Finite Group Databasebeta

← #10697058

Subgroups — #10697058

Order 768.

498 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 105 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×C3×#500C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic
3#83199C211yesmaximal, characteristic
4#71790C211yesmaximal, characteristic, complemented
5#71774C211yesmaximal, characteristic, complemented
6#69735C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic, complemented
8#10164—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#500—22no—
13C3×#500—22no—
14C2×C3×#107—22no—
15C2×C3×C4×C8C2211yescharacteristic
16#5093—21no—
17#5089—22no—
18#5088—21no—
19#5088—21no—
20#5084—22no—
21#5080—21no—
22#4968—22no—
23#4966—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4546—22no—
28#4530—22no—
29#4459—22no—
30#3227—31no—
31C2×#500D311yescharacteristic, complemented
32#3141—31no—
33#1997—31no—
34#1548—31no—
35#1492—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D412yes—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C42C2311yescharacteristic
54#930—21no—
55#930—21no—
56C2×#208—21no—
57C2×#208D412yes—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869D411yescharacteristic
63C3×#107—28no—
64C3×C4×C8—21no—
65C3×C4×C8D412yes—
66#830—21no—
67#810—24no—
68#809—24no—
69#809—24no—
70#808—24no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74#803—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#532—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4⋊Q8C2×D311yescharacteristic
80#500—22no—
81#500—22no—
82#498—61no—
83#489—62no—
84#487—61no—
85#486—61no—
86#486—61no—
87#481—62no—
88#476—62no—
89#431—31no—
90C2×#107—22no—
91#423—31no—
92#417—62no—
93C4×M16—31no—
94C2×C4×C8C2×D311yescharacteristic
95#343—62no—
96#333—62no—
97D4⋊C8—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—82no—
103C2×C3×Q8—82no—
104C22×C3×C4—22no—
105C22×C3×C4—41no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—41no—
110#235—42no—
111#235—42no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×C4×D3—41no—
116C2×Dic6—21no—
117C2×C3×C8—24no—
118C2×C3×C8—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135#208—22no—
136#208—22no—
137#208—42no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—41no—
141C2×#41—61no—
142C22×Q8—21no—
143C22×Q8—21no—
144C2×M16—32no—
145C2×M16—61no—
146C22×C8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—31no—
155C4⋊Q8#6712yes—
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—21no—
165C2×C4⋊C4#6712yes—
166C2×C42C22×D311yescharacteristic
167#107—28no—
168#106—64no—
169C4⋊C8—32no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#103—64no—
173#102—64no—
174#102—64no—
175C22⋊C8—61no—
176#97—61no—
177C4×C8—21no—
178C4×C8—61no—
179C4×C8—61no—
180C4×C8#6712yes—
181C23×C3#12111yescharacteristic
182C22×D3—41no—
183C3×Q8—82no—
184C3×Q8—82no—
185C3×Q8—82no—
186C3×Q8—82no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—82no—
192C2×C3×C4—82no—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12711yescharacteristic
199C2×C3×C4#13512yes—
200C2×C3×C4#13712yes—
201#67—82no—
202C2×Dic3—41no—
203D12—42no—
204D12—42no—
205D12—42no—
206C4×D3—81no—
207C4×D3—81no—
208Dic6—42no—
209C3×C8—24no—
210C3⋊C8—42no—
211C3⋊C8—42no—
212C3⋊C8—81no—
213#41—122no—
214#41—122no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—61no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—121no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—121no—
229C22×C4D3×D411yescharacteristic
230C22×C4#23711yescharacteristic
231C22×C4C2×#6711yescharacteristic
232M16—62no—
233M16—62no—
234M16—122no—
235C2×C8—24no—
236C2×C8—22no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8—121no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C22⋊C4—121no—
251C22⋊C4—121no—
252C42D3×D411yescharacteristic
253C42D3×D411yescharacteristic
254C42C2×#6711yescharacteristic
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—82no—
267C3×C4#45212yes—
268C3×C4#45612yes—
269C3×C4#47212yes—
270C3×C4#49511yescharacteristic
271C3×C4#49711yescharacteristic
272Dic3—81no—
273C23—121no—
274C23#93911yescharacteristic
275Q8—82no—
276Q8—82no—
277Q8—82no—
278Q8—82no—
279Q8—122no—
280D4—122no—
281D4—122no—
282D4—122no—
283D4—242no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—121no—
291C2×C4—241no—
292C2×C4—241no—
293C2×C4#91812yes—
294C2×C4#91912yes—
295C2×C4#93711yescharacteristic
296C2×C4#93711yescharacteristic
297C2×C4#93811yescharacteristic
298C2×C4#93911yescharacteristic
299C2×C4#94011yescharacteristic
300C2×C4#95311yescharacteristic
301C8—24no—
302C8—122no—
303C8—122no—
304C8—241no—
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#518512yes—
316C22#522311yescharacteristic
317C4—21no—
318C4—21no—
319C4—82no—
320C4—82no—
321C4—241no—
322C4#509512yes—
323C4#516012yes—
324C4#518712yes—
325C4#519011yescharacteristic
326C4#519111yescharacteristic
327C3#1016411yescharacteristic, complemented, Sylow
328C2—22no—
329C2—481no—
330C2#7179511yescharacteristic
331C2#7204611yescharacteristic
332C2#7207211yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#500C2
socleC22×C3#392
prime radical#5477C22
hypercenterC2×#500D3
hypocenterC3#10164
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
1. C2×C3×#500C2semidirect
4. #71790C2semidirect
5. #71774C2semidirect
7. #67656C2semidirect
31. C2×#500D3semidirect
327. C3#10164semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5002
#832162
#831992
#717902
#717742
#697352
#676562
#101643