Finite Group Databasebeta

← #10697053

Subgroups — #10697053

Order 768.

488 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 107 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#83207C211yesmaximal, characteristic
2C2×#5093C211yesmaximal, characteristic
3#79313C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic
5#70524C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic
7#67739C211yesmaximal, characteristic
8#10163—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5093—22no—
15#5093—22no—
16#5081—22no—
17#5080—21no—
18#5078—21no—
19C2×#810—22no—
20#4964—22no—
21C4×#208C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23#4849—22no—
24#4739—22no—
25#4699—21no—
26#4681—21no—
27C3×#333—22no—
28#4554—22no—
29#4484—22no—
30#3209—31no—
31#3165—31no—
32C2×#487—31no—
33#1997—31no—
34#1548D311yescharacteristic
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×Dic6D412yes—
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47#926—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50#924—21no—
51#924—21no—
52C2×#208D412yes—
53C22×C3⋊C8—21no—
54C2×Dic12—24no—
55#870—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69#816—24no—
70#810—28no—
71#808—24no—
72#807—24no—
73#803—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yes—
77#533—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#501—62no—
81#500—61no—
82#498—61no—
83#494—61no—
84#487—61no—
85#487—62no—
86#487—62no—
87#462—62no—
88#451—62no—
89#431C2×D311yescharacteristic
90#428—62no—
91#423C2×D311yescharacteristic
92C2×#103—62no—
93C4×M16—31no—
94C2×C4×C8—31no—
95#343—62no—
96#333—22no—
97Q8⋊C8—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—41no—
100C2×C3×Q8—82no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—22no—
106C2×Dic6—22no—
107C2×Dic6—22no—
108C2×Dic6—21no—
109C2×Dic6—21no—
110C2×Dic6—44no—
111C3×M16—42no—
112C2×C3×C8—22no—
113C2×C3×C8—41no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122#217—41no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—41no—
128C4⋊Dic3—41no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133C4×Dic3—41no—
134#208—22no—
135#208Q1614yescomplemented
136C2×C3⋊C8—22no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—42no—
139Dic12—84no—
140C22×Q8—21no—
141C22×Q8—32no—
142C2×Q16—64no—
143C2×M16—21no—
144C2×M16—32no—
145C22×C8—61no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—31no—
150C4⋊Q8—31no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8D1212yes—
156#125—61no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159C4×Q8—61no—
160C4×Q8—61no—
161#117—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#107—64no—
168#106—64no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8#6712yes—
172#103—68no—
173#103—64no—
174C22⋊C8—21no—
175#97—61no—
176C4×C8—21no—
177C4×C8—32no—
178C4×C8—61no—
179C4×C8—61no—
180C23×C3#12111yescharacteristic
181C3×Q8—82no—
182C3×Q8—82no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—42no—
186C2×C3×C4—82no—
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12711yescharacteristic
193C2×C3×C4C2×Q1612yes—
194C2×C3×C4#13712yes—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—42no—
198C2×Dic3—82no—
199Dic6—44no—
200Dic6—42no—
201Dic6—44no—
202Dic6—42no—
203C3×C8—42no—
204C3×C8—81no—
205C3⋊C8—24no—
206C3⋊C8—42no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—62no—
210C2×Q8—61no—
211C2×Q8—62no—
212C2×Q8—61no—
213C2×Q8—82no—
214C2×Q8—124no—
215C22×C4—41no—
216C22×C4—62no—
217C22×C4#23511yescharacteristic
218C22×C4D3×D411yescharacteristic
219C22×C4D3×D411yescharacteristic
220Q16—244no—
221M16—34no—
222M16—42no—
223M16—62no—
224C2×C8—22no—
225C2×C8—41no—
226C2×C8—62no—
227C2×C8—62no—
228C2×C8—122no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—121no—
244C42—121no—
245C42C2×D1211yescharacteristic
246C42D3×D411yescharacteristic
247C42D3×D411yescharacteristic
248C42C2×#6711yescharacteristic
249C22×C3—21no—
250C22×C3—21no—
251C22×C3#39211yescharacteristic
252C22×C3#46212yes—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—82no—
256C3×C4#45112yes—
257C3×C4#45512yes—
258C3×C4#47112yes—
259C3×C4#49411yescharacteristic
260C3×C4#49711yescharacteristic
261Dic3—82no—
262Dic3—81no—
263Dic3—81no—
264C23#88411yescharacteristic
265Q8—82no—
266Q8—82no—
267Q8—124no—
268Q8—122no—
269Q8—124no—
270Q8—122no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—82no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—122no—
278C2×C4—242no—
279C2×C4#88211yescharacteristic
280C2×C4#88211yescharacteristic
281C2×C4#88411yescharacteristic
282C2×C4#91512yes—
283C2×C4D3×Q1612yes—
284C2×C4#93011yescharacteristic
285C2×C4#93711yescharacteristic
286C2×C4#94011yescharacteristic
287C8—42no—
288C8—64no—
289C8—81no—
290C8—122no—
291C2×C3—22no—
292C2×C3#200411yescharacteristic
293C2×C3#201611yescharacteristic
294C2×C3#201911yescharacteristic
295C22—21no—
296C22—21no—
297C22#479512yes—
298C22#496011yescharacteristic
299C4—21no—
300C4—21no—
301C4—82no—
302C4—242no—
303C4—241no—
304C4—241no—
305C4#474812yes—
306C4#480912yes—
307C4#516711yescharacteristic
308C4#518311yescharacteristic
309C4#520212yes—
310C3#1016311yescharacteristic, complemented, Sylow
311C2—22no—
312C2#7009111yescharacteristic
313C2#7028311yescharacteristic
314C2#7034411yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1548D3
hypocenterC3#10163
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
135. #208Q16semidirect
310. C3#10163semidirect

Maximal subgroups

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

subgroupindex
#832072
C2×#50932
#793132
C3×#15482
#705242
#705012
#677392
#101633