Finite Group Databasebeta

← C2×#71330 #10776331

Subgroups — C2×#71330 #10776331

Order 768.

958 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5672C211yesmaximal, characteristic
2C2×C3×#497C211yesmaximal, characteristic
3C2×#5081C212yesmaximal, complemented
4C2×#4540C211yesmaximal, characteristic
5#71330C218yesmaximal, direct product factor
6C2×#4465C212yesmaximal, complemented
7C2×#1660—31nomaximal, Sylow
8C22×C3×Q16—21no—
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#124C2211yescharacteristic
12C2×#944—21no—
13C2×#935—21no—
14#5672—22no—
15#5672—24no—
16#5672C2214yes—
17C2×#879—22no—
18C2×#878—22no—
19C2×#870C2212yes—
20C3×#497—22no—
21C3×#497C2214yes—
22C2×C3×#97C2211yescharacteristic
23#5081—24no—
24#5081C2218yescomplemented
25C2×#808—22no—
26C2×#807—22no—
27C2×#803C2212yes—
28#4540C2214yes—
29#4465C2218yescomplemented
30C2×#563—31no—
31C2×#501—32no—
32C2×#497D311yescharacteristic
33#1660—38no—
34C2×#330—32no—
35C2×#329—31no—
36C22×C3×Q8—21no—
37C22×C3×Q8D411yescharacteristic
38C22×C3×D4D411yescharacteristic
39C23×Dic3—21no—
40C22×Dic6—21no—
41C2×C3×Q16—24no—
42C2×C3×Q16—44no—
43C2×C3×SD16—24no—
44C2×C3×SD16—44no—
45C22×C3×C8—21no—
46C3×C4⋊Q8—22no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8C2312yes—
49C3×#124—22no—
50C3×#124D412yes—
51C3×#124C2312yes—
52C2×C3×C4⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C42C2311yescharacteristic
55C2×#217—22no—
56C2×#217—21no—
57#944—24no—
58#944—24no—
59#935—24no—
60#935—24no—
61C2×C4⋊Dic3D412yes—
62C2×C4⋊Dic3D412yes—
63C2×Dic3⋊C4—22no—
64C2×Dic3⋊C4—22no—
65C2×Dic3⋊C4—22no—
66C2×C4×Dic3—21no—
67C22×C3⋊C8—22no—
68#879—28no—
69#879—28no—
70#878—28no—
71#878—28no—
72#870—22no—
73#870—22no—
74#870D414yes—
75#870C2314yes—
76C3×#97—21no—
77C3×#97C2312yes—
78#808—28no—
79#807—28no—
80#803—22no—
81#803C2314yes—
82C22×Q16—21no—
83C22×SD16—21no—
84#563—34no—
85#563—64no—
86#563—62no—
87C2×C4⋊Q8—61no—
88C2×C4⋊Q8C2×D311yescharacteristic
89C2×#126—62no—
90C2×#125—32no—
91C2×#124C2×D311yescharacteristic
92C2×#123—61no—
93C2×C22⋊Q8—62no—
94#501—38no—
95#501—64no—
96#497—22no—
97#497C2×D314yes—
98C2×#107—62no—
99C2×#106—62no—
100C2×#97—32no—
101C2×#97C2×D311yescharacteristic
102#330—38no—
103#329—34no—
104C24×C3—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—22no—
107C2×C3×Q8—42no—
108C2×C3×Q8—42no—
109C2×C3×Q8—42no—
110C2×C3×Q8—41no—
111C2×C3×Q8C2×D412yes—
112C2×C3×D4—21no—
113C2×C3×D4—42no—
114C2×C3×D4C2×D412yes—
115C22×C3×C4—21no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—22no—
120C22×Dic3—22no—
121C22×Dic3—24no—
122C22×Dic3—82no—
123C2×Dic6—24no—
124C2×Dic6—82no—
125C3×Q16—48no—
126C3×SD16—48no—
127C2×C3×C8—24no—
128C2×C3×C8—22no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—42no—
131C3×C22⋊C4—24no—
132C3×C22⋊C4—24no—
133C3×C42C2×D412yes—
134C3×C42C2×D411yescharacteristic
135C3×C42C2411yescharacteristic
136#217—28no—
137#217—24no—
138C4⋊Dic3—22no—
139C4⋊Dic3—22no—
140C4⋊Dic3C2×D414yes—
141C4⋊Dic3C2×D414yes—
142Dic3⋊C4—28no—
143Dic3⋊C4—24no—
144Dic3⋊C4—24no—
145Dic3⋊C4—24no—
146Dic3⋊C4—24no—
147C4×Dic3—24no—
148C2×C3⋊C8—28no—
149C2×C3⋊C8—24no—
150C22×Q8—21no—
151C22×Q8—61no—
152C22×Q8#6711yescharacteristic
153C22×D4#6711yescharacteristic
154C23×C4—61no—
155C2×Q16—24no—
156C2×Q16—44no—
157C2×SD16—24no—
158C2×SD16—44no—
159C22×C8—21no—
160C22×C8—62no—
161C4⋊Q8—22no—
162C4⋊Q8—21no—
163C4⋊Q8—64no—
164C4⋊Q8—64no—
165C4⋊Q8C22×D312yes—
166#126—68no—
167#126—68no—
168#125—34no—
