Finite Group Databasebeta

← #10698419

Subgroups — #10698419

Order 768.

602 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 75 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#74088C211yesmaximal, characteristic, complemented
2C3×#1782C211yesmaximal, characteristic
3#72138C212yesmaximal
4#67629C211yesmaximal, characteristic, complemented
5#67628C212yesmaximal, complemented
6#10386—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×#99—21no—
9C2×C3×#99—21no—
10C3×#381C2211yescharacteristic
11C3×#381C2211yescharacteristic
12C2×#837—22no—
13C2×#837—22no—
14C4×#217C2212yes—
15#5207C2212yes—
16#4949—21no—
17#4947—21no—
18#4655—22no—
19#4646—22no—
20#2607—31no—
21#1789—32no—
22#1782D311yescharacteristic
23#1306—31no—
24#1284—32no—
25C2×C3×#41—22no—
26C2×C3×#41—22no—
27C2×C3×#41—42no—
28C2×C3×#41—42no—
29C22×C3×Q8Q811yescharacteristic, complemented
30C22×C3×D4—21no—
31C22×C3×D4Q811yescharacteristic, complemented
32C23×C3×C4—21no—
33C23×C3×C4C2311yescharacteristic
34C22×Dic6—21no—
35C2×C3×C22⋊C4—21no—
36C2×C3×C22⋊C4D411yescharacteristic
37C2×C3×C22⋊C4D411yescharacteristic
38C2×#217—22no—
39C2×C4⋊Dic3—21no—
40C2×Dic3⋊C4—22no—
41C2×C4×Dic3D412yes—
42C2×C4×Dic3D412yes—
43C2×#208—22no—
44C3×#99—42no—
45C3×#99—42no—
46C3×#95—22no—
47C3×#95—22no—
48#837—44no—
49#837—44no—
50C3⋊#95—21no—
51C3⋊#95—22no—
52C3⋊#95—21no—
53C3⋊#95—22no—
54#830—22no—
55#830—22no—
56C22×#41C2×D311yescharacteristic
57C2×#104—62no—
58C2×#104—62no—
59C2×#99—21no—
60C2×#99—21no—
61#407—32no—
62#399—61no—
63#393—61no—
64#391—62no—
65#387—62no—
66#381C2×D311yescharacteristic
67#381C2×D311yescharacteristic
68C4×C22⋊C4—32no—
69C24×C3—21no—
70C24×C3C2×D411yescharacteristic
71C3×#41—24no—
72C3×#41—24no—
73C3×#41—42no—
74C3×#41—42no—
75C3×#41—44no—
76C3×#41—44no—
77C2×C3×Q8—22no—
78C2×C3×Q8—42no—
79C2×C3×D4—22no—
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—42no—
83C2×C3×D4—41no—
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C2×C3×D4—41no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4—42no—
95C22×C3×C4—41no—
96C22×C3×C4C2×D412yes—
97C22×C3×C4C2×Q811yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C22×C3×C4#4112yes—
100C22×Dic3—22no—
101C22×Dic3—22no—
102C2×Dic6—44no—
103C2×Dic6—81no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C22⋊C4—41no—
107C3×C22⋊C4—41no—
108#217—44no—
109C4⋊Dic3—81no—
110Dic3⋊C4—44no—
111C4×Dic3—42no—
112C4×Dic3—42no—
113#208—42no—
114#208—42no—
115C2×C3⋊C8—42no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—42no—
119C2×#41—42no—
120C22×Q8—61no—
121C22×Q8Dic611yescharacteristic, complemented
122C22×D4—21no—
123C22×D4Dic611yescharacteristic, complemented
124C23×C4—21no—
125C23×C4C22×D311yescharacteristic
126C2×M16—62no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—61no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—62no—
131C2×C22⋊C4#6711yescharacteristic
132C2×C22⋊C4#6711yescharacteristic
133C2×C42—32no—
134C2×C42—32no—
135#104—124no—
136#104—124no—
137#99—42no—
138#99—42no—
139C22⋊C8—62no—
140C22⋊C8—62no—
141#95—22no—
142#95—22no—
143#95—61no—
144#95—62no—
145#95—61no—
146#95—62no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—22no—
150C23×C3—42no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3—82no—
154C23×C3#12011yescharacteristic
155C23×C3C22⋊Q811yescharacteristic
156C23×C3#12411yescharacteristic
157C3×Q8—24no—
158C3×Q8—42no—
159C3×D4—24no—
160C3×D4—42no—
161C3×D4—42no—
162C3×D4—42no—
163C3×D4—42no—
164C3×D4—42no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—44no—
172C2×C3×C4—44no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—81no—
180C2×C3×C4—82no—
181C2×C3×C4C22⋊Q812yes—
182C2×C3×C4#12412yes—
183C2×Dic3—24no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189Dic6—84no—
190C3⋊C8—82no—
191C24—21no—
192C24C2×#6711yescharacteristic
193#41—24no—
194#41—24no—
195#41—42no—
196#41—42no—
197#41—44no—
198#41—44no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×Q8—124no—
202C2×Q8—241no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—62no—
221C22×C4—62no—
222C22×C4C2×Dic611yescharacteristic
223C22×C4#23511yescharacteristic
224C22×C4D3×D412yes—
225C22×C4#23712yes—
226M16—122no—
227M16—122no—
228C2×C8—122no—
229C4⋊C4—124no—
230C4⋊C4—241no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—124no—
236C42—122no—
237C42—122no—
238C22×C3—22no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—42no—
244C22×C3—81no—
245C22×C3—81no—
246C22×C3—82no—
247C22×C3—82no—
248C22×C3—82no—
249C22×C3#39311yescharacteristic
250C22×C3B(4,𝔽2)11yescharacteristic
251C22×C3B(4,𝔽2)11yescharacteristic
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—42no—
255C3×C4—81no—
256C3×C4—81no—
257C3×C4—161no—
258Dic3—44no—
259Dic3—82no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23—82no—
267C23#92411yescharacteristic
268C23#93611yescharacteristic
269C23#93711yescharacteristic
270Q8—24no—
271Q8—42no—
272Q8—244no—
273D4—24no—
274D4—42no—
275D4—42no—
276D4—42no—
277D4—42no—
278D4—42no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—64no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4#87812yes—
302C2×C4#88512yes—
303C8—242no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—82no—
309C2×C3#202911yescharacteristic
310C2×C3#203112yes—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22#494711yescharacteristic
323C22#503211yescharacteristic
324C22#503211yescharacteristic
325C4—22no—
326C4—22no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—124no—
331C4—161no—
332C4—242no—
333C3#1038611yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—82no—
339C2#6949711yescharacteristic
340C2#7003812yes—
subgroupG/H
centerC22#4947
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1782C2
socleC22×C3#393
prime radicalC22×C3×#41C22
hypercenter#1782D3
hypocenterC3#10386
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #74088C2semidirect
4. #67629C2semidirect
5. #67628C2semidirect
29. C22×C3×Q8Q8semidirect
31. C22×C3×D4Q8semidirect
121. C22×Q8Dic6semidirect
123. C22×D4Dic6semidirect
333. C3#10386semidirect

Maximal subgroups

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

subgroupindex
#740882
C3×#17822
#721382
#676292
#676282
#103863