Finite Group Databasebeta

← #10697052

Subgroups — #10697052

Order 768.

488 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 113 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×#3165C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic
3C2×#5093C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic
6#69879C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic
8#10163—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#501—22no—
13C3×#500—21no—
14C3×#498—21no—
15C3×#428—22no—
16C3×C4×M16C2211yescharacteristic
17#5093—21no—
18#5093—22no—
19#5093—22no—
20#5092—21no—
21#5050—22no—
22#5031—22no—
23C2×#810—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4554—22no—
28#4546—22no—
29#4535—22no—
30#3209—31no—
31#3165D311yescharacteristic
32C2×#487—31no—
33#1997—31no—
34#1548—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×Dic6D412yes—
39C2×C3×M16D412yes—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D412yes—
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#216—24no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×#208—21no—
59C22×C3⋊C8—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64C3×#107—24no—
65C3×#106—24no—
66C3×#97—21no—
67C3×C4×C8—21no—
68#830—21no—
69#810—28no—
70#810—24no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yes—
77#533C2×D311yescharacteristic
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#501—22no—
81#500—21no—
82#498—21no—
83#494—61no—
84#487—62no—
85#487—62no—
86#487—61no—
87#462—62no—
88#451—62no—
89#431—31no—
90#428—22no—
91#423—31no—
92C2×#103—62no—
93C4×M16C2×D311yescharacteristic
94C2×C4×C8—31no—
95#343—62no—
96#333—62no—
97Q8⋊C8—62no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—22no—
101C2×C3×Q8—41no—
102C2×C3×Q8—82no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—22no—
108C2×Dic6—22no—
109C2×Dic6—22no—
110C2×Dic6—44no—
111C3×M16—22no—
112C3×M16Q1614yescomplemented
113C2×C3×C8—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C42—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130#216—84no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135#208—42no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—42no—
139C2×C3⋊C8—41no—
140C22×Q8—21no—
141C22×Q8—32no—
142C2×Q16—64no—
143C2×M16—61no—
144C2×M16#6712yes—
145C22×C8—61no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8#6712yes—
154C4⋊Q8#6711yescharacteristic
155C4⋊Q8#6711yescharacteristic
156#125—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159C4×Q8—21no—
160C4×Q8—21no—
161#117—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#107—24no—
168#106—24no—
169C4⋊C8—32no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#103—68no—
173#103—64no—
174C22⋊C8—61no—
175#97—21no—
176C4×C8—21no—
177C4×C8—32no—
178C4×C8—61no—
179C4×C8—61no—
180C23×C3#12111yescharacteristic
181C3×Q8—42no—
182C3×Q8—42no—
183C3×Q8—82no—
184C3×Q8—82no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—82no—
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12711yescharacteristic
197C2×C3×C4C2×Q1612yes—
198C2×C3×C4#13712yes—
199C2×Dic3—42no—
200C2×Dic3—82no—
201Dic6—44no—
202Dic6—44no—
203C3×C8—24no—
204C3⋊C8—42no—
205C3⋊C8—42no—
206C3⋊C8—81no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—62no—
212C2×Q8—62no—
213C2×Q8—82no—
214C2×Q8—124no—
215C22×C4—41no—
216C22×C4—62no—
217C22×C4D3×D411yescharacteristic
218C22×C4#23711yescharacteristic
219C22×C4C2×#6711yescharacteristic
220Q16—244no—
221M16—22no—
222M16—122no—
223M16#21614yescomplemented
224C2×C8—22no—
225C2×C8—62no—
226C2×C8—62no—
227C2×C8—122no—
228C2×C8—121no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—41no—
244C42—41no—
245C42D3×D411yescharacteristic
246C42D3×D411yescharacteristic
247C42C2×#6711yescharacteristic
248C42C2×#6711yescharacteristic
249C22×C3—21no—
250C22×C3—21no—
251C22×C3#39211yescharacteristic
252C22×C3#46212yes—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—81no—
256C3×C4—82no—
257C3×C4—81no—
258C3×C4#45112yes—
259C3×C4#45512yes—
260C3×C4#47112yes—
261C3×C4#49411yescharacteristic
262C3×C4#49711yescharacteristic
263Dic3—82no—
264C23#93811yescharacteristic
265Q8—42no—
266Q8—42no—
267Q8—82no—
268Q8—82no—
269Q8—124no—
270Q8—124no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—122no—
278C2×C4—242no—
279C2×C4#93711yescharacteristic
280C2×C4#93711yescharacteristic
281C2×C4#93911yescharacteristic
282C2×C4#93911yescharacteristic
283C2×C4#94011yescharacteristic
284C2×C4#94212yes—
285C2×C4C2×#21612yes—
286C2×C4#95311yescharacteristic
287C8—24no—
288C8—122no—
289C8—122no—
290C8—241no—
291C2×C3—22no—
292C2×C3#200411yescharacteristic
293C2×C3#201611yescharacteristic
294C2×C3#201911yescharacteristic
295C22—21no—
296C22—21no—
297C22#509012yes—
298C22#522311yescharacteristic
299C4—21no—
300C4—21no—
301C4—82no—
302C4—81no—
303C4—81no—
304C4—242no—
305C4#508611yescharacteristic
306C4#509211yescharacteristic
307C4#520012yes—
308C4#520212yes—
309C4#522812yes—
310C3#1016311yescharacteristic, complemented, Sylow
311C2—22no—
312C2#7179311yescharacteristic
313C2#7209711yescharacteristic
314C2#7210611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3165D3
hypocenterC3#10163
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
112. C3×M16Q16semidirect
223. M16#216semidirect
310. C3#10163semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#833702
C2×#50932
#718262
#717742
#698792
#676672
#101633