Finite Group Databasebeta

← #60493

Subgroups — #60493

Order 256.

1015 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544C211yesmaximal, characteristic, complemented
2#3428C211yesmaximal, characteristic
3#3084C214yesmaximal, complemented
4#3083C214yesmaximal, complemented
5#3082C214yesmaximal, complemented
6#3081C214yesmaximal, complemented
7#3080C214yesmaximal, complemented
8#3079C214yesmaximal, complemented
9#2911C211yesmaximal, characteristic, complemented
10C2×#428C211yesmaximal, characteristic, complemented
11#2905C211yesmaximal, characteristic, complemented
12#2896C211yesmaximal, characteristic
13C2×#417C211yesmaximal, characteristic
14C22×#41C2211yescharacteristic
15C22×M16C2211yescharacteristic
16C22×M16C2211yescharacteristic
17#544—22no—
18#544—22no—
19#544C2214yes—
20#544C2214yescomplemented
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2212yescomplemented
23C2×#125—21no—
24C2×#125C2212yescomplemented
25C2×C22⋊Q8—22no—
26#520—21no—
27#520—21no—
28#520C2212yes—
29#520C2212yes—
30#519C2212yes—
31#518—22no—
32#518C2212yes—
33C4×#41—22no—
34C4×#41C2214yes—
35C2×#117—22no—
36C2×#117C2211yescharacteristic
37C2×#117C2211yescharacteristic
38C2×#117C2211yescharacteristic
39C22×C4⋊C4C2211yescharacteristic
40#479C2218yescomplemented
41#479C2218yescomplemented
42#478C2218yescomplemented
43#478C2218yescomplemented
44#477C2218yescomplemented
45#476C2218yescomplemented
46#475C2218yescomplemented
47#474C2218yescomplemented
48#428C2214yes—
49#428C2212yescomplemented
50#428C2214yescomplemented
51#428C2212yescomplemented
52#428C2212yescomplemented
53#428C2212yescomplemented
54#427C2214yes—
55C2×#107C2212yescomplemented
56C2×#106C2212yescomplemented
57#424C2214yes—
58#424C2214yescomplemented
59#423C2214yescomplemented
60#419C2212yes—
61#418C2212yes—
62#417C2212yes—
63#417C2214yes—
64#417C2212yes—
65#417C2214yes—
66#416C2214yes—
67C2×#103C2212yes—
68C2×#102C2212yes—
69C2×#41—21no—
70C2×#41—21no—
71C2×#41—24no—
72C2×#41Q818yescomplemented
73C2×#41C2312yes—
74C2×#41C2312yes—
75C22×Q8—22no—
76C22×Q8C2311yescharacteristic
77C22×D4—21no—
78C22×D4C2311yescharacteristic
79C23×C4—21no—
80C23×C4—21no—
81C23×C4C2311yescharacteristic
82C2×M16—24no—
83C2×M16—24no—
84C2×M16C2312yes—
85C2×M16C2312yes—
86C2×M16C2312yes—
87C2×M16C2312yes—
88C22×C8C2312yes—
89C22×C8C2312yes—
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92C4⋊Q8—24no—
93C4⋊Q8C2318yes—
94#125—24no—
95#125—24no—
96#125—24no—
97#125C2318yes—
98C22⋊Q8—28no—
99C22⋊Q8—28no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×Q8C2314yes—
103C4×Q8C2314yes—
104C4×D4—24no—
105C4×D4—24no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—24no—
109C4×D4—24no—
110C4×D4C2314yes—
111C4×D4C2314yes—
112#117—21no—
113#117—21no—
114#117—21no—
115#117—21no—
116#117—22no—
117#117—22no—
118#117—24no—
119#117—24no—
120#117D412yes—
121#117D412yes—
122#117C2312yes—
123#117C2312yes—
124#117C2312yes—
125#117C2312yes—
126#117C2312yes—
127#117C2312yes—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4D411yescharacteristic
137C2×C4⋊C4D411yescharacteristic
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2312yes—
140C2×C4⋊C4C2312yes—
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4D412yes—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—24no—
152C2×C42C2312yes—
153#107C2318yes—
154#107C2314yes—
155#107C2314yes—
156#106C2318yes—
157#106C2314yes—
158#106C2314yes—
159C4⋊C8C2318yes—
160C4⋊C8C2318yes—
161#103C2318yes—
162#103C2314yes—
163#103C2314yes—
164#102C2318yes—
165#102C2314yes—
166#102C2314yes—
167C24—21no—
168C24C2×D411yescharacteristic
