Finite Group Databasebeta

← #32373

Subgroups — #32373

Order 256.

332 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2563C211yesmaximal, characteristic
2#2507C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic
5#2085C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic
7#2057C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic
11#1925C211yesmaximal, characteristic
12#1917C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic
14#1864C211yesmaximal, characteristic
15#1847C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#97C2211yescharacteristic
30#398—21no—
31#398—21no—
32#393—21no—
33#393—21no—
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#389—21no—
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#387—21no—
42#384C2211yescharacteristic
43#382C2211yescharacteristic
44#382C2211yescharacteristic
45#380—21no—
46#380—21no—
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C4⋊Q8—41no—
67C4⋊Q8—41no—
68C4×D4—41no—
69C4×D4—41no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C42—21no—
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95#107—41no—
96#107—41no—
97#107—41no—
98#107—41no—
99#106—41no—
100#106—41no—
101C4⋊C8—22no—
102#102—22no—
103#102—41no—
104#102—41no—
105#102Q814yescomplemented
106#97—22no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95D411yescharacteristic
117#95D411yescharacteristic
118C24—21no—
119C2×Q8—42no—
120C2×Q8—81no—
121C2×Q8—81no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4C2×Q812yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2411yescharacteristic
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8C2×Q812yes—
156C2×C8#4112yes—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4C2×Q812yes—
180C22⋊C4—42no—
181C22⋊C4—81no—
182C22⋊C4—81no—
183C22⋊C4—81no—
184C22⋊C4—81no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—81no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23C22×D411yescharacteristic
200Q8—84no—
201D4—22no—
202D4—22no—
203D4—41no—
204D4—41no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—84no—
244C2×C4#13712yes—
245C2×C4C22×Q811yescharacteristic
246C2×C4C2×#4111yescharacteristic
247C2×C4H(5,𝔽2)11yescharacteristic
248C2×C4H(5,𝔽2)11yescharacteristic
249C2×C42−1+411yescharacteristic
250C2×C42−1+411yescharacteristic
251C8—22no—
252C8—22no—
253C8—41no—
254C8—41no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22#53511yescharacteristic
263C22#54711yescharacteristic
264C22D4×Q811yescharacteristic
265C22C2×#13711yescharacteristic
266C22#57511yescharacteristic
267C22#57611yescharacteristic
268C22#57611yescharacteristic
269C4—22no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—42no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4#53911yescharacteristic
283C4#54411yescharacteristic
284C4#55411yescharacteristic
285C4#56311yescharacteristic
286C2—82no—
287C2#272011yescharacteristic
288C2#321511yescharacteristic
289C2#322011yescharacteristic
290C2#334711yescharacteristic
291C2#335711yescharacteristic
292C2#338511yescharacteristic
293C2#338911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2507C2semidirect
3. #2101C2semidirect
5. #2085C2semidirect
7. #2057C2semidirect
8. #2057C2semidirect
9. #1949C2semidirect
12. #1917C2semidirect
15. #1847C2semidirect
105. #102Q8semidirect

Maximal subgroups

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

subgroupindex
#25632
#25072
#21012
#20932
#20852
#20672
#20572
#19492
#19362
#19252
#19172
#18642
#18472