Finite Group Databasebeta

← #10696996

Subgroups — #10696996

Order 768.

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

normal subgroups only

This group has 552 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×#5092C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic
5#71775C211yesmaximal, characteristic
6#69879C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic
8#10155—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C22×#216—22no—
12C2×#869C2211yescharacteristic
13C3×#501—22no—
14C3×#500—21no—
15C3×#498—21no—
16C3×#428—22no—
17C3×C4×M16C2211yescharacteristic
18#5093—21no—
19#5092—22no—
20#5092—21no—
21#5092—22no—
22#5050—22no—
23#5031—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4554—22no—
28#4547—22no—
29#4535—22no—
30#3209—31no—
31#3165D311yescharacteristic
32C2×#494—31no—
33#1997—31no—
34#1549—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⋊Q8D412yes—
44C3×C4⋊Q8D411yescharacteristic
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—28no—
56C2×#216—24no—
57C2×#216—48no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×#208—21no—
61C22×C3⋊C8—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66C3×#107—24no—
67C3×#106—24no—
68C3×#97—21no—
69C3×C4×C8—21no—
70#830—21no—
71#810—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8D412yes—
78C22×Q16—62no—
79#533C2×D311yescharacteristic
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82#501—22no—
83#500—21no—
84#498—21no—
85#494—62no—
86#494—62no—
87#494—61no—
88#487—61no—
89#462—62no—
90#451—62no—
91#431—31no—
92#428—22no—
93#423—31no—
94C4×M16C2×D311yescharacteristic
95C2×C4×C8—31no—
96#343—62no—
97#332—62no—
98Q8⋊C8—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—22no—
101C2×C3×Q8—21no—
102C2×C3×Q8—44no—
103C2×C3×Q8—41no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×Dic3—22no—
109C2×Dic6—22no—
110C2×Dic6—22no—
111C2×Dic6—44no—
112C3×M16—22no—
113C3×M16Q1614yescomplemented
114C2×C3×C8—22no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C42—41no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131#216—416no—
132#216—84no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137#208—42no—
138C2×C3⋊C8—24no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—41no—
142C22×Q8—21no—
143C22×Q8—32no—
144C2×Q16—68no—
145C2×Q16—64no—
146C2×Q16—128no—
147C2×M16—61no—
148C2×M16#6712yes—
149C22×C8—61no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8#6712yes—
158C4⋊Q8#6711yescharacteristic
159C4⋊Q8#6711yescharacteristic
160#125—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165#117—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C42C22×D311yescharacteristic
171#107—24no—
172#106—24no—
173C4⋊C8—32no—
174C4⋊C8—61no—
175C4⋊C8—61no—
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—42no—
186C3×Q8—42no—
187C3×Q8—82no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—82no—
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12711yescharacteristic
200C2×C3×C4C2×Q1612yes—
201C2×C3×C4#13712yes—
202C2×Dic3—42no—
203C2×Dic3—82no—
204Dic6—44no—
205Dic6—44no—
206C3×C8—24no—
207C3⋊C8—24no—
208C3⋊C8—42no—
209C3⋊C8—81no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—21no—
213C2×Q8—44no—
214C2×Q8—41no—
215C2×Q8—62no—
216C2×Q8—62no—
217C2×Q8—124no—
218C22×C4—41no—
219C22×C4—62no—
220C22×C4D3×D411yescharacteristic
221C22×C4#23711yescharacteristic
222C22×C4C2×#6711yescharacteristic
223Q16—1216no—
224Q16—244no—
225M16—22no—
226M16—122no—
227M16#21614yescomplemented
228C2×C8—22no—
229C2×C8—64no—
230C2×C8—62no—
231C2×C8—62no—
232C2×C8—121no—
233C4⋊C4—21no—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—22no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C22⋊C4—41no—
248C42—41no—
249C42D3×D411yescharacteristic
250C42D3×D411yescharacteristic
251C42C2×#6711yescharacteristic
252C42C2×#6711yescharacteristic
253C22×C3—21no—
254C22×C3—21no—
255C22×C3#39211yescharacteristic
256C22×C3#46212yes—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—82no—
260C3×C4—81no—
261C3×C4—81no—
262C3×C4#45112yes—
263C3×C4#45312yes—
264C3×C4#47312yes—
265C3×C4#49411yescharacteristic
266C3×C4#49711yescharacteristic
267Dic3—82no—
268C23#93811yescharacteristic
269Q8—44no—
270Q8—42no—
271Q8—42no—
272Q8—82no—
273Q8—124no—
274Q8—124no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—122no—
282C2×C4—242no—
283C2×C4#93711yescharacteristic
284C2×C4#93711yescharacteristic
285C2×C4#93911yescharacteristic
286C2×C4#93911yescharacteristic
287C2×C4#94011yescharacteristic
288C2×C4#94212yes—
289C2×C4C2×#21612yes—
290C2×C4#95311yescharacteristic
291C8—24no—
292C8—64no—
293C8—122no—
294C8—241no—
295C2×C3—22no—
296C2×C3#200211yescharacteristic
297C2×C3#201611yescharacteristic
298C2×C3#202011yescharacteristic
299C22—21no—
300C22—21no—
301C22#509012yes—
302C22#522311yescharacteristic
303C4—21no—
304C4—21no—
305C4—82no—
306C4—81no—
307C4—81no—
308C4—242no—
309C4#508611yescharacteristic
310C4#509211yescharacteristic
311C4#519812yes—
312C4#520412yes—
313C4#522812yes—
314C3#1015511yescharacteristic, complemented, Sylow
315C2—22no—
316C2#7179311yescharacteristic
317C2#7209411yescharacteristic
318C2#7210911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3165D3
hypocenterC3#10155
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
113. 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
C3×#31652
#833702
C2×#50922
#718262
#717752
#698792
#676672
#101553