Finite Group Databasebeta

← #10686255

Subgroups — #10686255

Order 768.

482 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83275C211yesmaximal, characteristic
2C3×#1502C211yesmaximal, characteristic
3#71804C211yesmaximal, characteristic
4#70740C211yesmaximal, characteristic, complemented
5C4⋊C4×C3⋊C8C211yesmaximal, characteristic, complemented
6#67745C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#8251—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#948—21no—
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5234—21no—
14#5012—21no—
15#5012C412yescomplemented
16D4×C3⋊C8—21no—
17D4×C3⋊C8C412yescomplemented
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4×C3⋊C8—21no—
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×#327C414yes—
24#4547C412yescomplemented
25#4544C412yescomplemented
26#4524—21no—
27#4454—21no—
28C8×C3⋊C8—21no—
29#2931—31no—
30C8×C4⋊C4—31no—
31#1540—31no—
32#1502D311yescharacteristic
33#1450—31no—
34#1303—31no—
35#1301—31no—
36C2×C3×#41—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#127C2×C411yescharacteristic
39C3×#124—21no—
40C3×#121—21no—
41C3×#121C2×C412yes—
42C3×C4×D4—21no—
43C3×C4×D4C2×C412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47#948—42no—
48#948—42no—
49C2×#208—21no—
50C2×#208—22no—
51C22×C3⋊C8—21no—
52C22×C3⋊C8—22no—
53C22×C3⋊C8D411yescharacteristic
54C22×C3⋊C8D411yescharacteristic
55C3×C4⋊C8—21no—
56C3×C22⋊C8C2×C412yes—
57C3×C4×C8—21no—
58C3×#95C2×C412yes—
59#830—21no—
60#830—22no—
61#830—22no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)C2×C412yes—
65C4⋊(C3⋊C8)C2×C412yes—
66C4⋊(C3⋊C8)C2×C412yes—
67C4×C3⋊C8—21no—
68C4×C3⋊C8—42no—
69C4×C3⋊C8C2×C411yescharacteristic
70C4×C3⋊C8C2×C411yescharacteristic
71C4×C3⋊C8D411yescharacteristic
72C4×C3⋊C8D411yescharacteristic
73C2×#131—61no—
74#532C2×D311yescharacteristic
75#436—32no—
76#436—61no—
77C8×D4—32no—
78C8×D4—61no—
79#431—31no—
80#431C2×D311yescharacteristic
81#423—31no—
82C2×C4⋊C8—31no—
83#408—61no—
84C2×C4×C8—31no—
85C2×C4×C8—61no—
86C4×C4⋊C4C2×D311yescharacteristic
87#336—61no—
88#332—32no—
89#331—32no—
90#327Dic314yes—
91#322—61no—
92C82—61no—
93C3×#41—41no—
94C3×#41—41no—
95C3×#41—41no—
96C3×#41—41no—
97C2×C3×Q8—21no—
98C2×C3×D4—22no—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4C4212yes—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×C3×C8—22no—
108C3×C4⋊C4—24no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4C4212yes—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C42—41no—
114C3×C42—41no—
115C3×C42C22×C411yescharacteristic
116C3×C42C22×C411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42#4111yescharacteristic
119#208—41no—
120#208—41no—
121#208—44no—
122C2×C3⋊C8—21no—
123C2×C3⋊C8—21no—
124C2×C3⋊C8—22no—
125C2×C3⋊C8—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—44no—
130C2×C3⋊C8C22⋊C411yescharacteristic
131C2×C3⋊C8C22⋊C411yescharacteristic
132C2×C3⋊C8C22⋊C411yescharacteristic
133C2×C3⋊C8C22⋊C411yescharacteristic
134C2×C3⋊C8D812yes—
135C2×C3⋊C8SD1612yes—
136C2×#41—21no—
137#131—122no—
138#131—122no—
139C2×M16—62no—
140C2×M16—61no—
141C22×C8—31no—
142C22×C8—31no—
143C22×C8—62no—
144C22×C8—61no—
145C4⋊Q8C4×D311yescharacteristic
146#127C4×D311yescharacteristic
147#124—21no—
148#121—21no—
149#121C2×Dic312yes—
150C4×D4—21no—
151C4×D4C4×D312yes—
152C2×C4⋊C4—21no—
153C2×C42—21no—
154C2×C42C22×D311yescharacteristic
155C4⋊C8—21no—
156C4⋊C8—32no—
157C4⋊C8—32no—
158C4⋊C8—32no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C22⋊C8—62no—
162C22⋊C8—62no—
163C22⋊C8—61no—
164C22⋊C8C2×Dic312yes—
165C4×C8—21no—
166C4×C8—31no—
167C4×C8—31no—
168C4×C8—31no—
169C4×C8—31no—
170C4×C8—61no—
171C4×C8—122no—
172#95C2×Dic312yes—
173C23×C3—22no—
174C23×C3C4×D411yescharacteristic
175C3×Q8—41no—
176C3×Q8—41no—
177C3×D4—42no—
178C3×D4—42no—
