Finite Group Databasebeta

← #10697305

Subgroups — #10697305

Order 768.

486 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 103 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#83222C211yesmaximal, characteristic
2#83207C211yesmaximal, characteristic
3#79037C211yesmaximal, characteristic
4C3×#1987C211yesmaximal, characteristic
5#69493C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic
7#67706C211yesmaximal, characteristic
8#10199—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5228—22no—
15#5116—22no—
16#5092—21no—
17#5090—22no—
18#5081—22no—
19#5080—21no—
20#5079—21no—
21#5078—21no—
22#4964—22no—
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4850—21no—
26#4850—21no—
27#4849—21no—
28#4849—21no—
29C3×#340—22no—
30#4523—22no—
31#4452—22no—
32#3249—31no—
33#3209—31no—
34#3165—31no—
35#1987D311yescharacteristic
36#1986—31no—
37#1503—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C22×C3×C8D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#216—24no—
51#926—21no—
52#926—21no—
53#924—21no—
54#924—21no—
55#924—21no—
56#924—21no—
57C2×#208D412yes—
58C2×Dic12—22no—
59#870—21no—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×#95—21no—
72#835—24no—
73#830D412yes—
74#818—22no—
75#816—22no—
76#816—22no—
77#808—24no—
78#807—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#533—31no—
85C2×C4⋊Q8C2×D311yescharacteristic
86#501—62no—
87#500—61no—
88#498—61no—
89#494—61no—
90#487—61no—
91#477—61no—
92#477—61no—
93#467—62no—
94#462—61no—
95#462—61no—
96#462—62no—
97#451—62no—
98#431—31no—
99#428—62no—
100C2×C4⋊C8C2×D311yescharacteristic
101C4×M16—31no—
102C4×C4⋊C4C2×D311yescharacteristic
103#340—22no—
104#339—62no—
105#328—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—41no—
108C2×C3×Q8—82no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×C3×C8—22no—
119C2×C3×C8—42no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42—42no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131#217—41no—
132#217—41no—
133#216—84no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147C4×Dic3—41no—
148#208—22no—
149#208Q1614yescomplemented
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152Dic12—82no—
153C22×Q8—21no—
154C2×Q16—62no—
155C2×Q16—64no—
156C2×M16—32no—
157C22×C8#6712yes—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8D1212yes—
166#125—61no—
167#125—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176#117—61no—
177#117—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C42—21no—
182C2×C42C22×D311yescharacteristic
183#107—64no—
184#107—62no—
185#106—64no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—61no—
189#103—62no—
190#103—62no—
191#103—64no—
192C22⋊C8—32no—
193#97—61no—
194C4×C8—61no—
195C4×C8—61no—
196#95—21no—
197C23×C3#12711yescharacteristic
198C3×Q8—44no—
199C3×Q8—82no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—81no—
206C2×C3×C4—82no—
207C2×C3×C4—81no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4C2×Q1612yes—
215C2×C3×C4#13712yes—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220Dic6—42no—
221Dic6—41no—
222Dic6—41no—
223Dic6—41no—
224Dic6—41no—
225Dic6—81no—
226C3×C8—42no—
227C3⋊C8—24no—
228C3⋊C8—42no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—82no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4D3×D411yescharacteristic
239C22×C4D3×D411yescharacteristic
240C22×C4C2×#6711yescharacteristic
241Q16—242no—
242Q16—244no—
243M16—34no—
244M16—62no—
245C2×C8—22no—
246C2×C8—42no—
247C2×C8—62no—
248C2×C8—62no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—42no—
270C42—121no—
271C42—121no—
272C42C2×D1211yescharacteristic
273C42#23511yescharacteristic
274C42D3×D411yescharacteristic
275C42D3×D411yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#39211yescharacteristic
279C22×C3#49411yescharacteristic
280C22×C3#49711yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—82no—
285C3×C4—82no—
286C3×C4#45112yes—
287C3×C4#45511yescharacteristic
288C3×C4#45611yescharacteristic
289C3×C4#46212yes—
290Dic3—81no—
291Dic3—81no—
292Dic3—81no—
293Dic3—81no—
294C23#94011yescharacteristic
295Q8—44no—
296Q8—82no—
297Q8—122no—
298Q8—121no—
299Q8—121no—
300Q8—121no—
301Q8—121no—
302Q8—241no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4#88211yescharacteristic
316C2×C4#88211yescharacteristic
317C2×C4#88411yescharacteristic
318C2×C4#88411yescharacteristic
319C2×C4#91512yes—
320C2×C4D3×Q1612yes—
321C2×C4#93011yescharacteristic
322C2×C4#93711yescharacteristic
323C8—42no—
324C8—64no—
325C8—122no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#200711yescharacteristic
329C2×C3#200811yescharacteristic
330C2×C3#201611yescharacteristic
331C22—21no—
332C22—21no—
333C22#496011yescharacteristic
334C22#516711yescharacteristic
335C22#518311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—82no—
340C4—82no—
341C4—241no—
342C4—241no—
343C4—241no—
344C4—241no—
345C4#479512yes—
346C4#480912yes—
347C4#506111yescharacteristic
348C4#509511yescharacteristic
349C3#1019911yescharacteristic, complemented, Sylow
350C2—21no—
351C2—21no—
352C2#6969411yescharacteristic
353C2#7028311yescharacteristic
354C2#7033111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1987C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1987D3
hypocenterC3#10199
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
149. #208Q16semidirect
349. C3#10199semidirect

Maximal subgroups

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

subgroupindex
#832222
#832072
#790372
C3×#19872
#694932
#694712
#677062
#101993