54812yes—
303C4C2×#13516yes—
304C4C22×#4111yescharacteristic
305C4C2×2−1+411yescharacteristic
306C2—22no—
307C2—42no—
308C2—83no—
309C2C2×#54811yescharacteristic
310C2C22×#13511yescharacteristic
311C2#358411yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutator, Frattini, cosocleC2×C4C25
socleC22C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3480C2semidirect
2. Q8×#41C2semidirect
3. #3365C2semidirect
4. #3364C2semidirect
5. #3363C2semidirect
6. #3362C2semidirect
7. #3361C2semidirect
8. #3360C2semidirect
9. #3359C2semidirect
10. #3358C2semidirect
11. #3131C2semidirect
12. #3076C2semidirect
13. #3035C2semidirect
14. C2×C8×Q8C2semidirect
15. C4×#135C2semidirect
169. #41#41semidirect
247. Q8#135semidirect

Maximal subgroups

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

subgroupindex
#34802
Q8×#412
#33652
#33642
#33632
#33622
#33612
#33602
#33592
#33582
#31312
#30762
#30352
C2×C8×Q82
C4×#1352