Finite Group Databasebeta

← #32353

Subgroups — #32353

Order 256.

331 conjugacy classes of subgroups.

The subgroup lattice has 293 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2563C211yesmaximal, characteristic
2#2508C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic
5#2085C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic
7#2058C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic
11#1926C211yesmaximal, characteristic
12#1916C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic
14#1862C211yesmaximal, characteristic
15#1846C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×C4×C8C2211yescharacteristic
30#400—21no—
31#398—21no—
32#393—21no—
33#393—21no—
34#390C2211yescharacteristic
35#390C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#387—21no—
41#382—21no—
42#382C2211yescharacteristic
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⋊C4D411yescharacteristic
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C42—21no—
90C2×C42C2311yescharacteristic
91C2×C42C2311yescharacteristic
92C2×C42C2311yescharacteristic
93#107—41no—
94#107—41no—
95#107—41no—
96#106—41no—
97#106—41no—
98#106—41no—
99C4⋊C8—22no—
100#102—22no—
101#102—41no—
102#102—41no—
103#102Q814yescomplemented
104C4×C8—22no—
105#95—21no—
106#95—21no—
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⋊C4C2×Q812yes—
178C22⋊C4—42no—
179C22⋊C4—81no—
180C22⋊C4—81no—
181C22⋊C4—81no—
182C22⋊C4—81no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—42no—
190C23—22no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23C22×D411yescharacteristic
198Q8—84no—
199D4—22no—
200D4—22no—
201D4—41no—
202D4—41no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
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—41no—
222C2×C4—41no—
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—84no—
242C2×C4#13512yes—
243C2×C4C22×Q811yescharacteristic
244C2×C4C2×#4111yescharacteristic
245C2×C4H(5,𝔽2)11yescharacteristic
246C2×C4H(5,𝔽2)11yescharacteristic
247C2×C42−1+411yescharacteristic
248C2×C42−1+411yescharacteristic
249C8—22no—
250C8—22no—
251C8—41no—
252C8—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22#53511yescharacteristic
261C22#54711yescharacteristic
262C22D4×Q811yescharacteristic
263C22C2×#13511yescharacteristic
264C22#57511yescharacteristic
265C22#57611yescharacteristic
266C22#57711yescharacteristic
267C4—22no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—41no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4#53911yescharacteristic
281C4#54411yescharacteristic
282C4#55411yescharacteristic
283C4#56311yescharacteristic
284C2—82no—
285C2#272011yescharacteristic
286C2#321111yescharacteristic
287C2#322411yescharacteristic
288C2#334111yescharacteristic
289C2#334811yescharacteristic
290C2#338111yescharacteristic
291C2#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. #2508C2semidirect
3. #2101C2semidirect
5. #2085C2semidirect
7. #2058C2semidirect
8. #2057C2semidirect
9. #1949C2semidirect
12. #1916C2semidirect
15. #1846C2semidirect
103. #102Q8semidirect

Maximal subgroups

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

subgroupindex
#25632
#25082
#21012
#20922
#20852
#20672
#20582
#20572
#19492
#19342
#19262
#19162
#18642
#18622
#18462