Finite Group Databasebeta

← #30191

Subgroups — #30191

Order 256.

370 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2629C211yesmaximal, characteristic, complemented
2#2620C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#2026C211yesmaximal, characteristic, complemented
10#2025C211yesmaximal, characteristic, complemented
11#1895C211yesmaximal, characteristic
12#1894C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic
14#1892C211yesmaximal, characteristic, complemented
15#1729C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31#398—21no—
32#398—21no—
33#396—21no—
34#396—21no—
35#396—21no—
36#396—21no—
37#394—21no—
38#394—21no—
39#391C2211yescharacteristic
40#390C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#384C2211yescharacteristic
44#384C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C2×M16—22no—
64C2×M16—41no—
65C2×M16—41no—
66C22×C8—22no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73#125—41no—
74#125—41no—
75#124—41no—
76#124—41no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79#121—22no—
80#121—22no—
81#117—41no—
82#117—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4D411yescharacteristic
109C2×C22⋊C4D411yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#103—41no—
113#103—41no—
114#103—41no—
115#103—41no—
116#102—41no—
117#102—41no—
118#102—41no—
119#102—41no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24—21no—
127C24C2×D411yescharacteristic
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—42no—
141C2×D4—81no—
142C2×D4—81no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4#4112yes—
160C22×C4#4112yes—
161C22×C4C2411yescharacteristic
162M16—44no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—44no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—81no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23#12412yes—
227C23C22×D411yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
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—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4#12412yes—
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C2×C4H(5,𝔽2)11yescharacteristic
281C2×C4H(5,𝔽2)11yescharacteristic
282C2×C42−1+411yescharacteristic
283C8—41no—
284C8—41no—
285C8—41no—
286C8—41no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22C2×#12411yescharacteristic
302C22#53411yescharacteristic
303C22#53711yescharacteristic
304C22#57611yescharacteristic
305C22#57711yescharacteristic
306C22#57711yescharacteristic
307C22#57811yescharacteristic
308C4—22no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4#53811yescharacteristic
321C4#53811yescharacteristic
322C4#53911yescharacteristic
323C4#53911yescharacteristic
324C2—42no—
325C2—161no—
326C2#263911yescharacteristic
327C2#313611yescharacteristic
328C2#313711yescharacteristic
329C2#320311yescharacteristic
330C2#320411yescharacteristic
331C2#325711yescharacteristic
332C2#325811yescharacteristic
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. #2629C2semidirect
3. #2100C2semidirect
4. #2099C2semidirect
5. #2072C2semidirect
6. #2071C2semidirect
7. #2042C2semidirect
8. #2042C2semidirect
9. #2026C2semidirect
10. #2025C2semidirect
12. #1894C2semidirect
14. #1892C2semidirect
15. #1729C2semidirect

Maximal subgroups

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

subgroupindex
#26292
#26202
#21002
#20992
#20722
#20712
#20422
#20262
#20252
#18952
#18942
#18932
#18922
#17292