Finite Group Databasebeta

← #10692541

Subgroups — #10692541

Order 768.

538 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 117 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#79286C211yesmaximal, characteristic
2#73807C211yesmaximal, characteristic
3#70779C211yesmaximal, characteristic, complemented
4#70527C211yesmaximal, characteristic
5#70507C211yesmaximal, characteristic, complemented
6C3×#1303C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9391—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#823—21no—
15#4987—22no—
16#4939—21no—
17#4936—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4834—22no—
21D3×C22⋊C8—22no—
22C8×D12—22no—
23#4673C414yes—
24#4554—21no—
25#4481—22no—
26#4481—22no—
27#2972—31no—
28#2563—31no—
29#1800—31no—
30#1548—31no—
31#1548—31no—
32#1303—31no—
33#1303D311yescharacteristic, complemented
34C22×D12D411yescharacteristic
35C22×C4×D3D412yes—
36C22×Dic6C2×C412yes—
37C2×C3×M16—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C42—21no—
40C2×C3×C42C2311yescharacteristic
41C2×(C2×D3)⋊C4—24no—
42C2×(C2×D3)⋊C4C2×C412yes—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×#208—21no—
47C2×D3×C8—24no—
48C4×D12—21no—
49C4×D12—21no—
50C4×D12—21no—
51C4×D12—21no—
52#869—21no—
53#869—21no—
54#869—21no—
55#869—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8D412yes—
59C3×C22⋊C8—21no—
60C3×C4×C8—21no—
61C3×#95C2×C412yes—
62C3⋊#95—24no—
63#830—21no—
64#823—42no—
65(C2×D3)⋊C8—24no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)D412yes—
69C4×C3⋊C8—21no—
70C2×C4⋊Q8—31no—
71C2×C4×D4—31no—
72C8×D4—62no—
73C8×D4—62no—
74#431—31no—
75#431C2×D311yescharacteristic
76#423—31no—
77#423C2×D311yescharacteristic
78C2×#100—61no—
79C2×C22⋊C8—62no—
80#393—34no—
81#390—61no—
82#382—61no—
83#380—62no—
84C4×C4⋊C4C2×D311yescharacteristic
85#343—61no—
86#333—62no—
87#333—62no—
88C23×D3C22⋊C412yes—
89C22×C3×C4—22no—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×Dic3—22no—
94C22×Dic3C22⋊C412yes—
95C2×D12—21no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12SD1612yes—
100C2×D12Q1612yes—
101C2×C4×D3—22no—
102C2×C4×D3—22no—
103C2×C4×D3—44no—
104C2×Dic6—22no—
105C2×Dic6—22no—
106C2×Dic6—82no—
107C3×M16—41no—
108C3×M16—41no—
109C2×C3×C8—22no—
110C2×C3×C8—41no—
111C3×C4⋊C4—42no—
112C3×C42—41no—
113C3×C42—41no—
114C3×C42C2×D411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42#4111yescharacteristic
117C3×C42#4111yescharacteristic
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—84no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125C4⋊Dic3—41no—
126C4⋊Dic3—41no—
127#208—41no—
128#208—41no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—41no—
131D3×C8—84no—
132C22×Q8—32no—
133C22×D4—31no—
134C23×C4—32no—
135C2×M16—21no—
136C2×M16—61no—
137C22×C8—64no—
138C4⋊Q8—61no—
139C4⋊Q8—61no—
140C4⋊Q8—61no—
141C4⋊Q8—61no—
142C4×D4—61no—
143C4×D4—61no—
144C4×D4—61no—
145C4×D4—61no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C22⋊C4—32no—
151C2×C22⋊C4—64no—
152C2×C42—21no—
153C2×C42C22×D311yescharacteristic
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—32no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8D1212yes—
160#100—122no—
161C22⋊C8—21no—
162C22⋊C8—64no—
163C22⋊C8—61no—
164C4×C8—21no—
165C4×C8—61no—
166#95—64no—
167#95C4×D312yes—
168C23×C3C2×C22⋊C411yescharacteristic
169C22×D3—22no—
170C22×D3—22no—
171C22×D3—42no—
172C22×D3—42no—
173C22×D3#10314yes—
174C2×C3×C4—22no—
175C2×C3×C4—41no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4C4×D411yescharacteristic
180C2×C3×C4C4×D411yescharacteristic
