Finite Group Databasebeta

← #10698416

Subgroups — #10698416

Order 768.

602 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 77 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×#2607C211yesmaximal, characteristic, complemented
2#69537C211yesmaximal, characteristic
3#69454C212yesmaximal
4#67629C211yesmaximal, characteristic, complemented
5#67616C212yesmaximal, complemented
6#10386—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C3×#399—21no—
9C3×#393—21no—
10C3×#391—22no—
11C3×#387—22no—
12C3×#381C2211yescharacteristic
13C3×C4×C22⋊C4C2212yes—
14#5211C2211yescharacteristic
15#5207C2212yes—
16C2×#806—21no—
17C2×#806—21no—
18C2×#805—22no—
19C2×#805—22no—
20#2607D311yescharacteristic, complemented
21#1789—32no—
22#1782—31no—
23#1306—31no—
24#1284—32no—
25C22×C3×Q8—21no—
26C23×C3×C4C2311yescharacteristic
27C22×#67—21no—
28C2×#235—22no—
29C2×#235—22no—
30C2×#235—42no—
31C2×#235—42no—
32C22×D12Q811yescharacteristic, complemented
33C22×C4×D3—21no—
34C22×Dic6Q811yescharacteristic, complemented
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C42D412yes—
41C2×C3×C42D412yes—
42C2×#217—21no—
43C2×#208—22no—
44C3×#95—21no—
45C3×#95—22no—
46C3×#95—21no—
47C3×#95—22no—
48C3×#95—22no—
49C3⋊#95—22no—
50#830—22no—
51#830—22no—
52#806—42no—
53#806—42no—
54#805—44no—
55#805—44no—
56C22×#41—31no—
57C2×#104—62no—
58C2×#104—62no—
59C2×#99—61no—
60C2×#99—61no—
61#407—32no—
62#399—21no—
63#393—21no—
64#391—22no—
65#387—22no—
66#381—31no—
67#381C2×D311yescharacteristic
68C4×C22⋊C4C2×D312yes—
69C24×C3C2×D411yescharacteristic
70C23×D3—21no—
71C2×C3×Q8—44no—
72C2×C3×Q8—81no—
73C22×C3×C4—22no—
74C22×C3×C4—21no—
75C22×C3×C4—22no—
76C22×C3×C4—21no—
77C22×C3×C4—41no—
78C22×C3×C4—41no—
79C22×C3×C4C2×D412yes—
80C22×C3×C4C2×Q811yescharacteristic
81C22×C3×C4#4112yes—
82C22×C3×C4#4111yescharacteristic
83C2×#67—22no—
84C2×#67—22no—
85C2×#67—41no—
86C2×#67—41no—
87C2×#67—41no—
88C2×#67—41no—
89C22×Dic3—21no—
90C22×Dic3—41no—
91#235—24no—
92#235—24no—
93#235—44no—
94#235—44no—
95#235—42no—
96#235—42no—
97C2×D12—22no—
98C2×D12—42no—
99C2×C4×D3—42no—
100C2×C4×D3—42no—
101C2×C4×D3—42no—
102C2×Dic6—22no—
103C2×Dic6—42no—
104C3×C4⋊C4—44no—
105C3×C4⋊C4—81no—
106C3×C22⋊C4—44no—
107C3×C22⋊C4—41no—
108C3×C22⋊C4—41no—
109C3×C42—42no—
110C3×C42—42no—
111#217—41no—
112#217—41no—
113#208—42no—
114#208—42no—
115C2×C3⋊C8—42no—
116C2×#41—62no—
117C2×#41—62no—
118C2×#41—122no—
119C2×#41—122no—
120C22×Q8—21no—
121C22×Q8—31no—
122C22×D4—31no—
123C22×D4—61no—
124C23×C4—61no—
125C23×C4C22×D311yescharacteristic
126C2×M16—62no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—61no—
131C2×C22⋊C4#6711yescharacteristic
132C2×C22⋊C4#6711yescharacteristic
133C2×C42#6712yes—
134C2×C42#6712yes—
135#104—124no—
136#104—124no—
137#99—122no—
138#99—122no—
139C22⋊C8—62no—
140C22⋊C8—62no—
141#95—21no—
142#95—22no—
143#95—21no—
144#95—22no—
145#95—22no—
146#95—62no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—42no—
150C23×C3#12011yescharacteristic
151C23×C3C22⋊Q811yescharacteristic
152C23×C3#12411yescharacteristic
153C22×D3—22no—
154C22×D3—41no—
155C22×D3—41no—
156C22×D3—82no—
