Finite Group Databasebeta

← #30350

Subgroups — #30350

Order 256.

335 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2645C211yesmaximal, characteristic, complemented
2#2640C211yesmaximal, characteristic
3#2068C211yesmaximal, characteristic, complemented
4#2067C211yesmaximal, characteristic
5#2057C211yesmaximal, characteristic, complemented
6#2056C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic
8#2042C211yesmaximal, characteristic
9#2024C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic
11#1973C211yesmaximal, characteristic
12#1969C211yesmaximal, characteristic, complemented
13#1935C211yesmaximal, characteristic, complemented
14#1934C211yesmaximal, characteristic
15#1768C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#401—21no—
35#400—21no—
36#400—21no—
37#400—21no—
38#397—21no—
39#397—21no—
40#393—21no—
41#393—21no—
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#389C2211yescharacteristic
52#376C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C2×Q16—22no—
64C2×Q16—41no—
65C2×Q16—41no—
66C2×SD16—22no—
67C2×SD16—41no—
68C2×SD16—41no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75#124—41no—
76#124—41no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
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×C22⋊C4—21no—
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95#107—41no—
96#107—41no—
97C4⋊C8—22no—
98C4⋊C8—22no—
99#103—22no—
100#103—41no—
101#103—41no—
102#103—41no—
103#103—41no—
104#102—22no—
105#102—41no—
106#102—41no—
107#97—41no—
108C4×C8—41no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
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#95D411yescharacteristic
123#95D411yescharacteristic
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×Q8—42no—
140C2×Q8—81no—
141C2×Q8—81no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2411yescharacteristic
162Q16—44no—
163SD16—44no—
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#4112yes—
175C2×C8#4112yes—
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—42no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—81no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23C22×D411yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—84no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—41no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
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—41no—
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×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4H(5,𝔽2)11yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C2×C42−1+411yescharacteristic
268C8—22no—
269C8—22no—
270C8—41no—
271C8—41no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22#53211yescharacteristic
280C22#53511yescharacteristic
281C22#53711yescharacteristic
282C22#57611yescharacteristic
283C22#57711yescharacteristic
284C22#57811yescharacteristic
285C22#57811yescharacteristic
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4#53811yescharacteristic
300C4#53811yescharacteristic
301C4#54211yescharacteristic
302C4#54211yescharacteristic
303C2—161no—
304C2#266211yescharacteristic
305C2#317511yescharacteristic
306C2#317811yescharacteristic
307C2#322311yescharacteristic
308C2#322411yescharacteristic
309C2#325811yescharacteristic
310C2#326211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2645C2semidirect
3. #2068C2semidirect
5. #2057C2semidirect
6. #2056C2semidirect
9. #2024C2semidirect
12. #1969C2semidirect
13. #1935C2semidirect
15. #1768C2semidirect

Maximal subgroups

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

subgroupindex
#26452
#26402
#20682
#20672
#20572
#20562
#20552
#20422
#20242
#20152
#19732
#19692
#19352
#19342
#17682