Finite Group Databasebeta

← C2×#3388 #61362

Subgroups — C2×#3388 #61362

Order 256.

911 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#557C211yesmaximal, characteristic, complemented
2C2×#554C211yesmaximal, characteristic
3#3388C2116yesmaximal, direct product factor
4C2×#501C211yesmaximal, characteristic, complemented
5C2×#501C211yesmaximal, characteristic, complemented
6C2×#499C211yesmaximal, characteristic, complemented
7C2×#479C211yesmaximal, characteristic, complemented
8C2×#479C211yesmaximal, characteristic, complemented
9C2×#475C211yesmaximal, characteristic, complemented
10C2×#478C211yesmaximal, characteristic
11C2×#478C211yesmaximal, characteristic
12C2×#476C211yesmaximal, characteristic
13C2×#446C211yesmaximal, characteristic, complemented
14C2×#440C211yesmaximal, characteristic
15C2×#440C211yesmaximal, characteristic
16C2×C4×SD16C211yesmaximal, characteristic
17C22×SD16C2211yescharacteristic
18#557—24no—
19#557C2218yescomplemented
20#554—24no—
21#554C2218yes—
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25C2×#125—21no—
26C2×#125—21no—
27C2×#125C2211yescharacteristic
28C2×#125C2211yescharacteristic
29C2×#125C2211yescharacteristic
30C2×#125C2211yescharacteristic
31C2×C22⋊Q8—21no—
32C2×C22⋊Q8—21no—
33C2×C22⋊Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8C2211yescharacteristic
36C2×C4×Q8C2211yescharacteristic
37C2×C4×Q8C2211yescharacteristic
38C2×C4×Q8C2211yescharacteristic
39C2×C4×D4C2211yescharacteristic
40C2×C4×D4C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic
42C22×C4⋊C4—21no—
43#501C2218yescomplemented
44#501C2218yescomplemented
45#499C2218yescomplemented
46#479C2218yescomplemented
47#479C2218yescomplemented
48#478C2218yes—
49#478C2218yes—
50#476C2218yes—
51#475C2218yescomplemented
52#446C2218yescomplemented
53#440C2218yes—
54#440C2218yes—
55C4×SD16C2218yes—
56C2×#107C2211yescharacteristic
57C2×#107C2211yescharacteristic
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#106C2211yescharacteristic
61C2×#106C2211yescharacteristic
62C2×#106C2211yescharacteristic
63C2×#106C2211yescharacteristic
64C2×#106C2211yescharacteristic
65C2×C4⋊C8C2211yescharacteristic
66C2×C4⋊C8C2211yescharacteristic
67C2×C4⋊C8C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#103C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×#97C2211yescharacteristic
75C2×#97C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C22×Q8—21no—
78C22×Q8—21no—
79C22×Q8D411yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×D4C2311yescharacteristic
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C2×SD16Q818yescomplemented
86C2×SD16C2314yes—
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C4⋊Q8—28no—
92C4⋊Q8—22no—
93C4⋊Q8—22no—
94C4⋊Q8C2314yes—
95C4⋊Q8C2314yes—
96#125—28no—
97#125—28no—
98#125—22no—
99#125—22no—
100#125—22no—
101#125—22no—
102#125C2314yes—
103#125C2314yes—
104#125C2314yes—
105#125C2314yes—
106C22⋊Q8—28no—
107C22⋊Q8—28no—
108C22⋊Q8—28no—
109C4×Q8—28no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×Q8D414yes—
114C4×Q8C2314yes—
115C4×Q8C2314yes—
116C4×Q8C2314yes—
117C4×Q8C2314yes—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4C2314yes—
122C4×D4C2314yes—
123C4×D4C2314yes—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4D411yescharacteristic
139C2×C4⋊C4D411yescharacteristic
140C2×C4⋊C4D411yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2311yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42C2311yescharacteristic
154C2×C42C2311yescharacteristic
155C2×C42C2311yescharacteristic
156#107C2314yes—
157#107C2314yes—
158#107C2314yes—
159#107C2314yes—
160#106C2314yes—
161#106C2314yes—
162#106C2314yes—
163#106C2314yes—
164#106C2314yes—
165C4⋊C8C2314yes—
166C4⋊C8C2314yes—
167C4⋊C8C2314yes—
168#103C2314yes—
169#103C2314yes—
170#103C2314yes—
171#102C2314yes—
172#102C2314yes—
173#102C2314yes—
174#97C2314yes—
175#97C2314yes—
176C4×C8C2314yes—
177C24—21no—
178C2×Q8—21no—
179C2×Q8—24no—
180C2×Q8—24no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—44no—
184C2×Q8—82no—
185C2×Q8C2×D414yes—
186C2×Q8C2×Q814yes—
187C2×Q8C2412yes—
188C2×D4—21no—
189C2×D4—22no—
190C2×D4C2×Q814yes—
191C2×D4C2412yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—44no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2411yescharacteristic
212SD16C2×Q8116yescomplemented
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8C2×Q814yes—
217C2×C8C2412yes—
218C2×C8C2412yes—
219C2×C8C2412yes—
220C2×C8C2412yes—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4C2×D414yes—
241C4⋊C4C2×D414yes—
242C4⋊C4C2×D414yes—
243C4⋊C4C2412yes—
244C4⋊C4C2412yes—
245C4⋊C4C2412yes—
246C4⋊C4C2412yes—
247C4⋊C4C2412yes—
248C4⋊C4C2412yes—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C42—24no—
253C42—24no—
254C42—24no—
255C42C2×D412yes—
256C42C2×D412yes—
257C42C2×D412yes—
258C42C2412yes—
259C42C2412yes—
260C42C2412yes—
261C23—24no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C22×D411yescharacteristic
266Q8—24no—
267Q8—22no—
268Q8—44no—
269Q8—48no—
270Q8—84no—
271Q8C22×Q814yes—
272D4—24no—
273D4—22no—
274D4C22×Q814yes—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—48no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4C22×D412yes—
299C2×C4C22×D412yes—
300C2×C4C22×D412yes—
301C2×C4C22×D412yes—
302C2×C4C22×D412yes—
303C2×C4C22×D412yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C22×Q812yes—
306C2×C42−1+412yes—
307C2×C4C2511yescharacteristic
308C8—22no—
309C8—22no—
310C8—22no—
311C8C22×Q814yes—
312C22—24no—
313C22—48no—
314C22—42no—
315C22—42no—
316C22D4×Q812yes—
317C22#57512yes—
318C22#57712yes—
319C22C23×D411yescharacteristic
320C4—24no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4D4×Q812yes—
334C4C23×Q811yescharacteristic
335C4C2×2−1+411yescharacteristic
336C2—44no—
337C2#338814yesdirect product factor
338C2C2×D4×Q811yescharacteristic
339C2C2×#57511yescharacteristic
340C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#557C2semidirect
3. #3388C2direct (N × H)
4. C2×#501C2semidirect
5. C2×#501C2semidirect
6. C2×#499C2semidirect
7. C2×#479C2semidirect
8. C2×#479C2semidirect
9. C2×#475C2semidirect
13. C2×#446C2semidirect
19. #557C22semidirect
43. #501C22semidirect
44. #501C22semidirect
45. #499C22semidirect
46. #479C22semidirect
47. #479C22semidirect
51. #475C22semidirect
52. #446C22semidirect
85. C2×SD16Q8semidirect
212. SD16C2×Q8semidirect
337. C2#3388direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#5542
#33882
C2×#5012
C2×#4992
C2×#4792
C2×#4752
C2×#4782
C2×#4762
C2×#4462
C2×#4402
C2×C4×SD162