Finite Group Databasebeta

← #10696743

Subgroups — #10696743

Order 768.

454 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C3×#3165C211yesmaximal, characteristic
2#83374C211yesmaximal, characteristic
3#83223C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic
5#71790C211yesmaximal, characteristic
6#69862C211yesmaximal, characteristic
7#67669C211yesmaximal, characteristic
8#10119—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#501—22no—
13C3×#500—21no—
14C3×#498—21no—
15C3×#428—22no—
16C3×C4×M16C2211yescharacteristic
17#5094—21no—
18#5094—22no—
19#5093—21no—
20#5090—21no—
21#5080—21no—
22#5052—22no—
23#5031—21no—
24#5020—21no—
25#4971—22no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4535—22no—
30#4530—22no—
31#4465—22no—
32#3255—31no—
33#3229—31no—
34#3165D311yescharacteristic
35#1997—31no—
36#1526—31no—
37#1485—31no—
38#1299—31no—
39C22×C3×Q8—21no—
40C2×C3×M16—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56C2×#216—22no—
57#926—21no—
58#924—21no—
59#924—21no—
60C2×#208—21no—
61C2×#208D412yes—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#107—24no—
68C3×#106—24no—
69C3×#97D412yes—
70C3×C4×C8—21no—
71#835—22no—
72#830—21no—
73#810—24no—
74#810—22no—
75#808—24no—
76#808—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#533C2×D311yescharacteristic
85C2×C4⋊Q8C2×D311yescharacteristic
86#501—22no—
87#500—21no—
88#498—21no—
89#498—61no—
90#487—61no—
91#483—62no—
92#475—62no—
93#475—61no—
94#475—61no—
95#470—62no—
96#462—61no—
97#462—61no—
98#431—31no—
99#428—22no—
100#423—31no—
101C4×M16—31no—
102C4×M16C2×D311yescharacteristic
103#343—62no—
104#330—62no—
105Q8⋊C8—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C22×C3×C4—22no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C2×Dic6—21no—
116C2×Dic6—21no—
117C3×M16—24no—
118C2×C3×C8—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C42—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135#217—41no—
136#216—82no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143C4×Dic3—41no—
144#208—22no—
145#208—41no—
146#208—42no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—41no—
150C22×Q8—21no—
151C2×Q16—62no—
152C2×M16—21no—
153C2×M16—32no—
154C2×M16—61no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8#6711yescharacteristic
162C4⋊Q8#6711yescharacteristic
163#125—21no—
164#125—61no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—61no—
172C4×Q8—61no—
173#117—21no—
174#117—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4#6712yes—
177C2×C42C22×D311yescharacteristic
178#107—24no—
179#106—24no—
180#106—62no—
181#106—64no—
182C4⋊C8—32no—
183C4⋊C8—61no—
184C4⋊C8—61no—
185#103—62no—
186#103—64no—
187#103—62no—
188C22⋊C8—61no—
189#97—61no—
190#97#6712yes—
191C4×C8—21no—
192C4×C8—61no—
193C4×C8—61no—
194C23×C3#12111yescharacteristic
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—42no—
198C3×Q8—82no—
199C3×Q8—82no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—82no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12711yescharacteristic
214C2×Dic3—41no—
215C2×Dic3—41no—
216Dic6—41no—
217Dic6—41no—
218Dic6—42no—
219C3×C8—24no—
220C3⋊C8—42no—
221C3⋊C8—42no—
222C3⋊C8—42no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—61no—
228C2×Q8—61no—
229C2×Q8—82no—
230C22×C4—22no—
231C22×C4D3×D411yescharacteristic
232C22×C4#23711yescharacteristic
233C22×C4C2×#6711yescharacteristic
234Q16—242no—
235M16—24no—
236M16—62no—
237M16—121no—
238M16—122no—
239C2×C8—22no—
240C2×C8—62no—
241C2×C8—62no—
242C2×C8—121no—
243C4⋊C4—22no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—22no—
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C22⋊C4—41no—
260C22⋊C4—121no—
261C42—41no—
262C42—121no—
263C42D3×D411yescharacteristic
264C42D3×D411yescharacteristic
265C42C2×#6711yescharacteristic
266C42C2×#6711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#39211yescharacteristic
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—81no—
275C3×C4—82no—
276C3×C4—81no—
277C3×C4#45511yescharacteristic
278C3×C4#45611yescharacteristic
279C3×C4#45612yes—
280C3×C4#47112yes—
281Dic3—81no—
282Dic3—81no—
283C23#93911yescharacteristic
284Q8—42no—
285Q8—41no—
286Q8—41no—
287Q8—82no—
288Q8—82no—
289Q8—121no—
290Q8—121no—
291Q8—122no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—82no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4#93711yescharacteristic
303C2×C4#93711yescharacteristic
304C2×C4#93811yescharacteristic
305C2×C4#93911yescharacteristic
306C2×C4#94011yescharacteristic
307C2×C4#95311yescharacteristic
308C8—24no—
309C8—122no—
310C8—122no—
311C8—122no—
312C2×C3—41no—
313C2×C3#200511yescharacteristic
314C2×C3#200711yescharacteristic
315C2×C3#201911yescharacteristic
316C22—21no—
317C22—21no—
318C22—21no—
319C22#522311yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—82no—
324C4—81no—
325C4—81no—
326C4—241no—
327C4—241no—
328C4#509512yes—
329C4#520012yes—
330C4#520211yescharacteristic
331C4#520511yescharacteristic
332C3#1011911yescharacteristic, complemented, Sylow
333C2—41no—
334C2#7179511yescharacteristic
335C2#7209711yescharacteristic
336C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3165D3
hypocenterC3#10119
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
332. C3#10119semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833742
#832232
#718252
#717902
#698622
#676692
#101193