Finite Group Databasebeta

← #22678

Subgroups — #22678

Order 256.

491 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#440C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic
3C2×C4×SD16C211yesmaximal, characteristic
4Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#2349C211yesmaximal, characteristic
6#1943C211yesmaximal, characteristic, complemented
7#1942C211yesmaximal, characteristic, complemented
8#1865C211yesmaximal, characteristic, complemented
9#1864C211yesmaximal, characteristic
10#1863C211yesmaximal, characteristic, complemented
11#1862C211yesmaximal, characteristic
12#1847C211yesmaximal, characteristic, complemented
13#1805C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic
15#1771C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#440—24no—
24#440C418yescomplemented
25C4×SD16—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#97C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#389—21no—
42#389—21no—
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#385—21no—
48#385—21no—
49#384—21no—
50#384C2211yescharacteristic
51#382C2211yescharacteristic
52#380—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4C2211yescharacteristic
55C2×#95—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C2×SD16C2×C414yes—
68C2×SD16D414yes—
69C2×SD16Q814yes—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C4×Q8—22no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8C2×C414yescomplemented
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—41no—
84C4×D4—41no—
85C4×D4C2×C414yes—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#107—22no—
113#107—41no—
114#107C2×C414yescomplemented
115#106—22no—
116#106—41no—
117C4⋊C8—22no—
118#103—22no—
119#103—22no—
120#103C2×C414yescomplemented
121#102—22no—
122#102—22no—
123#102C2×C414yes—
124#97—22no—
125#97C2×C414yescomplemented
126C4×C8—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95D411yescharacteristic
134#95D411yescharacteristic
135C24—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C22×C412yes—
140C2×Q8C2×D412yes—
141C2×Q8C2×Q812yes—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C22×C412yes—
146C2×D4C2×D412yes—
147C2×D4C2×Q812yes—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4—81no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171SD16C4⋊C4116yescomplemented
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8C22×C412yes—
181C2×C8C22×C412yes—
182C2×C8C2×D412yes—
183C2×C8C2×Q812yes—
184C4⋊C4—24no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4C22×C412yes—
206C4⋊C4C22×C412yes—
207C22⋊C4—24no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C42—24no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—42no—
216C42—42no—
217C42C22×C412yes—
218C42#4112yes—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23C22×D411yescharacteristic
227Q8—22no—
228Q8—22no—
229Q8—22no—
230Q8C2×C4⋊C414yes—
231D4—22no—
232D4—22no—
233D4—22no—
234D4C2×C4⋊C414yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4C4×D412yes—
275C2×C4C4×D412yes—
276C2×C4#13712yes—
277C2×C4C23×C411yescharacteristic
278C2×C4C22×D411yescharacteristic
279C2×C4C22×Q811yescharacteristic
280C2×C4C2×#4111yescharacteristic
281C2×C4H(5,𝔽2)11yescharacteristic
282C2×C42−1+411yescharacteristic
283C8—22no—
284C8—41no—
285C8—41no—
286C8C2×C4⋊C414yes—
287C22—24no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22C2×C4×D411yescharacteristic
295C22D4211yescharacteristic
296C22D4×Q811yescharacteristic
297C22C2×#13711yescharacteristic
298C22#57511yescharacteristic
299C22#57611yescharacteristic
300C22#57711yescharacteristic
301C4—24no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—42no—
306C4—42no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4C22×C4⋊C411yescharacteristic
315C4#52011yescharacteristic
316C4#54711yescharacteristic
317C4#55411yescharacteristic
318C2—44no—
319C2D4×C4⋊C411yescharacteristic
320C2C4×#13711yescharacteristic
321C2#294611yescharacteristic
322C2#326611yescharacteristic
323C2#326811yescharacteristic
324C2#338511yescharacteristic
325C2#338811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. Q8×C4⋊C4C2semidirect
6. #1943C2semidirect
7. #1942C2semidirect
8. #1865C2semidirect
10. #1863C2semidirect
12. #1847C2semidirect
13. #1805C2semidirect
15. #1771C2semidirect
24. #440C4semidirect
79. C4×Q8C2×C4semidirect
114. #107C2×C4semidirect
120. #103C2×C4semidirect
125. #97C2×C4semidirect
171. SD16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
C2×C4×SD162
Q8×C4⋊C42
#23492
#19432
#19422
#18652
#18642
#18632
#18622
#18472
#18052
#18022
#17712