Finite Group Databasebeta

← #10697477

Subgroups — #10697477

Order 768.

466 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3229C211yesmaximal, characteristic
2#83340C211yesmaximal, characteristic
3#83223C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic
5#71786C211yesmaximal, characteristic
6#70793C211yesmaximal, characteristic
7#67741C211yesmaximal, characteristic
8#10227—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#498—21no—
13C3×#487—21no—
14C3×#483—22no—
15C3×#475—22no—
16C3×#431C2211yescharacteristic
17C2×#835—22no—
18#5094—22no—
19#5093—21no—
20#5081—21no—
21#5081—21no—
22#5080—21no—
23#5064—21no—
24#5064—21no—
25#4998—22no—
26#4971—22no—
27#4936C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4538—22no—
31#4467—22no—
32#3229—31no—
33#3229D311yescharacteristic
34#3162—31no—
35#1960—31no—
36#1518—31no—
37#1518—31no—
38#1281—31no—
39C22×C3×Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D412yes—
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55#926—21no—
56C2×C4⋊Dic3—21no—
57#924—21no—
58#924—21no—
59C22×C3⋊C8D412yes—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×#106—24no—
66C3×C4⋊C8—21no—
67C3×#103—24no—
68C3×C22⋊C8D412yes—
69C3×C4×C8—21no—
70#835—44no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73#830D412yes—
74#810—24no—
75#808—24no—
76#808—22no—
77#808—22no—
78#807—22no—
79#807—22no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#533C2×D311yescharacteristic
86C2×C4⋊Q8C2×D311yescharacteristic
87#501—61no—
88#501—61no—
89#501—61no—
90#501—61no—
91#498—21no—
92#498—61no—
93#487—21no—
94#487—61no—
95#483—22no—
96#483—62no—
97#475—22no—
98#475—62no—
99#431—31no—
100#431C2×D311yescharacteristic
101#427—62no—
102C2×#103—62no—
103C2×#97—31no—
104#382—31no—
105#328—62no—
106#328—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—41no—
111C2×C3×Q8—42no—
112C2×C3×Q8—42no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—41no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×C3×C8—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#217—41no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—42no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8SD1614yes—
150C22×Q8—21no—
151C22×C8—32no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157C4⋊Q8#6712yes—
158C4⋊Q8#6711yescharacteristic
159C4⋊Q8#6711yescharacteristic
160#125—21no—
161#125—61no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—61no—
169C4×Q8—61no—
170#117—21no—
171#117—61no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—61no—
175C2×C42C22×D311yescharacteristic
176#107—62no—
177#107—62no—
178#106—24no—
179#106—62no—
180#106—62no—
181#106—64no—
182C4⋊C8—21no—
183C4⋊C8—61no—
184#103—24no—
185#103—64no—
186#103—124no—
187C22⋊C8—32no—
188C22⋊C8#6712yes—
189#97—61no—
190#97—61no—
191C4×C8—21no—
192C4×C8—61no—
193#95—61no—
194#95—61no—
195C23×C3#12711yescharacteristic
196C3×Q8—42no—
197C3×Q8—44no—
198C3×Q8—81no—
199C3×Q8—82no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4C2×SD1612yes—
214C2×C3×C4#13712yes—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C2×Dic3—81no—
220Dic6—42no—
221Dic6—81no—
222C3×C8—42no—
223C3⋊C8—24no—
224C3⋊C8—42no—
225C2×Q8—22no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—61no—
232C2×Q8—61no—
233C22×C4—41no—
234C22×C4—121no—
235C22×C4D3×D411yescharacteristic
236C22×C4D3×D411yescharacteristic
237C22×C4C2×#6711yescharacteristic
238C2×C8—22no—
239C2×C8—34no—
240C2×C8—62no—
241C2×C8—62no—
242C2×C8—62no—
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—
259C4⋊C4—122no—
260C22⋊C4—41no—
261C22⋊C4—121no—
262C42—41no—
263C42—121no—
264C42D3×D411yescharacteristic
265C42#23711yescharacteristic
266C42C2×#6711yescharacteristic
267C42C2×#6711yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#39211yescharacteristic
271C22×C3#49211yescharacteristic
272C22×C3#49711yescharacteristic
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—81no—
278C3×C4—81no—
279C3×C4—82no—
280C3×C4#44812yes—
281C3×C4#46112yes—
282Dic3—81no—
283Dic3—81no—
284Dic3—161no—
285C23#94011yescharacteristic
286Q8—42no—
287Q8—44no—
288Q8—81no—
289Q8—82no—
290Q8—122no—
291Q8—241no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—241no—
302C2×C4—241no—
303C2×C4—241no—
304C2×C4D3×SD1612yes—
305C2×C4#91512yes—
306C2×C4#93711yescharacteristic
307C2×C4#93711yescharacteristic
308C2×C4#93811yescharacteristic
309C2×C4#93911yescharacteristic
310C2×C4#93911yescharacteristic
311C2×C4#95311yescharacteristic
312C8—42no—
313C8—64no—
314C8—122no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3#200911yescharacteristic
318C2×C3#200911yescharacteristic
319C2×C3#201311yescharacteristic
320C22—21no—
321C22—21no—
322C22#516711yescharacteristic
323C22#517511yescharacteristic
324C22#522311yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—82no—
330C4—81no—
331C4—81no—
332C4—241no—
333C4—241no—
334C4—481no—
335C4#516212yes—
336C4#517012yes—
337C3#1022711yescharacteristic, complemented, Sylow
338C2—21no—
339C2—21no—
340C2#7186911yescharacteristic
341C2#7205711yescharacteristic
342C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#3229D3
hypocenterC3#10227
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
337. C3#10227semidirect

Maximal subgroups

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

subgroupindex
C3×#32292
#833402
#832232
#718382
#717862
#707932
#677412
#102273