Finite Group Databasebeta

← #30901

Subgroups — #30901

Order 256.

416 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic
3#2547C211yesmaximal, characteristic
4#2451C211yesmaximal, characteristic, complemented
5#2066C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic
8#2024C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic
11#1971C211yesmaximal, characteristic, complemented
12#1917C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic
15#1864C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#489—24no—
26#464—24no—
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#97C2211yescharacteristic
40#398—21no—
41#397—21no—
42#397—21no—
43#395—21no—
44#391—21no—
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#387—21no—
52#380—21no—
53#380—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—41no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84#124—22no—
85#124—22no—
86#124—41no—
87#124—41no—
88C4×Q8—22no—
89C4×Q8—41no—
90C4×D4—41no—
91C4×D4—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4D411yescharacteristic
99C2×C4⋊C4D411yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#106—41no—
114C4⋊C8—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—41no—
119#103—41no—
120#103—41no—
121#103—41no—
122#102—22no—
123#102—41no—
124#102D414yes—
125#97—22no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—81no—
159C2×Q8#4112yes—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4C2×D412yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2411yescharacteristic
183Q16—44no—
184Q16—44no—
185SD16—44no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8C2×D412yes—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4C2×D412yes—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—42no—
226C23—22no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23C22×D411yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—41no—
237Q8—41no—
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—84no—
247D4—22no—
248D4—22no—
249D4—41no—
250D4—41no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
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—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—84no—
290C2×C4#13712yes—
291C2×C4C22×D411yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4H(5,𝔽2)11yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C42−1+411yescharacteristic
297C8—22no—
298C8—41no—
299C8—41no—
300C8—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22#53511yescharacteristic
309C22D4211yescharacteristic
310C22#54811yescharacteristic
311C22C2×#13711yescharacteristic
312C22#57511yescharacteristic
313C22#57711yescharacteristic
314C22#57811yescharacteristic
315C4—22no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—42no—
321C4—41no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4#53711yescharacteristic
329C4#53811yescharacteristic
330C4#54611yescharacteristic
331C4#56211yescharacteristic
332C2—82no—
333C2#267611yescharacteristic
334C2#321511yescharacteristic
335C2#322511yescharacteristic
336C2#326711yescharacteristic
337C2#327611yescharacteristic
338C2#336911yescharacteristic
339C2#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
4. #2451C2semidirect
5. #2066C2semidirect
6. #2043C2semidirect
8. #2024C2semidirect
9. #2016C2semidirect
11. #1971C2semidirect
12. #1917C2semidirect
13. #1878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4642
#25472
#24512
#20662
#20432
#20422
#20242
#20162
#20152
#19712
#19172
#18782
#18772
#18642