Finite Group Databasebeta

← #10697921

Subgroups — #10697921

Order 768.

528 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 109 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×#3209C211yesmaximal, characteristic
2#83370C211yesmaximal, characteristic
3#83252C211yesmaximal, characteristic
4#71839C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic
6#70779C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic
8#10297—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#494—21no—
13C3×#487—21no—
14C3×#462—22no—
15C3×#451—22no—
16C3×#431C2211yescharacteristic
17#5229—22no—
18#5093—21no—
19#5092—21no—
20#5050—22no—
21#5031—22no—
22#4977—22no—
23#4973—22no—
24C42×Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4546—22no—
28#4532—22no—
29#4481—22no—
30#3209—31no—
31#3209D311yescharacteristic
32#2862—31no—
33#1994—31no—
34#1548—31no—
35#1548—31no—
36#1276—31no—
37C22×C3×Q8D412yes—
38C22×Dic6D412yes—
39C2×C3×Q16—24no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
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—24no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3D412yes—
61C2×#208—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66C3×C4⋊C8D412yes—
67C3×#103—24no—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70C3⋊#95—28no—
71#830—21no—
72#810—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77#533C2×D311yescharacteristic
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#494—21no—
81#494—61no—
82#487—21no—
83#487—61no—
84#462—22no—
85#462—62no—
86#451—22no—
87#451—62no—
88#431—31no—
89#431C2×D311yescharacteristic
90#423—31no—
91#391—62no—
92#391—62no—
93#382—62no—
94C43—31no—
95#339—62no—
96#333—62no—
97#333—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—22no—
102C2×C3×Q8—44no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—22no—
108C22×Dic3—22no—
109C2×Dic6—22no—
110C2×Dic6—22no—
111C2×Dic6—44no—
112C3×Q16—84no—
113C2×C3×C8—22no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
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×C42#4111yescharacteristic
130#216—84no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4×Dic3—22no—
136C4×Dic3—42no—
137C4×Dic3—42no—
138C4×Dic3Q1614yescomplemented
139#208—42no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—41no—
142C22×Q8—32no—
143C22×Q8#6712yes—
144C2×Q16—24no—
145C2×Q16—64no—
146C2×M16—61no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8#6711yescharacteristic
156C4⋊Q8#6711yescharacteristic
157#125—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162#117—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C42—32no—
168C2×C42—61no—
169C2×C42—61no—
170C2×C42C22×D311yescharacteristic
171C4⋊C8—32no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174C4⋊C8#6712yes—
175#103—24no—
176#103—64no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179C4×C8—21no—
180C4×C8—61no—
181#95—68no—
182C23×C3#12011yescharacteristic
183C3×Q8—42no—
184C3×Q8—42no—
185C3×Q8—44no—
186C3×Q8—82no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—44no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12711yescharacteristic
199C2×C3×C4C2×Q1612yes—
200C2×C3×C4#13712yes—
201C2×Dic3—24no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204C2×Dic3—44no—
205C2×Dic3—42no—
206Dic6—44no—
207Dic6—82no—
208C3×C8—42no—
209C3⋊C8—42no—
210C3⋊C8—81no—
211C2×Q8—21no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—21no—
215C2×Q8—44no—
216C2×Q8—62no—
217C2×Q8—62no—
218C2×Q8—124no—
219C22×C4—22no—
220C22×C4—62no—
221C22×C4—62no—
222C22×C4D3×D411yescharacteristic
223C22×C4D3×D411yescharacteristic
224C22×C4C2×#6711yescharacteristic
225Q16—84no—
226Q16—244no—
227M16—122no—
228C2×C8—22no—
229C2×C8—62no—
230C2×C8—121no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C22⋊C4—41no—
246C42—34no—
247C42—41no—
248C42—62no—
249C42—122no—
250C42—122no—
251C42D3×D411yescharacteristic
252C42#23711yescharacteristic
253C42C2×#6711yescharacteristic
254C42C2×#6711yescharacteristic
255C22×C3—21no—
256C22×C3—21no—
257C22×C3#39211yescharacteristic
258C22×C3#45112yes—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—82no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4#45512yes—
265C3×C4#45512yes—
266C3×C4#46212yes—
267C3×C4#49411yescharacteristic
268C3×C4#49711yescharacteristic
269Dic3—44no—
270Dic3—82no—
271C23#93711yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—44no—
275Q8—82no—
276Q8—124no—
277Q8—242no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—64no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—124no—
288C2×C4—122no—
289C2×C4D3×Q1612yes—
290C2×C4#91812yes—
291C2×C4#93711yescharacteristic
292C2×C4#93811yescharacteristic
293C2×C4#93911yescharacteristic
294C2×C4#93911yescharacteristic
295C2×C4#94011yescharacteristic
296C2×C4#95311yescharacteristic
297C8—42no—
298C8—122no—
299C8—241no—
300C2×C3—22no—
301C2×C3#200411yescharacteristic
302C2×C3#200411yescharacteristic
303C2×C3#201611yescharacteristic
304C22—21no—
305C22—21no—
306C22#518612yes—
307C22#522311yescharacteristic
308C4—21no—
309C4—21no—
310C4—81no—
311C4—82no—
312C4—81no—
313C4—124no—
314C4—242no—
315C4#518212yes—
316C4#518311yescharacteristic
317C4#519011yescharacteristic
318C4#520212yes—
319C4#524212yes—
320C3#1029711yescharacteristic, complemented, Sylow
321C2—22no—
322C2#7185711yescharacteristic
323C2#7207311yescharacteristic
324C2#7210611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3209C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3209D3
hypocenterC3#10297
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
138. C4×Dic3Q16semidirect
320. C3#10297semidirect

Maximal subgroups

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

subgroupindex
C3×#32092
#833702
#832522
#718392
#717742
#707792
#677362
#102973