169#125—34no—
170#125—62no—
171#125—62no—
172#124—22no—
173#124#6712yes—
174#124C22×D312yes—
175#123—64no—
176#123—64no—
177C22⋊Q8—68no—
178C22⋊Q8—68no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—32no—
181C2×C4⋊C4—32no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—62no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4—61no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—68no—
191#106—68no—
192#97—21no—
193#97—34no—
194#97—62no—
195#97C22×D312yes—
196C23×C3—22no—
197C23×C3—42no—
198C23×C3—44no—
199C23×C3#12011yescharacteristic
200C3×Q8—42no—
201C3×Q8—42no—
202C3×Q8—42no—
203C3×Q8—81no—
204C3×Q8—82no—
205C3×D4—42no—
206C3×D4—81no—
207C2×C3×C4—22no—
208C2×C3×C4—24no—
209C2×C3×C4—22no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—82no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12012yes—
215C2×C3×C4C22×D412yes—
216C2×C3×C4C22×D411yescharacteristic
217C2×Dic3—24no—
218C2×Dic3—24no—
219C2×Dic3—44no—
220C2×Dic3—44no—
221C2×Dic3—84no—
222Dic6—84no—
223C3×C8—24no—
224C3⋊C8—28no—
225C24—21no—
226C2×Q8—21no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—41no—
232C2×Q8—64no—
233C2×Q8—242no—
234C2×Q8C2×#6712yes—
235C2×D4—21no—
236C2×D4—42no—
237C2×D4C2×#6712yes—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—64no—
243C22×C4—242no—
244C22×C4D3×D412yes—
245C22×C4C2×#6711yescharacteristic
246Q16—48no—
247SD16—48no—
248C2×C8—24no—
249C2×C8—22no—
250C2×C8—68no—
251C2×C8—64no—
252C4⋊C4—24no—
253C4⋊C4—34no—
254C4⋊C4—34no—
255C4⋊C4—42no—
256C4⋊C4—68no—
257C4⋊C4—62no—
258C4⋊C4—62no—
259C4⋊C4—64no—
260C4⋊C4—64no—
261C4⋊C4—64no—
262C4⋊C4—64no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—68no—
266C22⋊C4—64no—
267C42—64no—
268C42D3×D412yes—
269C42C2×#6711yescharacteristic
270C42C23×D311yescharacteristic
271C22×C3—42no—
272C22×C3—44no—
273C22×C3—84no—
274C22×C3#45312yes—
275C22×C3#45614yes—
276C22×C3C2×#12011yescharacteristic
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—42no—
282C3×C4—82no—
283Dic3—44no—
284Dic3—44no—
285C23—22no—
286C23—44no—
287C23—42no—
288C23#93711yescharacteristic
289Q8—42no—
290Q8—42no—
291Q8—42no—
292Q8—81no—
293Q8—82no—
294Q8—244no—
295D4—42no—
296D4—81no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—64no—
303C2×C4—64no—
304C2×C4—82no—
305C2×C4—124no—
306C2×C4—124no—
307C2×C4—244no—
308C2×C4#93711yescharacteristic
309C2×C4#93712yes—
310C2×C4C2×D3×D412yes—
311C2×C4C22×#6711yescharacteristic
312C8—24no—
313C8—68no—
314C2×C3—82no—
315C2×C3#166014yescomplemented
316C2×C3C2×#45311yescharacteristic
317C2×C3C2×#45612yes—
318C22—44no—
319C22—42no—
320C22—84no—
321C22#507712yes—
322C22#520514yes—
323C22C2×#93711yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—82no—
330C4—124no—
331C4—124no—
332C3C2×#166011yescharacteristic, complemented, Sylow
333C2—82no—
334C2#7133014yesdirect product factor
335C2C2×#507711yescharacteristic
336C2C2×#520512yes—
subgroupG/H
centerC23#937
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC2×C3×#497C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenterC2×#497D3
hypocenterC3C2×#1660
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
3. C2×#5081C2semidirect
5. #71330C2direct (N × H)
6. C2×#4465C2semidirect
24. #5081C22semidirect
29. #4465C22semidirect
315. C2×C3#1660semidirect
332. C3C2×#1660semidirect
334. C2#71330direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56722
C2×C3×#4972
C2×#50812
C2×#45402
#713302
C2×#44652
C2×#16603