Finite Group Databasebeta

← #10686236

Subgroups — #10686236

Order 768.

392 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83126C211yesmaximal, characteristic
2C3×#1524C211yesmaximal, characteristic
3#71754C211yesmaximal, characteristic
4#69847C211yesmaximal, characteristic, complemented
5#69806C211yesmaximal, characteristic, complemented
6#67665C211yesmaximal, characteristic, complemented
7#67662C211yesmaximal, characteristic, complemented
8#8248—31nomaximal, Sylow
9C3×#529C2211yescharacteristic
10C3×C4×M16C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12#5024C411yescharacteristic, complemented
13Q8×C3⋊C8C411yescharacteristic, complemented
14#5012C411yescharacteristic, complemented
15#5011—21no—
16D4×C3⋊C8—21no—
17D4×C3⋊C8C411yescharacteristic, complemented
18#5000—21no—
19#4997—21no—
20#4932C2211yescharacteristic
21#4926C2211yescharacteristic
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#330C412yescomplemented
25C3×#329C412yescomplemented
26#4550—21no—
27#4550—21no—
28#4538C412yes—
29#4537C412yes—
30#4454—21no—
31#4454—21no—
32#2966—31no—
33#1746—31no—
34#1524D311yescharacteristic
35#1500—31no—
36#1451—31no—
37#1302—31no—
38#1300—31no—
39C2×C3×M16—21no—
40C22×C3×C8—21no—
41C3×#126—21no—
42C3×#125C2×C411yescharacteristic
43C3×#124C2×C411yescharacteristic
44C3×#123—21no—
45C3×C22⋊Q8C2×C411yescharacteristic
46C3×#121C2×C411yescharacteristic
47C3×C4×Q8C2×C411yescharacteristic
48C3×C4×D4—21no—
49C3×C4×D4C2×C411yescharacteristic
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C2×C4×Dic3—21no—
53C2×#208—21no—
54C22×C3⋊C8—21no—
55C22×C3⋊C8D411yescharacteristic
56C22×C3⋊C8D411yescharacteristic
57C3×#97—21no—
58C3×#97C2×C412yes—
59C3×#97C2×C412yes—
60C3×C4×C8—21no—
61C3⋊#95—21no—
62C3⋊#95C2×C412yes—
63#830—21no—
64#830—21no—
65#830—21no—
66#830C2×C412yes—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)C2×C411yescharacteristic
71C4⋊(C3⋊C8)C2×C411yescharacteristic
72#803—21no—
73#803—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8C2×C411yescharacteristic
77C4×C3⋊C8C2×C411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79C4×C3⋊C8D411yescharacteristic
80#529C2×D311yescharacteristic
81#446—31no—
82C8×Q8—31no—
83#436—31no—
84#435—61no—
85C8×D4—31no—
86C8×D4—61no—
87#433—61no—
88#432—31no—
89#431—31no—
90#405—61no—
91C4×M16C2×D311yescharacteristic
92C2×#97C2×D311yescharacteristic
93C2×C4×C8—31no—
94#376—31no—
95#336—61no—
96#336—61no—
97#330Dic312yescomplemented
98#329Dic312yescomplemented
99#328—32no—
100#327—32no—
101#322—61no—
102#322—61no—
103C2×C3×Q8C4211yescharacteristic
104C2×C3×D4—21no—
105C2×C3×D4C4211yescharacteristic
106C22×C3×C4—21no—
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—22no—
111C3×M16—42no—
112C2×C3×C8—22no—
113C2×C3×C8—22no—
114C2×C3×C8—41no—
115C2×C3×C8—41no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4C4211yescharacteristic
121C3×C4⋊C4C4211yescharacteristic
122C3×C22⋊C4—21no—
123C3×C22⋊C4—21no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4—21no—
126C3×C22⋊C4—21no—
127C3×C42—21no—
128C3×C42C22×C411yescharacteristic
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C4×Dic3—41no—
133C4×Dic3—41no—
134#208—42no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—42no—
141C2×C3⋊C8C22⋊C411yescharacteristic
142C2×C3⋊C8C22⋊C411yescharacteristic
143C2×C3⋊C8C22⋊C411yescharacteristic
144C2×C3⋊C8C22⋊C411yescharacteristic
145C2×M16—21no—
146C2×M16—61no—
147C22×C8—21no—
148C22×C8—31no—
149C22×C8—31no—
150C22×C8—61no—
151#126—21no—
152#125C2×Dic311yescharacteristic
153#124C2×Dic311yescharacteristic
154#123—21no—
155C22⋊Q8C4×D311yescharacteristic
156#121C4×D311yescharacteristic
157C4×Q8C4×D311yescharacteristic