157C3×Q8—84no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—24no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4C22⋊Q812yes—
174C2×C3×C4#12412yes—
175#67—42no—
176#67—42no—
177#67—42no—
178#67—42no—
179C2×Dic3—22no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—81no—
183C2×Dic3—82no—
184D12—24no—
185D12—42no—
186C4×D3—44no—
187C4×D3—44no—
188Dic6—24no—
189Dic6—42no—
190C3⋊C8—82no—
191C24—61no—
192C24C2×#6711yescharacteristic
193#41—64no—
194#41—64no—
195#41—122no—
196#41—122no—
197#41—124no—
198#41—124no—
199C2×Q8—44no—
200C2×Q8—62no—
201C2×Q8—81no—
202C2×Q8—122no—
203C2×D4—62no—
204C2×D4—62no—
205C2×D4—62no—
206C2×D4—122no—
207C2×D4—121no—
208C2×D4—121no—
209C2×D4—121no—
210C2×D4—121no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—61no—
218C22×C4—122no—
219C22×C4—122no—
220C22×C4—122no—
221C22×C4—121no—
222C22×C4D3×Q811yescharacteristic
223C22×C4#23912yes—
224C22×C4#23911yescharacteristic
225C22×C4C2×#6712yes—
226M16—122no—
227M16—122no—
228C2×C8—122no—
229C4⋊C4—44no—
230C4⋊C4—81no—
231C22⋊C4—44no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—121no—
235C22⋊C4—121no—
236C42—42no—
237C42—42no—
238C22×C3—22no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—81no—
244C22×C3—81no—
245C22×C3#39311yescharacteristic
246C22×C3B(4,𝔽2)11yescharacteristic
247C22×C3B(4,𝔽2)11yescharacteristic
248C2×D3—42no—
249C2×D3—82no—
250C2×D3—82no—
251C2×D3—82no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—44no—
255C3×C4—82no—
256C3×C4—81no—
257C3×C4—81no—
258Dic3—42no—
259Dic3—161no—
260C23—21no—
261C23—21no—
262C23—42no—
263C23—62no—
264C23—121no—
265C23—121no—
266C23—242no—
267C23#94611yescharacteristic
268C23#94711yescharacteristic
269C23#95311yescharacteristic
270Q8—64no—
271Q8—84no—
272Q8—122no—
273D4—64no—
274D4—122no—
275D4—122no—
276D4—122no—
277D4—122no—
278D4—122no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—62no—
295C2×C4—124no—
296C2×C4—124no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—241no—
300C2×C4—242no—
301C2×C4#94612yes—
302C2×C4#94712yes—
303C8—242no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3#202911yescharacteristic
309C2×C3#203112yes—
310D3—82no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—21no—
315C22—21no—
316C22—81no—
317C22—81no—
318C22—122no—
319C22—242no—
320C22—242no—
321C22—242no—
322C22#523111yescharacteristic
323C22#524411yescharacteristic
324C22#524411yescharacteristic
325C4—22no—
326C4—22no—
327C4—44no—
328C4—81no—
329C4—82no—
330C4—81no—
331C4—122no—
332C4—481no—
333C3#1038611yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—242no—
339C2#7184212yes—
340C2#7189711yescharacteristic
subgroupG/H
centerC22#5231
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2607C2
socleC22×C3#393
prime radicalC22×#235C22
hypercenter#2607D3
hypocenterC3#10386
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
1. C3×#2607C2semidirect
4. #67629C2semidirect
5. #67616C2semidirect
20. #2607D3semidirect
32. C22×D12Q8semidirect
34. C22×Dic6Q8semidirect
333. C3#10386semidirect

Maximal subgroups

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

subgroupindex
C3×#26072
#695372
#694542
#676292
#676162
#103863