Finite Group Databasebeta

← #25916

Subgroups — #25916

Order 256.

526 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544C211yesmaximal, characteristic, complemented
2C2×#489C211yesmaximal, characteristic, complemented
3C2×#488C211yesmaximal, characteristic, complemented
4#2620C211yesmaximal, characteristic
5#2026C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1992C211yesmaximal, characteristic, complemented
10#1990C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic
12#1874C211yesmaximal, characteristic
13#1873C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16#544—24no—
17#544—24no—
18C2×C4⋊Q8—21no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125—21no—
22C2×#125C2211yescharacteristic
23C2×#125C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8—21no—
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32#489—24no—
33#488—24no—
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#396—21no—
47#396—21no—
48#394—21no—
49#394—21no—
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#384C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8—21no—
65C22×Q8C2311yescharacteristic
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C4⋊Q8—28no—
75C4⋊Q8—22no—
76C4⋊Q8—22no—
77C4⋊Q8—22no—
78C4⋊Q8D414yescomplemented
79#125—28no—
80#125—22no—
81#125—22no—
82#125—22no—
83#125D414yescomplemented
84#124—22no—
85#124—22no—
86C22⋊Q8—28no—
87C22⋊Q8—28no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90#121—22no—
91#121—22no—
92#117—22no—
93#117—22no—
94#117—22no—
95#117—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4D411yescharacteristic
124C2×C22⋊C4D411yescharacteristic
125C2×C42—21no—
126C2×C42C2311yescharacteristic
127C2×C42C2311yescharacteristic
128#103—22no—
129#103—22no—
130#103—22no—
131#103—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138#97—22no—
139C4×C8—22no—
140#95—21no—
141#95—21no—
142C24—21no—
143C24C2×D411yescharacteristic
144C2×Q8—24no—
145C2×Q8—24no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—42no—
163C2×D4—81no—
164C2×D4—81no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—24no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4#4112yes—
185C22×C4#4112yes—
186C22×C4C2411yescharacteristic
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4C2×D412yes—
220C4⋊C4C2×D412yes—
221C4⋊C4C2×D412yes—
222C4⋊C4C2×D412yes—
223C22⋊C4—24no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C42—24no—
233C42—21no—
234C42—21no—
235C42C2×D412yes—
236C42C2×D412yes—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23#12412yes—
251C23C22×D411yescharacteristic
252Q8—41no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—84no—
257Q8—84no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—84no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—84no—
301C2×C4#12012yes—
302C2×C4#12012yes—
303C2×C4#12412yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C22×D411yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C42−1+411yescharacteristic
309C2×C42−1+411yescharacteristic
310C8—41no—
311C8—41no—
312C8—41no—
313C8—41no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C2×#12011yescharacteristic
329C22C2×#12411yescharacteristic
330C22#53611yescharacteristic
331C22#57611yescharacteristic
332C22#57711yescharacteristic
333C22#57711yescharacteristic
334C22#57811yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—41no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—41no—
343C4—42no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4#54811yescharacteristic
348C4#54811yescharacteristic
349C4#54811yescharacteristic
350C4#54811yescharacteristic
351C2—42no—
352C2—161no—
353C2#243611yescharacteristic
354C2#301211yescharacteristic
355C2#301311yescharacteristic
356C2#313611yescharacteristic
357C2#313711yescharacteristic
358C2#324311yescharacteristic
359C2#324411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#544C2semidirect
2. C2×#489C2semidirect
3. C2×#488C2semidirect
5. #2026C2semidirect
6. #2025C2semidirect
7. #2024C2semidirect
8. #2023C2semidirect
9. #1992C2semidirect
10. #1990C2semidirect
13. #1873C2semidirect
14. #1872C2semidirect
15. #1750C2semidirect
78. C4⋊Q8D4semidirect
83. #125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4892
C2×#4882
#26202
#20262
#20252
#20242
#20232
#19922
#19902
#18752
#18742
#18732
#18722
#17502