Finite Group Databasebeta

← #21871

Subgroups — #21871

Order 256.

543 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic, complemented
3#2317C211yesmaximal, characteristic, complemented
4#1993C211yesmaximal, characteristic, complemented
5#1990C211yesmaximal, characteristic, complemented
6#1959C211yesmaximal, characteristic, complemented
7#1958C211yesmaximal, characteristic, complemented
8#1875C211yesmaximal, characteristic
9#1874C211yesmaximal, characteristic
10#1873C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1805C211yesmaximal, characteristic, complemented
13#1804C211yesmaximal, characteristic, complemented
14#1803C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16#542—24no—
17#542C418yescomplemented
18C2×C4⋊Q8C2211yescharacteristic
19C2×#126—21no—
20C2×#124—21no—
21C2×#124C2211yescharacteristic
22C2×#123—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#433—24no—
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
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#389—21no—
47#389—21no—
48#385—21no—
49#385—21no—
50#384C2211yescharacteristic
51#384C2211yescharacteristic
52#382C2211yescharacteristic
53#382C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C4⋊Q8—22no—
74C4⋊Q8C2×C414yescomplemented
75#126—28no—
76#124—28no—
77#124—22no—
78#124C2×C414yescomplemented
79#123—28no—
80C22⋊Q8—22no—
81C22⋊Q8—28no—
82C22⋊Q8C2×C414yes—
83#121—22no—
84#121C2×C414yes—
85C4×Q8—22no—
86C4×Q8C2×C414yescomplemented
87C4×D4—22no—
88C4×D4C2×C414yescomplemented
89#117—41no—
90#117—41no—
91#117C2×C414yes—
92#117D414yes—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42—21no—
121C2×C42C2311yescharacteristic
122C2×C42C2311yescharacteristic
123C4⋊C8—22no—
124C4⋊C8—22no—
125#103—41no—
126#103—41no—
127#103—41no—
128#103—41no—
129#102—41no—
130#102—41no—
131#102—41no—
132#102—41no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135#97—22no—
136C4×C8—22no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141C24—21no—
142C24C2×D411yescharacteristic
143C2×Q8—24no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—81no—
150C2×Q8—81no—
151C2×Q8C22×C412yes—
152C2×D4—24no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—81no—
159C2×D4—81no—
160C2×D4C22×C412yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4C22×C412yes—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D412yes—
185C22×C4C2411yescharacteristic
186C2×C8—21no—
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—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—42no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4C22⋊C414yes—
217C4⋊C4C22⋊C414yes—
218C4⋊C4C22×C412yes—
219C4⋊C4C22×C412yes—
220C22⋊C4—24no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4C22⋊C414yes—
227C22⋊C4C22⋊C414yes—
228C42—24no—
229C42—42no—
230C42C22×C412yes—
231C42C22×C412yes—
232C42C2×D412yes—
233C42C2×D412yes—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊C412yes—
248C23C22×D411yescharacteristic
249Q8—22no—
250Q8—22no—
251Q8—41no—
252Q8—41no—
253Q8—84no—
254D4—22no—
255D4—22no—
256D4—41no—
257D4—41no—
258D4—84no—
259C2×C4—21no—
260C2×C4—21no—
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—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
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—84no—
295C2×C4—84no—
296C2×C4C2×C22⋊C412yes—
297C2×C4C2×C22⋊C412yes—
298C2×C4C2×C22⋊C412yes—
299C2×C4C2×C22⋊C412yes—
300C2×C4C2×C22⋊C412yes—
301C2×C4C23×C411yescharacteristic
302C2×C4C22×D411yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C4H(5,𝔽2)11yescharacteristic
305C2×C42−1+411yescharacteristic
306C2×C42−1+411yescharacteristic
307C8—41no—
308C8—41no—
309C8—41no—
310C8—41no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22C22×C22⋊C411yescharacteristic
326C22#53511yescharacteristic
327C22#53611yescharacteristic
328C22#57611yescharacteristic
329C22#57711yescharacteristic
330C22#57711yescharacteristic
331C22#57811yescharacteristic
332C4—22no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4#52011yescharacteristic
345C4#52011yescharacteristic
346C4#53711yescharacteristic
347C4#53711yescharacteristic
348C2—42no—
349C2—82no—
350C2#231111yescharacteristic
351C2#289711yescharacteristic
352C2#289811yescharacteristic
353C2#322411yescharacteristic
354C2#322511yescharacteristic
355C2#324311yescharacteristic
356C2#324411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×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
2. C2×#433C2semidirect
3. #2317C2semidirect
4. #1993C2semidirect
5. #1990C2semidirect
6. #1959C2semidirect
7. #1958C2semidirect
10. #1873C2semidirect
11. #1872C2semidirect
12. #1805C2semidirect
13. #1804C2semidirect
14. #1803C2semidirect
15. #1802C2semidirect
17. #542C4semidirect
74. C4⋊Q8C2×C4semidirect
78. #124C2×C4semidirect
86. C4×Q8C2×C4semidirect
88. C4×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4332
#23172
#19932
#19902
#19592
#19582
#18752
#18742
#18732
#18722
#18052
#18042
#18032
#18022