169#41—28no—
170#41—28no—
171#41—24no—
172#41—24no—
173#41C2×Q818yes—
174#41C2×Q818yescomplemented
175C2×Q8—21no—
176C2×Q8—28no—
177C2×Q8—22no—
178C2×Q8—84no—
179C2×Q8C2×Q814yescomplemented
180C2×Q8C2412yes—
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—22no—
186C2×D4C2×Q814yescomplemented
187C2×D4C2412yes—
188C22×C4—21no—
189C22×C4—28no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—82no—
202C22×C4C2×D412yes—
203C22×C4C2×D412yes—
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×Q814yes—
208C22×C4#4114yes—
209C22×C4C2411yescharacteristic
210C22×C4C2411yescharacteristic
211C22×C4C2411yescharacteristic
212M16—24no—
213M16—24no—
214M16—24no—
215M16—24no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8C2414yes—
219C2×C8C2414yes—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—28no—
228C4⋊C4—28no—
229C4⋊C4—28no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4C2×D412yes—
233C4⋊C4C2×D412yes—
234C4⋊C4C2×D412yes—
235C4⋊C4C2×D412yes—
236C4⋊C4C2414yes—
237C4⋊C4C2412yes—
238C4⋊C4C2412yes—
239C4⋊C4C2412yes—
240C4⋊C4C2412yes—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4C2×D414yes—
248C22⋊C4C2×D414yes—
249C42—22no—
250C42—24no—
251C42—24no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42C2×D414yes—
256C42C2414yes—
257C23—22no—
258C23—24no—
259C23—41no—
260C23—41no—
261C23—81no—
262C23—81no—
263C23C22⋊Q814yes—
264C23C22×D411yescharacteristic
265C23C22×D411yescharacteristic
266C23C22×D411yescharacteristic
267Q8—24no—
268Q8—22no—
269Q8—88no—
270Q8C22×Q814yes—
271D4—28no—
272D4—28no—
273D4—24no—
274D4—22no—
275D4C22×Q814yes—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4C22⋊Q814yes—
297C2×C4C22⋊Q814yes—
298C2×C4C22⋊Q814yes—
299C2×C4C22×D414yes—
300C2×C4C22×D414yes—
301C2×C4C22×D411yescharacteristic
302C2×C4C22×D411yescharacteristic
303C2×C4C22×D411yescharacteristic
304C2×C4C22×Q812yes—
305C2×C4C22×Q812yes—
306C2×C4C22×Q812yes—
307C2×C4C2×#4112yes—
308C2×C4C2×#4112yes—
309C2×C4C2×#4112yes—
310C2×C4C2511yescharacteristic
311C8—24no—
312C8—24no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—24no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—84no—
324C22C2×C22⋊Q812yes—
325C22C2×C22⋊Q812yes—
326C22C2×C22⋊Q812yes—
327C22C23×D411yescharacteristic
328C4—24no—
329C4—24no—
330C4—24no—
331C4—44no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4C2×C22⋊Q812yes—
337C4C2×C22⋊Q812yes—
338C4C2×C22⋊Q812yes—
339C4C23×Q811yescharacteristic
340C4C22×#4111yescharacteristic
341C2—22no—
342C2—22no—
343C2—22no—
344C2—44no—
345C2C22×C22⋊Q811yescharacteristic
346C2#358511yescharacteristic
347C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC22×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#544C2semidirect
3. #3084C2semidirect
4. #3083C2semidirect
5. #3082C2semidirect
6. #3081C2semidirect
7. #3080C2semidirect
8. #3079C2semidirect
9. #2911C2semidirect
10. C2×#428C2semidirect
11. #2905C2semidirect
20. #544C22semidirect
22. C2×C4⋊Q8C22semidirect
24. C2×#125C22semidirect
40. #479C22semidirect
41. #479C22semidirect
42. #478C22semidirect
43. #478C22semidirect
44. #477C22semidirect
45. #476C22semidirect
46. #475C22semidirect
47. #474C22semidirect
49. #428C22semidirect
50. #428C22semidirect
51. #428C22semidirect
52. #428C22semidirect
53. #428C22semidirect
55. C2×#107C22semidirect
56. C2×#106C22semidirect
58. #424C22semidirect
59. #423C22semidirect
72. C2×#41Q8semidirect
174. #41C2×Q8semidirect
179. C2×Q8C2×Q8semidirect
186. C2×D4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#34282
#30842
#30832
#30822
#30812
#30802
#30792
#29112
C2×#4282
#29052
#28962
C2×#4172