Finite Group Databasebeta

← #10696735

Subgroups — #10696735

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
1#79272C211yesmaximal, characteristic
2#79069C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic
4#70769C211yesmaximal, characteristic
5#70522C211yesmaximal, characteristic
6#69862C211yesmaximal, characteristic
7C3×#1299C211yesmaximal, characteristic
8#10117—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5137—22no—
16C2×#824—22no—
17#5121—22no—
18C4×#208C2211yescharacteristic
19#4851—21no—
20#4851—21no—
21#4850—22no—
22#4849—21no—
23#4849—21no—
24#4722—22no—
25#4702—22no—
26#4682—21no—
27#4682—21no—
28#4681—21no—
29#4680—21no—
30#4484—22no—
31#4465—22no—
32#3255—31no—
33#3230—31no—
34#3165—31no—
35#1966—31no—
36#1526—31no—
37#1485—31no—
38#1299D311yescharacteristic
39C22×Dic6—21no—
40C2×C3×M16—21no—
41C2×C3×M16D412yes—
42C2×C3×C42C2311yescharacteristic
43#926—21no—
44#926—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3D412yes—
47#924—21no—
48#924—21no—
49#924—21no—
50#924—21no—
51C2×#208—21no—
52C2×Dic12—22no—
53#870—21no—
54#870—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×C22⋊C8—21no—
71C3×#97D412yes—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74#824—44no—
75#818—24no—
76#818—24no—
77#817—24no—
78#817—22no—
79#816—22no—
80#816—22no—
81#816—24no—
82#803—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#533—31no—
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—61no—
95#475—61no—
96#470—62no—
97#462—61no—
98#462—61no—
99#431C2×D311yescharacteristic
100#428—62no—
101#423C2×D311yescharacteristic
102C2×#101—62no—
103C4×M16—31no—
104C4×M16C2×D311yescharacteristic
105#330—62no—
106Q8⋊C8—62no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—22no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—42no—
116C2×Dic6—42no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C3×M16—24no—
120C3×M16—42no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128#217—41no—
129#217—41no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—22no—
133C4⋊Dic3—21no—
134C4⋊Dic3—22no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147C4×Dic3—41no—
148#208—22no—
149#208—41no—
150C2×C3⋊C8—41no—
151Dic12—82no—
152C22×Q8—61no—
153C2×Q16—62no—
154C2×M16—21no—
155C2×M16—61no—
156C2×M16D1212yes—
157C4⋊Q8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#125—61no—
166#125—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175#117—61no—
176#117—61no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—61no—
179C2×C42C22×D311yescharacteristic
180#107—64no—
181#107—64no—
182#106—64no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8D1212yes—
187#103—62no—
188#103—62no—
189#103—64no—
190#101—124no—
191C22⋊C8—21no—
192#97—61no—
193#97D1212yes—
194C4×C8—21no—
195C4×C8—21no—
196C4×C8—61no—
197C23×C3#12111yescharacteristic
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12711yescharacteristic
208C2×Dic3—44no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—42no—
214Dic6—41no—
215Dic6—41no—
216Dic6—41no—
217Dic6—41no—
218Dic6—42no—
219Dic6—42no—
220Dic6—82no—
221Dic6—81no—
222Dic6—81no—
223C3×C8—24no—
224C3×C8—42no—
225C3×C8—42no—
226C3⋊C8—42no—
227C2×Q8—61no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—122no—
232C2×Q8—122no—
233C2×Q8—121no—
234C2×Q8—121no—
235C22×C4—62no—
236C22×C4C2×D1211yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4#23911yescharacteristic
239Q16—242no—
240M16—24no—
241M16—42no—
242M16—62no—
243M16—121no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—121no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—62no—
251C4⋊C4—61no—
252C4⋊C4—62no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—121no—
266C42—121no—
267C42—121no—
268C42C2×D1211yescharacteristic
269C42C2×D1211yescharacteristic
270C42D3×D411yescharacteristic
271C42D3×D411yescharacteristic
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3#39211yescharacteristic
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4#45311yescharacteristic
280C3×C4#45411yescharacteristic
281C3×C4#45612yes—
282C3×C4#47312yes—
283Dic3—81no—
284Dic3—81no—
285Dic3—81no—
286Dic3—82no—
287Dic3—81no—
288C23#89511yescharacteristic
289Q8—122no—
290Q8—121no—
291Q8—121no—
292Q8—121no—
293Q8—121no—
294Q8—122no—
295Q8—242no—
296Q8—241no—
297Q8—241no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—124no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—122no—
308C2×C4#87411yescharacteristic
309C2×C4#88211yescharacteristic
310C2×C4#88211yescharacteristic
311C2×C4#88211yescharacteristic
312C2×C4#89511yescharacteristic
313C2×C4#89511yescharacteristic
314C8—24no—
315C8—42no—
316C8—42no—
317C8—122no—
318C2×C3—41no—
319C2×C3#200511yescharacteristic
320C2×C3#200611yescharacteristic
321C2×C3#202011yescharacteristic
322C22—21no—
323C22—21no—
324C22—21no—
325C22#467111yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—241no—
330C4—242no—
331C4—241no—
332C4—241no—
333C4—241no—
334C4#474711yescharacteristic
335C4#475311yescharacteristic
336C4#482612yes—
337C4#489612yes—
338C3#1011711yescharacteristic, complemented, Sylow
339C2—41no—
340C2#6876811yescharacteristic
341C2#6894411yescharacteristic
342C2#6900811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1299C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1299D3
hypocenterC3#10117
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1299

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
#792722
#790692
#785062
#707692
#705222
#698622
C3×#12992
#101173