Finite Group Databasebeta

← #22950

Subgroups — #22950

Order 256.

384 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3091C211yesmaximal, characteristic
2#2933C211yesmaximal, characteristic
3#2924C211yesmaximal, characteristic, complemented
4#1946C211yesmaximal, characteristic, complemented
5#1938C211yesmaximal, characteristic, complemented
6#1929C211yesmaximal, characteristic, complemented
7#1922C211yesmaximal, characteristic, complemented
8#1922C211yesmaximal, characteristic, complemented
9#1896C211yesmaximal, characteristic
10#1866C211yesmaximal, characteristic, complemented
11#1855C211yesmaximal, characteristic, complemented
12#1773C211yesmaximal, characteristic, complemented
13#1765C211yesmaximal, characteristic, complemented
14#1749C211yesmaximal, characteristic, complemented
15C2×#339C211yesmaximal, characteristic, complemented
16C2×#131C2211yescharacteristic
17C22×M16C2211yescharacteristic
18#525C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C42C2211yescharacteristic
21#485C411yescharacteristic, complemented
22#485C411yescharacteristic, complemented
23#484C411yescharacteristic, complemented
24#483C411yescharacteristic, complemented
25#482C411yescharacteristic, complemented
26#482C411yescharacteristic, complemented
27#481C411yescharacteristic, complemented
28#480C411yescharacteristic, complemented
29#436—21no—
30#435—21no—
31#435—21no—
32C8×D4—21no—
33#433—21no—
34#433—21no—
35#432—21no—
36#431—21no—
37C2×#108C2211yescharacteristic
38C2×#108C2211yescharacteristic
39#427C2211yescharacteristic
40#427C2211yescharacteristic
41#424C2211yescharacteristic
42#424C2211yescharacteristic
43#424C2211yescharacteristic
44C2×#104C2211yescharacteristic
45C2×#104C2211yescharacteristic
46#416C2211yescharacteristic
47#416C2211yescharacteristic
48#408C2211yescharacteristic
49#407C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52#405C2211yescharacteristic
53#405C2211yescharacteristic
54#405C2211yescharacteristic
55C2×#97C2211yescharacteristic
56C2×#97C2211yescharacteristic
57#341C2211yescharacteristic
58#341C2211yescharacteristic
59#339C2211yescharacteristic
60#339C2211yescharacteristic
61#339C2211yescharacteristic
62#339C2211yescharacteristic
63#339C2211yescharacteristic
64#339C2211yescharacteristic
65#336C2211yescharacteristic
66#336C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70#131—22no—
71#131—41no—
72C2×M16—21no—
73C2×M16—21no—
74C2×M16—22no—
75C2×M16—41no—
76C2×M16C2311yescharacteristic
77C2×M16C2311yescharacteristic
78C2×M16C2311yescharacteristic
79C2×M16C2311yescharacteristic
80C22×C8—21no—
81C22×C8—21no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#123—21no—
87C22⋊Q8C2×C411yescharacteristic
88C22⋊Q8C2×C411yescharacteristic
89#121C2×C411yescharacteristic
90#121C2×C411yescharacteristic
91#120—21no—
92C4×D4—21no—
93C4×D4—21no—
94#117—21no—
95#117—21no—
96#117C2×C411yescharacteristic
97#117C2×C411yescharacteristic
98#117C2311yescharacteristic
99#117C2311yescharacteristic
100#117C2311yescharacteristic
101C2×C4⋊C4C2×C411yescharacteristic
102C2×C4⋊C4C2×C411yescharacteristic
103C2×C22⋊C4—21no—
104C2×C42—21no—
105C2×C42—22no—
106C2×C42—21no—
107C2×C42—41no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#108—22no—
111#108—41no—
112#107C2×C411yescharacteristic, complemented
113#107C2×C411yescharacteristic, complemented
114#107C2×C411yescharacteristic, complemented
115#107C2×C411yescharacteristic, complemented
116#106C2×C411yescharacteristic, complemented
117#106C2×C411yescharacteristic, complemented
118#106C2×C411yescharacteristic, complemented
119#106C2×C411yescharacteristic, complemented
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—21no—
126#104—22no—
127#104—41no—
128#103C2×C411yescharacteristic, complemented
129#103C2×C411yescharacteristic, complemented
130#103C2×C411yescharacteristic, complemented
131#103C2×C411yescharacteristic, complemented
132#102C2×C411yescharacteristic, complemented
133#102C2×C411yescharacteristic, complemented
134#102C2×C411yescharacteristic, complemented
135#102C2×C411yescharacteristic, complemented
136C22⋊C8—21no—
