Finite Group Databasebeta

← #10696730

Subgroups — #10696730

Order 768.

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

normal subgroups only

This group has 470 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#83324C211yesmaximal, characteristic
4#71825C211yesmaximal, characteristic
5#71821C211yesmaximal, characteristic
6#70522C211yesmaximal, characteristic
7#67710C211yesmaximal, characteristic
8#10117—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
18C2×#824—22no—
19#5080—21no—
20#5079—21no—
21#5078—21no—
22#5078—21no—
23#5064—22no—
24#5044—22no—
25#5021—22no—
26#4995—22no—
27#4925C2211yescharacteristic
28C4×#208C2211yescharacteristic
29C4×#208C2211yescharacteristic
30#4543—22no—
31#4535—22no—
32#3255D311yescharacteristic
33#3230—31no—
34#3165—31no—
35#1966—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#824—44no—
75#808—24no—
76#807—24no—
77#807—24no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#533C2×D311yescharacteristic
85#533C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87#501—62no—
88#500—61no—
89#500—61no—
90#498—61no—
91#487—61no—
92#484—62no—
93#477—62no—
94#475—21no—
95#475—21no—
96#470—22no—
97#462—21no—
98#462—21no—
99#431—31no—
100#428—62no—
101#423C2×D311yescharacteristic
102C2×#101—62no—
103C4×M16—31no—
104C4×M16—31no—
105#330—62no—
106Q8⋊C8—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—22no—
115C2×Dic6—42no—
116C2×Dic6—42no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C3×Q16—82no—
120C3×M16—42no—
121C2×C3×C8—22no—
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—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C4⋊Dic3—22no—
143C4⋊Dic3—22no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146#208—24no—
147#208—22no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C22×Q8—61no—
153C2×Q16—22no—
154C2×M16—61no—
155C2×M16—61no—
156C2×M16#6712yes—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8#6711yescharacteristic
162C4⋊Q8#6711yescharacteristic
163C4⋊Q8#6711yescharacteristic
164C4⋊Q8#6711yescharacteristic
165#125—21no—
166#125—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175#117—21no—
176#117—21no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—61no—
179C2×C42C22×D311yescharacteristic
180#107—64no—
181#107—64no—
182#106—22no—
183#106—64no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—32no—
187#103—22no—
188#103—22no—
189#103—64no—
190#101—124no—
191C22⋊C8—61no—
192#97—32no—
193#97—61no—
194C4×C8—61no—
195C4×C8—61no—
196C4×C8—61no—
197C23×C3#12111yescharacteristic
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—42no—
203C3×Q8—42no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218C2×Dic3—44no—
219C2×Dic3—42no—
220Dic6—82no—
221Dic6—81no—
222Dic6—81no—
223C3×C8—42no—
224C3⋊C8—24no—
225C3⋊C8—42no—
226C3⋊C8—42no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—122no—
232C2×Q8—122no—
233C2×Q8—121no—
234C2×Q8—121no—
235C22×C4—62no—
236C22×C4D3×D411yescharacteristic
237C22×C4#23711yescharacteristic
238C22×C4C2×#6711yescharacteristic
239Q16—82no—
240M16—42no—
241M16—64no—
242M16—62no—
243M16—121no—
244C2×C8—22no—
245C2×C8—62no—
246C2×C8—62no—
247C2×C8—121no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C42—41no—
267C42—41no—
268C42D3×D411yescharacteristic
269C42D3×D411yescharacteristic
270C42C2×#6711yescharacteristic
271C42C2×#6711yescharacteristic
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3#39211yescharacteristic
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4—81no—
282C3×C4—81no—
283C3×C4#45311yescharacteristic
284C3×C4#45411yescharacteristic
285C3×C4#45612yes—
286C3×C4#47312yes—
287Dic3—82no—
288C23#93811yescharacteristic
289Q8—42no—
290Q8—42no—
291Q8—41no—
292Q8—41no—
293Q8—41no—
294Q8—41no—
295Q8—242no—
296Q8—241no—
297Q8—241no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—124no—
307C2×C4—122no—
308C2×C4#93711yescharacteristic
309C2×C4#93711yescharacteristic
310C2×C4#93911yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#94011yescharacteristic
313C2×C4#95311yescharacteristic
314C8—42no—
315C8—64no—
316C8—122no—
317C8—122no—
318C2×C3—41no—
319C2×C3#200511yescharacteristic
320C2×C3#200611yescharacteristic
321C2×C3#202011yescharacteristic
322C22—21no—
323C22—21no—
324C22—21no—
325C22#522311yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—242no—
334C4#520412yes—
335C4#520512yes—
336C4#524111yescharacteristic
337C4#524511yescharacteristic
338C3#1011711yescharacteristic, complemented, Sylow
339C2—41no—
340C2#7187911yescharacteristic
341C2#7210811yescharacteristic
342C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3255C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3255D3
hypocenterC3#10117
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
338. C3#10117semidirect

Maximal subgroups

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

subgroupindex
C3×#32552
#833412
#833242
#718252
#718212
#705222
#677102
#101173