Finite Group Databasebeta

← #30644

Subgroups — #30644

Order 256.

397 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#488C211yesmaximal, characteristic, complemented
2C2×#485C211yesmaximal, characteristic, complemented
3#2441C211yesmaximal, characteristic, complemented
4#2426C211yesmaximal, characteristic
5#2094C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2059C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2034C211yesmaximal, characteristic
11#1941C211yesmaximal, characteristic, complemented
12#1923C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic
14#1865C211yesmaximal, characteristic, complemented
15#1803C211yesmaximal, characteristic, complemented
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#488—24no—
23#485—24no—
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#399—21no—
41#396—21no—
42#396—21no—
43#394—21no—
44#390—21no—
45#390C2211yescharacteristic
46#389—21no—
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#388—21no—
51#387—21no—
52#384C2211yescharacteristic
53#380C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69#125—22no—
70#125—41no—
71#124—22no—
72#124—41no—
73C22⋊Q8—22no—
74C22⋊Q8—41no—
75#121—22no—
76#121—41no—
77C4×Q8—22no—
78C4×Q8—41no—
79#117—22no—
80#117—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4D411yescharacteristic
105C2×C22⋊C4D411yescharacteristic
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—22no—
111#106—22no—
112#106—22no—
113#106—41no—
114C4⋊C8—41no—
115#103—22no—
116#103—22no—
117#103—41no—
118#103D414yescomplemented
119#102—22no—
120#102—22no—
121#102—41no—
122#102—41no—
123C22⋊C8—22no—
124C22⋊C8—41no—
125C4×C8—22no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C2×D412yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—81no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2411yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8C2×D412yes—
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—21no—
189C4⋊C4—21no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4C2×D412yes—
200C4⋊C4#4112yes—
201C22⋊C4—22no—
202C22⋊C4—41no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—42no—
216C42—42no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23C22×D411yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—41no—
235Q8—41no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—84no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
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—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4#13512yes—
281C2×C4C22×D411yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4H(5,𝔽2)11yescharacteristic
284C2×C4H(5,𝔽2)11yescharacteristic
285C2×C42−1+411yescharacteristic
286C2×C42−1+411yescharacteristic
287C8—22no—
288C8—41no—
289C8—41no—
290C8—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22#53611yescharacteristic
306C22D4211yescharacteristic
307C22#54611yescharacteristic
308C22C2×#13511yescharacteristic
309C22#57511yescharacteristic
310C22#57711yescharacteristic
311C22#57711yescharacteristic
312C4—22no—
313C4—41no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—42no—
318C4—41no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4#53711yescharacteristic
325C4#54211yescharacteristic
326C4#54711yescharacteristic
327C4#56211yescharacteristic
328C2—81no—
329C2—161no—
330C2#267211yescharacteristic
331C2#323311yescharacteristic
332C2#324411yescharacteristic
333C2#327111yescharacteristic
334C2#328411yescharacteristic
335C2#330111yescharacteristic
336C2#332311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#488C2semidirect
2. C2×#485C2semidirect
3. #2441C2semidirect
5. #2094C2semidirect
6. #2067C2semidirect
7. #2059C2semidirect
8. #2042C2semidirect
9. #2035C2semidirect
11. #1941C2semidirect
12. #1923C2semidirect
14. #1865C2semidirect
15. #1803C2semidirect
118. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
C2×#4852
#24412
#24262
#20942
#20672
#20592
#20422
#20352
#20342
#19412
#19232
#18752
#18652
#18032