158C4×D4—21no—
159C4×D4C4×D311yescharacteristic
160#117—21no—
161C2×C42—61no—
162C2×C42C22×D311yescharacteristic
163C4⋊C8—31no—
164C4⋊C8—31no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168C22⋊C8—32no—
169C22⋊C8—61no—
170C22⋊C8—61no—
171C22⋊C8—61no—
172#97—21no—
173#97—61no—
174#97—61no—
175#97C2×Dic312yes—
176#97C2×Dic312yes—
177C4×C8—21no—
178C4×C8—31no—
179C4×C8—31no—
180C4×C8—31no—
181C4×C8—31no—
182C4×C8—61no—
183C4×C8—61no—
184#95—32no—
185#95—61no—
186C23×C3—21no—
187C23×C3C4×D411yescharacteristic
188C3×Q8—41no—
189C3×D4—41no—
190C3×D4—42no—
191C2×C3×C4—22no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4C2×C4211yescharacteristic
201C2×C3×C4C2×C22⋊C411yescharacteristic
202C2×C3×C4#11711yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—42no—
209C3×C8—42no—
210C3×C8—42no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214C3⋊C8—21no—
215C3⋊C8—42no—
216C3⋊C8—41no—
217C3⋊C8—41no—
218C2×Q8C4×Dic311yescharacteristic
219C2×D4—21no—
220C2×D4C4×Dic311yescharacteristic
221C22×C4—21no—
222C22×C4—62no—
223C22×C4C2×C4×D311yescharacteristic
224C22×C4D3×D411yescharacteristic
225C22×C4#23711yescharacteristic
226M16—42no—
227M16—122no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—31no—
233C2×C8—31no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—61no—
237C2×C8—61no—
238C2×C8—62no—
239C2×C8—121no—
240C2×C8—121no—
241C2×C8—122no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4C4×Dic311yescharacteristic
247C4⋊C4C4×Dic311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C42—21no—
254C42—121no—
255C42—121no—
256C42C2×C4×D311yescharacteristic
257C42D3×D411yescharacteristic
258C42#23711yescharacteristic
259C42C22×Dic311yescharacteristic
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3C4×C22⋊C411yescharacteristic
266C22×C3#44412yes—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4#42111yescharacteristic
276C3×C4#42111yescharacteristic
277Dic3—82no—
278C23—21no—
279C23#67⋊C411yescharacteristic
280Q8—41no—
281D4—41no—
282D4—42no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4#87711yescharacteristic
296C2×C4D3×C22⋊C411yescharacteristic
297C2×C4#67⋊C411yescharacteristic
298C2×C4C2×C4×Dic311yescharacteristic
299C2×C4D4×Dic311yescharacteristic
300C2×C4D4×Dic311yescharacteristic
301C8—42no—
302C8—42no—
303C8—61no—
304C8—61no—
305C8—61no—
306C8—61no—
307C8—122no—
308C8—121no—
309C8—121no—
310C2×C3—22no—
311C2×C3—81no—
312C2×C3#175711yescharacteristic
313C2×C3#175911yescharacteristic
314C2×C3#176511yescharacteristic
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22#482312yes—
321C22Dic3×C22⋊C411yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—242no—
331C4#482111yescharacteristic
332C4#482111yescharacteristic
333C3#824811yescharacteristic, complemented, Sylow
334C2—22no—
335C2—81no—
336C2#7013311yescharacteristic
337C2#7035411yescharacteristic
338C2#7037011yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C2×C42
FrattiniC2×C42C22×D3
FittingC3×#1524C2
socleC22×C3C4×C22⋊C4
prime radicalC3×#121C2×C4
cosocleC2×C3×C42C23
hypercenter#1524D3
hypocenterC3#8248
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1524

Complements

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

NH ≅ G/Ntype
4. #69847C2semidirect
5. #69806C2semidirect
6. #67665C2semidirect
7. #67662C2semidirect
12. #5024C4semidirect
13. Q8×C3⋊C8C4semidirect
14. #5012C4semidirect
17. D4×C3⋊C8C4semidirect
24. C3×#330C4semidirect
25. C3×#329C4semidirect
97. #330Dic3semidirect
98. #329Dic3semidirect
333. C3#8248semidirect

Maximal subgroups

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

subgroupindex
#831262
C3×#15242
#717542
#698472
#698062
#676652
#676622
#82483