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8C2×C411yescharacteristic
141C22⋊C8C2×C411yescharacteristic
142C22⋊C8C2×C411yescharacteristic
143C22⋊C8C2×C411yescharacteristic
144C22⋊C8D411yescharacteristic
145C22⋊C8D411yescharacteristic
146C22⋊C8D411yescharacteristic
147C22⋊C8D411yescharacteristic
148#97—21no—
149#97—21no—
150#97—21no—
151#97—22no—
152#97—41no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—21no—
156C24#4111yescharacteristic
157#41—21no—
158#41—21no—
159#41—21no—
160#41—21no—
161C2×Q8C22×C411yescharacteristic
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4C22×C411yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4C22×C411yescharacteristic
179C22×C4C22×C411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4#4111yescharacteristic
183C22×C4C2411yescharacteristic
184M16—21no—
185M16—21no—
186M16—21no—
187M16—21no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16—42no—
192M16—42no—
193M16#4112yes—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8C22×C411yescharacteristic
209C2×C8C22×C411yescharacteristic
210C2×C8C22×C411yescharacteristic
211C2×C8C22×C411yescharacteristic
212C2×C8C2×D411yescharacteristic
213C2×C8C2×D411yescharacteristic
214C2×C8C2×D411yescharacteristic
215C2×C8C2×D411yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4C22×C411yescharacteristic
221C4⋊C4C22×C411yescharacteristic
222C4⋊C4C22×C411yescharacteristic
223C4⋊C4C22×C411yescharacteristic
224C4⋊C4C22×C411yescharacteristic
225C4⋊C4C22×C411yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—21no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—21no—
232C22⋊C4—21no—
233C22⋊C4—21no—
234C22⋊C4—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—42no—
244C42#4112yes—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—21no—
249C23—41no—
250C23C4×D411yescharacteristic
251C23C4×D411yescharacteristic
252C23C2×#4111yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—21no—
256D4—21no—
257D4—41no—
258D4—41no—
259D4—42no—
260D4—81no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—48no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4C4×D411yescharacteristic
290C2×C4C4×D411yescharacteristic
291C2×C4C23×C411yescharacteristic
292C2×C4C22×D411yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C42−1+411yescharacteristic
297C8—21no—
298C8—21no—
299C8—21no—
300C8—21no—
301C8—21no—
302C8—21no—
303C8—22no—
304C8—21no—
305C8—21no—
306C8—41no—
307C8—41no—
308C8—41no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22#44412yes—
322C22C2×C4×D411yescharacteristic
323C22#55111yescharacteristic
324C22#55211yescharacteristic
325C4—21no—
326C4—21no—
327C4—42no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4C4×#4111yescharacteristic
337C4#52011yescharacteristic
338C4#54611yescharacteristic
339C4#54711yescharacteristic
340C2—21no—
341C2—21no—
342C2—22no—
343C2—41no—
344C2—81no—
345C2#235911yescharacteristic
346C2C2×#44411yescharacteristic
347C2#295611yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×C4×D4
prime radical#525C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2924C2semidirect
4. #1946C2semidirect
5. #1938C2semidirect
6. #1929C2semidirect
7. #1922C2semidirect
8. #1922C2semidirect
10. #1866C2semidirect
11. #1855C2semidirect
12. #1773C2semidirect
13. #1765C2semidirect
14. #1749C2semidirect
15. C2×#339C2semidirect
21. #485C4semidirect
22. #485C4semidirect
23. #484C4semidirect
24. #483C4semidirect
25. #482C4semidirect
26. #482C4semidirect
27. #481C4semidirect
28. #480C4semidirect
112. #107C2×C4semidirect
113. #107C2×C4semidirect
114. #107C2×C4semidirect
115. #107C2×C4semidirect
116. #106C2×C4semidirect
117. #106C2×C4semidirect
118. #106C2×C4semidirect
119. #106C2×C4semidirect
128. #103C2×C4semidirect
129. #103C2×C4semidirect
130. #103C2×C4semidirect
131. #103C2×C4semidirect
132. #102C2×C4semidirect
133. #102C2×C4semidirect
134. #102C2×C4semidirect
135. #102C2×C4semidirect

Maximal subgroups

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

subgroupindex
#30912
#29332
#29242
#19462
#19382
#19292
#19222
#18962
#18662
#18552
#17732
#17652
#17492
C2×#3392