Finite Group Databasebeta

← #10691879

Subgroups — #10691879

Order 768.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2961C211yesmaximal, characteristic, complemented
2#83355C211yesmaximal, characteristic
3#71838C211yesmaximal, characteristic, complemented
4#70502C211yesmaximal, characteristic
5#69823C211yesmaximal, characteristic
6#67713C211yesmaximal, characteristic, complemented
7#67660C211yesmaximal, characteristic, complemented
8#9294—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#874C2211yescharacteristic
11C3×#446—22no—
12C3×#446—22no—
13C3×#424—21no—
14C3×#423C2211yescharacteristic
15C3×#405—21no—
16C2×C3×#97C2211yescharacteristic
17C2×#821—21no—
18C2×#821—21no—
19#5088C412yescomplemented
20#5085C412yescomplemented
21#5046—22no—
22#5027—22no—
23#4936C2211yescharacteristic
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4538—22no—
27#4485—22no—
28#3208—31no—
29#2961D311yescharacteristic, complemented
30#1961—31no—
31#1545—31no—
32#1518—31no—
33#1286—31no—
34#1275—31no—
35C22×D12—21no—
36C22×D12—21no—
37C22×D12—21no—
38C2×C3×M16—21no—
39C22×C3×C8D411yescharacteristic
40C22×C3×C8D411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8D412yes—
46C3×C4×Q8—21no—
47C3×C4×Q8D412yes—
48C3×#117—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—24no—
51C2×C4⋊Dic3—21no—
52#874—21no—
53#874—21no—
54#874—21no—
55#874C2×C412yes—
56C3×C4⋊C8—24no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8D412yes—
59C3×#97—22no—
60C3×#97—21no—
61C3×#97—21no—
62C3×C4×C8—22no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—21no—
66#830—21no—
67#821—42no—
68#821—42no—
69#809—24no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8C2×C412yes—
74#533C2×D311yescharacteristic
75C2×#127—31no—
76#492—32no—
77#486—32no—
78#460—62no—
79#450—62no—
80#446—22no—
81#446—22no—
82#431—31no—
83#431—31no—
84#424—21no—
85#423C2×D311yescharacteristic
86C2×#102—61no—
87C2×#102—61no—
88#405—21no—
89C2×#97C2×D311yescharacteristic
90#382—31no—
91#331—62no—
92#328—62no—
93C23×D3—41no—
94C2×C3×Q8—21no—
95C2×C3×Q8C22⋊C412yes—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C22×Dic3—41no—
100C2×D12—22no—
101C2×D12—22no—
102C2×D12—22no—
103C2×D12—41no—
104C2×D12—41no—
105C2×D12—41no—
106C2×D12—41no—
107C2×D12—41no—
108C2×D12—41no—
109C2×D12—41no—
110C2×D12—82no—
111C2×D12—82no—
112C3×M16—41no—
113C3×M16—41no—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8—22no—
117C2×C3×C8—21no—
118C2×C3×C8—21no—
119C2×C3×C8—21no—
120C2×C3×C8SD1612yescomplemented
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4C22⋊C412yes—
127C3×C22⋊C4—22no—
128C3×C42—22no—
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133#215—84no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—41no—
138C22×D4—61no—
139C22×D4—61no—
140C22×D4—61no—
141C2×SD16—64no—
142C2×M16—21no—
143C22×C8#6711yescharacteristic
144C22×C8#6711yescharacteristic
145C4⋊Q8C4×D311yescharacteristic
146C4⋊Q8C4×D311yescharacteristic
147#127—32no—
148#127—61no—
149#127—61no—
150#127—61no—
151#125—21no—
152C22⋊Q8—21no—
153C22⋊Q8D1212yes—
154C4×Q8—21no—
155C4×Q8#6712yes—
156#117—21no—
157C2×C4⋊C4—61no—
158C2×C42C22×D311yescharacteristic
159C4⋊C8—24no—
160C4⋊C8—21no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163C4⋊C8#6712yes—
164#102—64no—
165#102—122no—
166#102—122no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169#97—22no—
170#97—21no—
171#97—21no—
172C4×C8—22no—
173C4×C8—32no—
174C4×C8—61no—
175#95—61no—
176#95—61no—
177C23×C3#12111yescharacteristic
178C22×D3—42no—
179C22×D3—82no—
180C22×D3—82no—
181C22×D3—81no—
182C22×D3—81no—
183C3×Q8—42no—