181C2×C3×C4#12011yescharacteristic
182C2×C3×C4#12111yescharacteristic
183C2×C3×C4#12111yescharacteristic
184C2×C3×C4#12311yescharacteristic
185C2×C3×C4C2×SD1611yescharacteristic
186C2×C3×C4C2×Q1611yescharacteristic
187C2×C3×C4#13511yescharacteristic
188C2×C3×C4#13511yescharacteristic
189C2×Dic3—22no—
190C2×Dic3—42no—
191C2×Dic3—44no—
192C2×Dic3—42no—
193D12—22no—
194D12—22no—
195D12—22no—
196D12—22no—
197C4×D3—44no—
198C4×D3—82no—
199Dic6—82no—
200Dic6—82no—
201C3×C8—42no—
202C3×C8—81no—
203C3⋊C8—42no—
204C3⋊C8—81no—
205C24—32no—
206C2×Q8—62no—
207C2×Q8—62no—
208C2×Q8—242no—
209C2×D4—32no—
210C2×D4—32no—
211C2×D4—61no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C22×C4—22no—
216C22×C4—32no—
217C22×C4—62no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4—124no—
221C22×C4C2×C4×D311yescharacteristic
222C22×C4D3×D411yescharacteristic
223C22×C4D3×D411yescharacteristic
224M16—41no—
225M16—41no—
226M16—121no—
227M16—121no—
228C2×C8—22no—
229C2×C8—41no—
230C2×C8—62no—
231C2×C8—121no—
232C2×C8—244no—
233C4⋊C4—42no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C22⋊C4—62no—
241C22⋊C4—62no—
242C22⋊C4—244no—
243C42—41no—
244C42—41no—
245C42C2×D1211yescharacteristic
246C42#23511yescharacteristic
247C42D3×D411yescharacteristic
248C42#23911yescharacteristic
249C22×C3—21no—
250C22×C3—21no—
251C22×C3#38611yescharacteristic
252C22×C3C2×#10311yescharacteristic
253C22×C3#41611yescharacteristic
254C2×D3—24no—
255C2×D3—42no—
256C2×D3—42no—
257C2×D3—44no—
258C2×D3—82no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—82no—
263C3×C4#45512yes—
264C3×C4#46511yescharacteristic
265C3×C4#46511yescharacteristic
266C3×C4#46711yescharacteristic
267C3×C4#46711yescharacteristic
268Dic3—42no—
269Dic3—82no—
270C23—34no—
271C23—62no—
272C23—62no—
273C23—122no—
274C23—122no—
275C23D3×C22⋊C411yescharacteristic
276Q8—242no—
277Q8—242no—
278D4—62no—
279D4—62no—
280D4—62no—
281D4—62no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—62no—
288C2×C4—124no—
289C2×C4—122no—
290C2×C4—124no—
291C2×C4—122no—
292C2×C4—242no—
293C2×C4C4×D1211yescharacteristic
294C2×C4#88211yescharacteristic
295C2×C4#88411yescharacteristic
296C2×C4#89311yescharacteristic
297C2×C4#89411yescharacteristic
298C2×C4#89511yescharacteristic
299C2×C4#91211yescharacteristic
300C2×C4D3×SD1611yescharacteristic
301C2×C4#91611yescharacteristic
302C2×C4D3×Q1611yescharacteristic
303C8—42no—
304C8—81no—
305C8—122no—
306C8—241no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3#188511yescharacteristic
310C2×C3#189211yescharacteristic
311C2×C3#191411yescharacteristic
312D3—44no—
313C22—21no—
314C22—21no—
315C22—64no—
316C22—124no—
317C22—122no—
318C22—122no—
319C22—242no—
320C22#466511yescharacteristic
321C22#477211yescharacteristic
322C22D3×#10311yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—122no—
328C4—242no—
329C4#477811yescharacteristic
330C4#481311yescharacteristic
331C4#482512yes—
332C4#486411yescharacteristic
333C4#488711yescharacteristic
334C3#939111yescharacteristic, complemented, Sylow
335C2—21no—
336C2—21no—
337C2—124no—
338C2#6875911yescharacteristic
339C2#6888211yescharacteristic
340C2#6904011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#9391
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #70779C2semidirect
5. #70507C2semidirect
6. C3×#1303C2semidirect
7. #67620C2semidirect
33. #1303D3semidirect
334. C3#9391semidirect

Maximal subgroups

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

subgroupindex
#792862
#738072
#707792
#705272
#705072
C3×#13032
#676202
#93913