Finite Group Databasebeta

← #10697898

Subgroups — #10697898

Order 768.

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

normal subgroups only

This group has 588 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×#2868C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic
3#83222C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic
7#67611C211yesmaximal, characteristic
8#10293—31nomaximal, Sylow
9C2×C3×C4⋊Q8—22no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#384—22no—
14C3×#384—22no—
15C3×C43C2211yescharacteristic
16#5228—22no—
17#5093—21no—
18#5092—21no—
19#5091—22no—
20#5090—22no—
21#5079—21no—
22#5078—21no—
23#4970—22no—
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4547—22no—
28#4546—22no—
29#4523—22no—
30#3230—31no—
31#3209—31no—
32#2868D311yescharacteristic
33#1994—31no—
34#1549—31no—
35#1548—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8D412yes—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—48no—
48C2×C3×C4⋊C4—28no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C42—21no—
54C2×C3×C42—21no—
55C2×C3×C42D412yes—
56C2×C3×C42C2311yescharacteristic
57C2×#216—24no—
58#926—21no—
59#924—21no—
60#924—21no—
61C2×#208—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67#835—24no—
68#830—21no—
69#830—21no—
70#810—24no—
71#807—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#533—31no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×C4⋊Q8C2×D311yescharacteristic
82#500—61no—
83#494—61no—
84#487—61no—
85#487—61no—
86#484—62no—
87#477—62no—
88#462—62no—
89#451—62no—
90#431—31no—
91#431—31no—
92#423—31no—
93#384—22no—
94#384—22no—
95C43C2×D311yescharacteristic
96#339—62no—
97#333—62no—
98#332—62no—
99C2×C3×Q8—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—44no—
102C2×C3×Q8—44no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C2×Dic6—21no—
112C2×Dic6—21no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—416no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C42—24no—
123C3×C42—24no—
124C3×C42—22no—
125C3×C42Q1614yescomplemented
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130#217—41no—
131#216—84no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139#208—42no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—41no—
143C22×Q8—21no—
144C22×Q8#6712yes—
145C2×Q16—64no—
146C2×M16—61no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—48no—
158#125—61no—
159C22⋊Q8—61no—
160C22⋊Q8—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163#117—61no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4#6712yes—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42#6712yes—
172C2×C42C22×D311yescharacteristic
173#107—64no—
174C4⋊C8—32no—
175C4⋊C8—32no—
176C4⋊C8—61no—
177C4⋊C8—61no—
178#103—64no—
179#103—64no—
180C22⋊C8—61no—
181C22⋊C8—61no—
182C4×C8—61no—
183C4×C8—61no—
184C23×C3#12011yescharacteristic
185C3×Q8—44no—
186C3×Q8—82no—
187C3×Q8—82no—
188C3×Q8—82no—
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—24no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—44no—
195C2×C3×C4—44no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12711yescharacteristic
204C2×C3×C4C2×Q1612yes—
205C2×C3×C4#13712yes—
206C2×Dic3—41no—
207C2×Dic3—41no—
208Dic6—42no—
209Dic6—42no—
210C3⋊C8—42no—
211C3⋊C8—42no—
212C3⋊C8—81no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—44no—
216C2×Q8—44no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—61no—
220C2×Q8—61no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4C2×#6711yescharacteristic
225C22×C4C2×#6711yescharacteristic
226C22×C4C2×#6711yescharacteristic
227Q16—244no—
228M16—122no—
229C2×C8—62no—
230C2×C8—62no—
231C2×C8—121no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—416no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C22⋊C4—121no—
248C42—24no—
249C42—24no—
250C42—22no—
251C42—121no—
252C42#21614yescomplemented
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C42D3×D411yescharacteristic
256C42#23711yescharacteristic
257C22×C3—21no—
258C22×C3—21no—
259C22×C3#39211yescharacteristic
260C22×C3#45112yes—
261C3×C4—28no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—82no—
265C3×C4—82no—
266C3×C4#45312yes—
267C3×C4#45612yes—
268C3×C4#46212yes—
269C3×C4#49411yescharacteristic
270C3×C4#49711yescharacteristic
271Dic3—81no—
272Dic3—81no—
273C23#95311yescharacteristic
274Q8—44no—
275Q8—82no—
276Q8—82no—
277Q8—82no—
278Q8—122no—
279Q8—122no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4#93711yescharacteristic
292C2×C4#93711yescharacteristic
293C2×C4#93811yescharacteristic
294C2×C4#93911yescharacteristic
295C2×C4#93911yescharacteristic
296C2×C4#94011yescharacteristic
297C2×C4#94212yes—
298C2×C4C2×#21612yes—
299C8—122no—
300C8—122no—
301C8—241no—
302C2×C3—22no—
303C2×C3#200211yescharacteristic
304C2×C3#200511yescharacteristic
305C2×C3#201611yescharacteristic
306C22—21no—
307C22—21no—
308C22#522311yescharacteristic
309C22#522812yes—
310C4—28no—
311C4—21no—
312C4—21no—
313C4—82no—
314C4—82no—
315C4—241no—
316C4—241no—
317C4#507712yes—
318C4#508611yescharacteristic
319C4#509012yes—
320C4#509211yescharacteristic
321C4#509512yes—
322C3#1029311yescharacteristic, complemented, Sylow
323C2—22no—
324C2#7176911yescharacteristic
325C2#7179311yescharacteristic
326C2#7179511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2868C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2868D3
hypocenterC3#10293
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
125. C3×C42Q16semidirect
252. C42#216semidirect
322. C3#10293semidirect

Maximal subgroups

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

subgroupindex
C3×#28682
#832242
#832222
#717752
#717742
#693962
#676112
#102933