184C3×Q8—81no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4C2×C22⋊C411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×C3×C4C2×SD1611yescharacteristic
196C2×C3×C4#13511yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
198C2×C3×C4#13711yescharacteristic
199C2×Dic3—81no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202D12—44no—
203D12—42no—
204D12—42no—
205D12—82no—
206D12—82no—
207D12—82no—
208D12—81no—
209D12—81no—
210C3×C8—22no—
211C3×C8—41no—
212C3×C8—42no—
213C3⋊C8—42no—
214C3⋊C8—81no—
215C24—121no—
216C2×Q8—21no—
217C2×Q8(C2×D3)⋊C412yes—
218C2×D4—62no—
219C2×D4—62no—
220C2×D4—62no—
221C2×D4—121no—
222C2×D4—121no—
223C2×D4—121no—
224C2×D4—121no—
225C2×D4—121no—
226C2×D4—121no—
227C2×D4—121no—
228C2×D4—242no—
229C2×D4—242no—
230C22×C4—121no—
231C22×C4C2×D1211yescharacteristic
232C22×C4#23511yescharacteristic
233C22×C4C2×#6711yescharacteristic
234SD16—244no—
235M16—41no—
236M16—41no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—22no—
242C2×C8—21no—
243C2×C8—62no—
244C2×C8—121no—
245C2×C8#21512yescomplemented
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4(C2×D3)⋊C412yes—
254C22⋊C4—22no—
255C42—22no—
256C42C2×C4×D311yescharacteristic
257C42#23511yescharacteristic
258C42C2×#6711yescharacteristic
259C42C2×#6711yescharacteristic
260C22×C3—21no—
261C22×C3—21no—
262C22×C3#38611yescharacteristic
263C22×C3#46011yescharacteristic
264C22×C3#46411yescharacteristic
265C2×D3—82no—
266C2×D3—82no—
267C2×D3—82no—
268C2×D3—162no—
269C2×D3—162no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—42no—
275C3×C4—42no—
276C3×C4C4×SD1611yescharacteristic
277C3×C4#44011yescharacteristic
278C3×C4#44811yescharacteristic
279C3×C4#44911yescharacteristic
280Dic3—161no—
281C23—122no—
282C23—242no—
283C23—242no—
284C23—241no—
285C23—241no—
286C23#93011yescharacteristic
287Q8—42no—
288Q8—81no—
289D4—124no—
290D4—122no—
291D4—122no—
292D4—242no—
293D4—242no—
294D4—242no—
295D4—241no—
296D4—241no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—241no—
302C2×C4—241no—
303C2×C4—241no—
304C2×C4C2×(C2×D3)⋊C411yescharacteristic
305C2×C4C4×#6711yescharacteristic
306C2×C4C4×#6711yescharacteristic
307C2×C4#92911yescharacteristic
308C2×C4#93011yescharacteristic
309C2×C4C2×#21511yescharacteristic
310C2×C4#94211yescharacteristic
311C2×C4#94911yescharacteristic
312C2×C4#95011yescharacteristic
313C2×C4#95311yescharacteristic
314C8—22no—
315C8—41no—
316C8—42no—
317C8—122no—
318C8—241no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3#188011yescharacteristic
322C2×C3#190311yescharacteristic
323C2×C3#191011yescharacteristic
324D3—162no—
325C22—21no—
326C22—21no—
327C22—242no—
328C22—242no—
329C22—242no—
330C22—482no—
331C22—482no—
332C22#502711yescharacteristic
333C22#502811yescharacteristic
334C22#521311yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—42no—
340C4—42no—
341C4—481no—
342C4C4×#21511yescharacteristic
343C4#502611yescharacteristic
344C4#522711yescharacteristic
345C4#523811yescharacteristic
346C3#929411yescharacteristic, complemented, Sylow
347C2—21no—
348C2—21no—
349C2—482no—
350C2#7163311yescharacteristic
351C2#7202311yescharacteristic
352C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2961C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#2961D3
hypocenterC3#9294
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
1. C3×#2961C2semidirect
3. #71838C2semidirect
6. #67713C2semidirect
7. #67660C2semidirect
19. #5088C4semidirect
20. #5085C4semidirect
29. #2961D3semidirect
120. C2×C3×C8SD16semidirect
245. C2×C8#215semidirect
346. C3#9294semidirect

Maximal subgroups

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

subgroupindex
C3×#29612
#833552
#718382
#705022
#698232
#677132
#676602
#92943