179C3×D4—42no—
180C3×D4—41no—
181C3×D4—41no—
182C3×D4—42no—
183C2×C3×C4—22no—
184C2×C3×C4—24no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4C2×C4211yescharacteristic
192C2×C3×C4C2×C22⋊C411yescharacteristic
193C2×C3×C4#11711yescharacteristic
194C2×C3×C4C4×D411yescharacteristic
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4C2×D811yescharacteristic
198C2×C3×C4C2×SD1611yescharacteristic
199C2×C3×C4#13511yescharacteristic
200C2×C3×C4#13511yescharacteristic
201C3×C8—42no—
202C3⋊C8—21no—
203C3⋊C8—21no—
204C3⋊C8—42no—
205C3⋊C8—42no—
206C3⋊C8—42no—
207C3⋊C8#10212yes—
208C3⋊C8#10212yes—
209#41—41no—
210#41—41no—
211#41—41no—
212#41—41no—
213C2×Q8—21no—
214C2×D4—22no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4C4×Dic312yes—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4D3×D411yescharacteristic
221C22×C4#23711yescharacteristic
222C22×C4C22×Dic311yescharacteristic
223M16—124no—
224M16—121no—
225M16—121no—
226C2×C8—22no—
227C2×C8—31no—
228C2×C8—31no—
229C2×C8—32no—
230C2×C8—32no—
231C2×C8—31no—
232C2×C8—31no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—62no—
236C2×C8—62no—
237C2×C8—62no—
238C2×C8—124no—
239C2×C8—121no—
240C2×C8—121no—
241C4⋊C4—24no—
242C4⋊C4—21no—
243C4⋊C4C4×Dic312yes—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C42—41no—
247C42—41no—
248C42C2×C4×D311yescharacteristic
249C42C2×C4×D311yescharacteristic
250C42D3×D411yescharacteristic
251C42#23711yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3C4×C22⋊C411yescharacteristic
258C22×C3C4×D811yescharacteristic
259C22×C3C4×SD1611yescharacteristic
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—44no—
264C3×C4—42no—
265C3×C4C2×#10211yescharacteristic
266C3×C4#41611yescharacteristic
267C3×C4C4×D811yescharacteristic
268C3×C4C4×SD1611yescharacteristic
269C3×C4#44312yes—
270C23—22no—
271C23D4×Dic311yescharacteristic
272Q8—41no—
273Q8—41no—
274D4—42no—
275D4—42no—
276D4—42no—
277D4—41no—
278D4—41no—
279D4—42no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4#87711yescharacteristic
289C2×C4D3×C22⋊C411yescharacteristic
290C2×C4#67⋊C411yescharacteristic
291C2×C4#67⋊C411yescharacteristic
292C2×C4D3×D811yescharacteristic
293C2×C4#91211yescharacteristic
294C2×C4D3×SD1611yescharacteristic
295C2×C4#91611yescharacteristic
296C2×C4C2×C4×Dic311yescharacteristic
297C2×C4D4×Dic311yescharacteristic
298C8—32no—
299C8—32no—
300C8—42no—
301C8—61no—
302C8—61no—
303C8—122no—
304C8—122no—
305C8—122no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—82no—
309C2×C3#175711yescharacteristic
310C2×C3C4×#10211yescharacteristic
311C2×C3#176411yescharacteristic
312C22—21no—
313C22—21no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22Dic3×C22⋊C411yescharacteristic
318C22Dic3×D811yescharacteristic
319C22Dic3×SD1611yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—44no—
324C4—42no—
325C4#475611yescharacteristic
326C4#475811yescharacteristic
327C4D3×#10211yescharacteristic
328C4#477211yescharacteristic
329C4#482412yes—
330C3#825111yescharacteristic, complemented, Sylow
331C2—21no—
332C2—21no—
333C2—82no—
334C2#7004011yescharacteristic
335C2Dic3×#10211yescharacteristic
336C2#7036911yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C2×C42
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3C4×C22⋊C4
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1502D3
hypocenterC3#8251
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
4. #70740C2semidirect
5. C4⋊C4×C3⋊C8C2semidirect
6. #67745C2semidirect
7. #67620C2semidirect
15. #5012C4semidirect
17. D4×C3⋊C8C4semidirect
24. #4547C4semidirect
25. #4544C4semidirect
330. C3#8251semidirect

Maximal subgroups

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

subgroupindex
#832752
C3×#15022
#718042
#707402
C4⋊C4×C3⋊C82
#677452
#676202
#82513