Finite Group Databasebeta

← #10696999

Subgroups — #10696999

Order 768.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 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
2C2×#5092C211yesmaximal, characteristic
3#78506C211yesmaximal, characteristic
4C3×#1997C211yesmaximal, characteristic
5#70508C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic
8#10155—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216—22no—
11#5470C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5228—22no—
16#5121—22no—
17#5092—22no—
18#5092—21no—
19#5092—22no—
20#5090—22no—
21#5079—21no—
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4702—22no—
25#4682—21no—
26#4680—21no—
27C3×#343—22no—
28#4486—22no—
29#4484—22no—
30#3209—31no—
31#3165—31no—
32C2×#494—31no—
33#1997D311yescharacteristic
34#1549—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8D412yes—
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C2×C3×M16D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#216—28no—
49C2×#216—24no—
50C2×#216—48no—
51#926—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54#924—21no—
55#924—21no—
56C22×C3⋊C8—21no—
57#870—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D412yes—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×#97—21no—
69C3×C4×C8—21no—
70#835—24no—
71#830—21no—
72#818—24no—
73#817—24no—
74C4⋊(C3⋊C8)D412yes—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8D412yes—
78C22×Q16—62no—
79#533—31no—
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82#501—62no—
83#500—61no—
84#498—61no—
85#494—62no—
86#494—62no—
87#494—61no—
88#487—61no—
89#462—62no—
90#451—62no—
91#431—31no—
92#428—62no—
93#423C2×D311yescharacteristic
94C4×M16C2×D311yescharacteristic
95C2×C4×C8—31no—
96#343—22no—
97#332—62no—
98Q8⋊C8—62no—
99C2×C3×Q8—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—44no—
102C22×C3×C4—22no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×Dic3—41no—
107C2×Dic6—21no—
108C2×Dic6—22no—
109C2×Dic6—21no—
110C2×Dic6—44no—
111C2×Dic6—41no—
112C3×M16—22no—
113C3×M16—42no—
114C3×M16Q1614yescomplemented
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125#217—41no—
126#216—416no—
127#216—84no—
128C4⋊Dic3—21no—
129C4⋊Dic3—22no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—22no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139C2×C3⋊C8—24no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142C22×Q8—61no—
143C22×Q8D1212yes—
144C2×Q16—68no—
145C2×Q16—64no—
146C2×Q16—128no—
147C2×M16—21no—
148C2×M16#6712yes—
149C22×C8—61no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—32no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160#125—61no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163C4×Q8—61no—
164C4×Q8—61no—
165#117—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C42C22×D311yescharacteristic
171#107—64no—
172#106—64no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—32no—
176#103—64no—
177C22⋊C8—61no—
178#97—21no—
179C4×C8—21no—
180C4×C8—32no—
181C4×C8—61no—
182C4×C8—61no—
183C23×C3#12111yescharacteristic
184C3×Q8—44no—
185C3×Q8—44no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—42no—
189C2×C3×C4—82no—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12711yescharacteristic
196C2×C3×C4C2×Q1612yes—
197C2×C3×C4#13712yes—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—42no—
201C2×Dic3—82no—
202Dic6—44no—
203Dic6—42no—
204Dic6—42no—
205Dic6—82no—
206C3×C8—24no—
207C3×C8—81no—
208C3⋊C8—24no—
209C3⋊C8—42no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—44no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8—61no—
216C2×Q8—124no—
217C2×Q8—121no—
218C22×C4—22no—
219C22×C4—121no—
220C22×C4C2×D1211yescharacteristic
221C22×C4#23511yescharacteristic
222C22×C4C2×#6711yescharacteristic
223Q16—1216no—
224Q16—244no—
225M16—22no—
226M16—42no—
227M16#21614yescomplemented
228C2×C8—22no—
229C2×C8—41no—
230C2×C8—64no—
231C2×C8—62no—
232C2×C8—62no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—61no—
238C4⋊C4—62no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—62no—
242C4⋊C4—61no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C22⋊C4—121no—
248C42—121no—
249C42D3×D411yescharacteristic
250C42D3×D411yescharacteristic
251C42D3×D411yescharacteristic
252C42D3×D411yescharacteristic
253C22×C3—21no—
254C22×C3—21no—
255C22×C3#39211yescharacteristic
256C22×C3#46212yes—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—82no—
260C3×C4#45112yes—
261C3×C4#45312yes—
262C3×C4#47312yes—
263C3×C4#49411yescharacteristic
264C3×C4#49711yescharacteristic
265Dic3—82no—
266Dic3—81no—
267Dic3—81no—
268C23#93011yescharacteristic
269Q8—44no—
270Q8—44no—
271Q8—124no—
272Q8—122no—
273Q8—122no—
274Q8—242no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—122no—
282C2×C4—242no—
283C2×C4#88211yescharacteristic
284C2×C4#88211yescharacteristic
285C2×C4#88411yescharacteristic
286C2×C4#88411yescharacteristic
287C2×C4#93711yescharacteristic
288C2×C4#94011yescharacteristic
289C2×C4C2×#21612yes—
290C2×C4#95112yes—
291C8—24no—
292C8—64no—
293C8—81no—
294C8—122no—
295C2×C3—22no—
296C2×C3#200211yescharacteristic
297C2×C3#201611yescharacteristic
298C2×C3#202011yescharacteristic
299C22—21no—
300C22—21no—
301C22#496011yescharacteristic
302C22#503112yes—
303C4—21no—
304C4—21no—
305C4—82no—
306C4—242no—
307C4—241no—
308C4—241no—
309C4#474712yes—
310C4#475512yes—
311C4#505012yes—
312C4#506011yescharacteristic
313C4#509211yescharacteristic
314C3#1015511yescharacteristic, complemented, Sylow
315C2—22no—
316C2#6969111yescharacteristic
317C2#7008211yescharacteristic
318C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1997D3
hypocenterC3#10155
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
114. C3×M16Q16semidirect
227. M16#216semidirect
314. C3#10155semidirect

Maximal subgroups

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

subgroupindex
#832222
C2×#50922
#785062
C3×#19972
#705082
#705012
#676672
#101553