Finite Group Databasebeta

← #32087

Subgroups — #32087

Order 256.

349 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2547C211yesmaximal, characteristic
2#2509C211yesmaximal, characteristic, complemented
3#2097C211yesmaximal, characteristic, complemented
4#2068C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic
8#2042C211yesmaximal, characteristic
9#1983C211yesmaximal, characteristic
10#1958C211yesmaximal, characteristic
11#1933C211yesmaximal, characteristic, complemented
12#1927C211yesmaximal, characteristic, complemented
13#1916C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic
15#1862C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#400—21no—
37#400—21no—
38#397—21no—
39#397—21no—
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#389—21no—
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#387—21no—
48#382C2211yescharacteristic
49#380—21no—
50#380—21no—
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C4⋊Q8—41no—
68C4⋊Q8—41no—
69#124—22no—
70#124—22no—
71#124—41no—
72#124—41no—
73C4×D4—41no—
74C4×D4—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97#107—41no—
98#106—41no—
99C4⋊C8—41no—
100#103—22no—
101#103—22no—
102#103—41no—
103#103—41no—
104#103—41no—
105#103—41no—
106#103—41no—
107#103—41no—
108#102—22no—
109#102—22no—
110#102—41no—
111#102—41no—
112C4×C8—41no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4#4112yes—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2411yescharacteristic
162C2×C8—21no—
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—
174C4⋊C4—22no—
175C4⋊C4—21no—
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—41no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4#4112yes—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—81no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23C22×D411yescharacteristic
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220D4—22no—
221D4—22no—
222D4—41no—
223D4—41no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—84no—
263C2×C4#13512yes—
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C4H(5,𝔽2)11yescharacteristic
269C2×C42−1+411yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22#53511yescharacteristic
282C22#54611yescharacteristic
283C22#54811yescharacteristic
284C22C2×#13511yescharacteristic
285C22#57511yescharacteristic
286C22#57711yescharacteristic
287C22#57811yescharacteristic
288C4—22no—
289C4—42no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4#53911yescharacteristic
302C4#54011yescharacteristic
303C4#55311yescharacteristic
304C4#56211yescharacteristic
305C2—82no—
306C2#271111yescharacteristic
307C2#321111yescharacteristic
308C2#322511yescharacteristic
309C2#330211yescharacteristic
310C2#330711yescharacteristic
311C2#336011yescharacteristic
312C2#337711yescharacteristic
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. #2509C2semidirect
3. #2097C2semidirect
4. #2068C2semidirect
5. #2043C2semidirect
6. #2043C2semidirect
11. #1933C2semidirect
12. #1927C2semidirect
13. #1916C2semidirect

Maximal subgroups

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

subgroupindex
#25472
#25092
#20972
#20682
#20432
#20422
#19832
#19582
#19332
#19272
#19162
#18642
#18622