Finite Group Databasebeta

← #10697060

Subgroups — #10697060

Order 768.

498 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 103 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×#5078C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic
3#79269C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic, complemented
5#70524C211yesmaximal, characteristic, complemented
6#70498C211yesmaximal, characteristic
7#67739C211yesmaximal, characteristic, complemented
8#10164—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5093—21no—
15#5089—22no—
16#5088—21no—
17#5078—22no—
18#5078—22no—
19#4966—22no—
20C2×#807—22no—
21C4×#208C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4842—22no—
24#4735—22no—
25#4694—21no—
26#4680—21no—
27C3×#333—22no—
28#4554—22no—
29#4483—22no—
30#3227—31no—
31C2×#500—31no—
32#3141—31no—
33#1997—31no—
34#1548D311yescharacteristic, complemented
35#1492—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C2×#235—21no—
39C22×Dic6—21no—
40C2×C3×M16—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48#930—21no—
49#930—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D412yes—
52C2×#208D412yes—
53C22×C3⋊C8—21no—
54#875—21no—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8D412yes—
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68#821—24no—
69#817—24no—
70#810—24no—
71#809—24no—
72#807—28no—
73#803—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yes—
77#532—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#500—62no—
81#500—62no—
82#498—61no—
83#489—62no—
84#487—61no—
85#486—61no—
86#486—61no—
87#481—62no—
88#476—62no—
89#431C2×D311yescharacteristic
90C2×#107—62no—
91#423C2×D311yescharacteristic
92#417—62no—
93C4×M16—31no—
94C2×C4×C8—31no—
95#343—62no—
96#333—22no—
97D4⋊C8—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—41no—
100C2×C3×Q8—82no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×#67—41no—
106C22×Dic3—22no—
107#235—42no—
108#235—42no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×C4×D3—41no—
113C2×Dic6—21no—
114C2×Dic6—41no—
115C2×Dic6—41no—
116C2×Dic6—82no—
117C3×M16—42no—
118C2×C3×C8—22no—
119C2×C3×C8—41no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—41no—
130C4⋊Dic3—22no—
131C4⋊Dic3—21no—
132C4⋊Dic3—22no—
133C4⋊Dic3—21no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136#208—22no—
137#208—22no—
138C2×C3⋊C8—24no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C2×#41—61no—
142C22×Q8—21no—
143C22×Q8—61no—
144C2×M16—21no—
145C2×M16—32no—
146C22×C8—61no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—31no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8D1212yes—
156#127—31no—
157#124—61no—
158#121—61no—
159#121—61no—
160C4×D4—61no—
161C4×D4—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—32no—
165C2×C4⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#107—68no—
168#106—64no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8#6712yes—
172#103—64no—
173#102—64no—
174#102—64no—
175C22⋊C8—21no—
176#97—61no—
177C4×C8—21no—
178C4×C8—32no—
179C4×C8—61no—
180C4×C8—61no—
181C23×C3#12111yescharacteristic
182C22×D3—41no—
183C3×Q8—82no—
184C3×Q8—82no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—42no—
188C2×C3×C4—82no—
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12711yescharacteristic
195C2×C3×C4#13512yes—
196C2×C3×C4#13712yes—
197#67—82no—
198C2×Dic3—42no—
199C2×Dic3—41no—
200C2×Dic3—82no—
201D12—42no—
202D12—42no—
203D12—42no—
204C4×D3—81no—
205C4×D3—81no—
206Dic6—42no—
207Dic6—82no—
208Dic6—82no—
209C3×C8—42no—
210C3×C8—81no—
211C3⋊C8—24no—
212C3⋊C8—42no—
213#41—122no—
214#41—122no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—61no—
218C2×Q8—82no—
219C2×Q8—121no—
220C2×Q8—121no—
221C2×Q8—242no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—121no—
226C22×C4—41no—
227C22×C4—62no—
228C22×C4—121no—
229C22×C4#23511yescharacteristic
230C22×C4D3×D411yescharacteristic
231C22×C4D3×D411yescharacteristic
232M16—42no—
233M16—62no—
234M16—62no—
235C2×C8—22no—
236C2×C8—41no—
237C2×C8—64no—
238C2×C8—62no—
239C2×C8—62no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—62no—
245C4⋊C4—61no—
246C4⋊C4—62no—
247C4⋊C4—61no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—121no—
251C22⋊C4—121no—
252C42C2×D1211yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C42C2×#6711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#39211yescharacteristic
259C22×C3#46412yes—
260C2×D3—81no—
261C2×D3—81no—
262C2×D3—81no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—82no—
266C3×C4#45212yes—
267C3×C4#45612yes—
268C3×C4#47212yes—
269C3×C4#49511yescharacteristic
270C3×C4#49711yescharacteristic
271Dic3—81no—
272Dic3—82no—
273C23—121no—
274C23#88411yescharacteristic
275Q8—82no—
276Q8—82no—
277Q8—122no—
278Q8—242no—
279Q8—242no—
280D4—122no—
281D4—122no—
282D4—122no—
283D4—242no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—82no—
288C2×C4—121no—
289C2×C4—122no—
290C2×C4—241no—
291C2×C4—241no—
292C2×C4—242no—
293C2×C4#88211yescharacteristic
294C2×C4#88211yescharacteristic
295C2×C4#88411yescharacteristic
296C2×C4#91612yes—
297C2×C4#91812yes—
298C2×C4#93011yescharacteristic
299C2×C4#93711yescharacteristic
300C2×C4#94011yescharacteristic
301C8—42no—
302C8—64no—
303C8—81no—
304C8—122no—
305C2×C3—22no—
306C2×C3#200511yescharacteristic
307C2×C3#201511yescharacteristic
308C2×C3#201811yescharacteristic
309D3—161no—
310C22—21no—
311C22—21no—
312C22—241no—
313C22—241no—
314C22—241no—
315C22#479812yes—
316C22#496011yescharacteristic
317C4—21no—
318C4—21no—
319C4—82no—
320C4—241no—
321C4—242no—
322C4#474912yes—
323C4#480812yes—
324C4#516811yescharacteristic
325C4#519011yescharacteristic
326C4#520512yes—
327C3#1016411yescharacteristic, complemented, Sylow
328C2—22no—
329C2—481no—
330C2#7009011yescharacteristic
331C2#7027811yescharacteristic
332C2#7034611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1548D3
hypocenterC3#10164
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. C2×#5078C2semidirect
4. C3×#1548C2semidirect
5. #70524C2semidirect
7. #67739C2semidirect
34. #1548D3semidirect
327. C3#10164semidirect

Maximal subgroups

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

subgroupindex
C2×#50782
#831992
#792692
C3×#15482
#705242
#704982
#677392
#101643