Finite Group Databasebeta

← #10696744

Subgroups — #10696744

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×#3255C211yesmaximal, characteristic
2#83341C211yesmaximal, characteristic
3#83323C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic
5#71821C211yesmaximal, characteristic
6#70524C211yesmaximal, characteristic
7#67710C211yesmaximal, characteristic
8#10119—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#533C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#475—21no—
13C3×#475—21no—
14C3×#470—22no—
15C3×#462—21no—
16C3×#462—21no—
17C3×#423C2211yescharacteristic
18#5080—21no—
19#5080—21no—
20#5079—21no—
21#5078—21no—
22#5064—22no—
23#5043—22no—
24#5020—22no—
25#4995—22no—
26#4925C2211yescharacteristic
27C4×#208C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4554—22no—
30#4543—22no—
31#4535—22no—
32#3255D311yescharacteristic
33#3229—31no—
34#3165—31no—
35#1997—31no—
36#1526—31no—
37#1485—31no—
38#1299—31no—
39C22×Dic6—21no—
40C2×C3×Q16—22no—
41C2×C3×M16D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×#117—21no—
57C3×#117—21no—
58C2×C3×C42C2311yescharacteristic
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3D412yes—
61C2×#208—21no—
62C2×#208—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67C3×#106—22no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×#103—22no—
71C3×#103—22no—
72#835—24no—
73#830—21no—
74#808—24no—
75#808—24no—
76#807—24no—
77C4⋊(C3⋊C8)D412yes—
78#803—21no—
79#803D412yes—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84#533C2×D311yescharacteristic
85C2×C4⋊Q8—31no—
86#501—62no—
87#500—61no—
88#498—61no—
89#498—61no—
90#487—61no—
91#483—62no—
92#475—21no—
93#475—21no—
94#475—62no—
95#470—22no—
96#462—21no—
97#462—21no—
98#431—31no—
99#428—62no—
100#423C2×D311yescharacteristic
101C4×M16—31no—
102C4×M16—31no—
103#343—62no—
104#330—62no—
105Q8⋊C8—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×Dic3—22no—
114C2×Dic6—41no—
115C2×Dic6—41no—
116C2×Dic6—82no—
117C3×Q16—82no—
118C3×M16—42no—
119C2×C3×C8—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C4⋊Dic3—22no—
141C4⋊Dic3—22no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144#208—24no—
145#208—22no—
146#208—41no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—41no—
150C22×Q8—61no—
151C2×Q16—22no—
152C2×M16—61no—
153C2×M16—61no—
154C2×M16#6712yes—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8#6711yescharacteristic
160C4⋊Q8#6711yescharacteristic
161C4⋊Q8#6711yescharacteristic
162C4⋊Q8#6711yescharacteristic
163#125—21no—
164#125—21no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173#117—21no—
174#117—21no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178#107—64no—
179#106—22no—
180#106—64no—
181#106—64no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—32no—
185#103—22no—
186#103—22no—
187#103—64no—
188C22⋊C8—61no—
189#97—32no—
190#97—61no—
191C4×C8—61no—
192C4×C8—61no—
193C4×C8—61no—
194C23×C3#12111yescharacteristic
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—42no—
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—42no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—21no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12711yescharacteristic
215C2×Dic3—42no—
216C2×Dic3—82no—
217Dic6—82no—
218Dic6—82no—
219C3×C8—42no—
220C3⋊C8—24no—
221C3⋊C8—42no—
222C3⋊C8—42no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—121no—
228C2×Q8—121no—
229C2×Q8—242no—
230C22×C4—62no—
231C22×C4D3×D411yescharacteristic
232C22×C4#23711yescharacteristic
233C22×C4C2×#6711yescharacteristic
234Q16—82no—
235M16—42no—
236M16—64no—
237M16—62no—
238M16—121no—
239C2×C8—22no—
240C2×C8—62no—
241C2×C8—62no—
242C2×C8—121no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—62no—
256C4⋊C4—62no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C42—41no—
262C42—41no—
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—81no—
276C3×C4—81no—
277C3×C4—81no—
278C3×C4#45511yescharacteristic
279C3×C4#45612yes—
280C3×C4#45611yescharacteristic
281C3×C4#47112yes—
282Dic3—82no—
283C23#93811yescharacteristic
284Q8—42no—
285Q8—42no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—242no—
291Q8—242no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—122no—
301C2×C4—242no—
302C2×C4#93711yescharacteristic
303C2×C4#93711yescharacteristic
304C2×C4#93911yescharacteristic
305C2×C4#93911yescharacteristic
306C2×C4#94011yescharacteristic
307C2×C4#95311yescharacteristic
308C8—42no—
309C8—64no—
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—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—242no—
328C4#520312yes—
329C4#520512yes—
330C4#524211yescharacteristic
331C4#524611yescharacteristic
332C3#1011911yescharacteristic, complemented, Sylow
333C2—41no—
334C2#7188111yescharacteristic
335C2#7210711yescharacteristic
336C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3255C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3255D3
hypocenterC3#10119
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

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×#32552
#833412
#833232
#718252
#718212
#705242
#677102
#101193