Finite Group Databasebeta

← #28326

Subgroups — #28326

Order 256.

523 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic, complemented
2C2×#483C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic
4C2×#452C211yesmaximal, characteristic
5#2621C211yesmaximal, characteristic, complemented
6#2311C211yesmaximal, characteristic
7#2097C211yesmaximal, characteristic, complemented
8#2092C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2013C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1971C211yesmaximal, characteristic, complemented
13#1969C211yesmaximal, characteristic, complemented
14#1968C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#484—24no—
27#483—24no—
28#452—24no—
29#452—24no—
30C2×#107C2211yescharacteristic
31C2×#107C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#398—22no—
47#397—22no—
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#390C2211yescharacteristic
53#390C2211yescharacteristic
54#386—22no—
55#382C2211yescharacteristic
56#380—22no—
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C2×#41—22no—
60C2×#41—22no—
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C22×Q8C2311yescharacteristic
66C22×Q8C2311yescharacteristic
67C22×Q8C2311yescharacteristic
68C22×D4—21no—
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×Q16—22no—
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—41no—
84C2×SD16—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C4⋊Q8—41no—
90C4⋊Q8—41no—
91C22⋊Q8—22no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95#117—41no—
96#117—41no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4D412yes—
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4D412yes—
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115#107—22no—
116#107—22no—
117#106—22no—
118#106—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#102—22no—
126#102—22no—
127C22⋊C8D414yes—
128C22⋊C8D414yes—
129#97—41no—
130C4×C8—41no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95—22no—
136C24—21no—
137C24C2×D411yescharacteristic
138#41—44no—
139#41—44no—
140#41—44no—
141#41—44no—
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—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8#4112yes—
160C2×D4—24no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4#4112yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—24no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—82no—
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D412yes—
189C22×C4C2×D412yes—
190C22×C4C2411yescharacteristic
191Q16—44no—
192Q16—44no—
193SD16—44no—
194SD16—44no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8C2×D412yes—
204C2×C8C2×D412yes—
205C2×C8C2×D412yes—
206C2×C8C2×D412yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—82no—
224C22⋊C4—82no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—41no—
235C23—41no—
236C23—42no—
237C23—42no—
238C23#12712yes—
239C23C22×D411yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252D4—22no—
253D4—22no—
254D4—48no—
255D4—41no—
256D4—41no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—48no—
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—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—84no—
288C2×C4#12712yes—
289C2×C4C22×D411yescharacteristic
290C2×C4C22×D411yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4H(5,𝔽2)11yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C8—22no—
296C8—22no—
297C8—22no—
298C8—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—42no—
303C22—42no—
304C22—44no—
305C22—42no—
306C22—81no—
307C22—81no—
308C22—82no—
309C22#52511yescharacteristic
310C22C2×#12711yescharacteristic
311C22#53411yescharacteristic
312C22#57712yes—
313C22#57812yes—
314C4—22no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4#54611yescharacteristic
325C4#54611yescharacteristic
326C4#54611yescharacteristic
327C4#54611yescharacteristic
328C2—42no—
329C2—82no—
330C2#253911yescharacteristic
331C2#312012yes—
332C2#315411yescharacteristic
333C2#315611yescharacteristic
334C2#320112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
2. C2×#483C2semidirect
5. #2621C2semidirect
7. #2097C2semidirect
8. #2092C2semidirect
9. #2016C2semidirect
10. #2013C2semidirect
11. #1973C2semidirect
12. #1971C2semidirect
13. #1969C2semidirect
14. #1968C2semidirect
15. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4832
C2×#4522
#26212
#23112
#20972
#20922
#20162
#20132
#19732
#19712
#19692
#19682
#17502