Finite Group Databasebeta

← #10692681

Subgroups — #10692681

Order 768.

402 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 91 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#83347C211yesmaximal, characteristic
2#83333C211yesmaximal, characteristic, complemented
3C3×#1515C211yesmaximal, characteristic, complemented
4#70811C211yesmaximal, characteristic
5#69564C211yesmaximal, characteristic
6#67753C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9418—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5088C411yescharacteristic, complemented
14#5085C411yescharacteristic, complemented
15#5079C411yescharacteristic, complemented
16#5078C411yescharacteristic, complemented
17#5049—22no—
18#5026—22no—
19#5024—22no—
20#5024—22no—
21#5000—22no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24C4×#208C2211yescharacteristic
25C3×#328—22no—
26#4464—22no—
27#4451—22no—
28#3182—31no—
29#2987—31no—
30#1536—31no—
31#1515D311yescharacteristic, complemented
32#1514—31no—
33#1291—31no—
34#1288—31no—
35C2×#235—21no—
36C3×C4⋊Q8C2×C411yescharacteristic
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8D412yes—
41C3×C4×Q8—21no—
42C3×C4×Q8D412yes—
43C3×#117—21no—
44C2×C3×C42C2311yescharacteristic
45C2×#215—22no—
46#930—21no—
47#930—21no—
48C2×#208—21no—
49#875—21no—
50#874C2×C411yescharacteristic
51C4×D12—21no—
52C4×D12—21no—
53#869C2×C411yescharacteristic
54C3×C4⋊C8—21no—
55C3×C22⋊C8—21no—
56C3×#97—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60#835—22no—
61#830—21no—
62#830D412yes—
63#809—22no—
64#807—22no—
65C4⋊(C3⋊C8)—24no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68#803—22no—
69#803—21no—
70#803—21no—
71#803D412yes—
72C4×C3⋊C8—22no—
73C4×C3⋊C8C2×C411yescharacteristic
74C4×C3⋊C8C2×C411yescharacteristic
75#533C2×D311yescharacteristic
76#532—31no—
77#500—31no—
78#492—31no—
79#487—31no—
80#486—31no—
81#468—62no—
82#446—62no—
83#446—62no—
84#440—62no—
85#433—62no—
86#432—31no—
87#432—31no—
88#432C2×D311yescharacteristic
89C4×M16—31no—
90#383C2×D311yescharacteristic
91#329—62no—
92#328—22no—
93#327—62no—
94C2×C3×Q8—21no—
95C2×C3×Q8C22⋊C412yes—
96C22×C3×C4—41no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C2×#67—41no—
101#235—82no—
102C2×D12—21no—
103C2×D12—21no—
104C2×D12—21no—
105C2×C4×D3—41no—
106C2×Dic6—21no—
107C2×C3×C8—41no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4C22⋊C412yes—
114C3×C22⋊C4—22no—
115C3×C42—22no—
116C3×C42C22×C411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42#4111yescharacteristic
120#215—44no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125#208—42no—
126C2×C3⋊C8—21no—
127C2×C3⋊C8—21no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8—22no—
130C2×#41—61no—
131C2×SD16—62no—
132C2×M16—61no—
133C4⋊Q8—31no—
134C4⋊Q8C4×D311yescharacteristic
135C4⋊Q8C4×D311yescharacteristic
136#127—31no—
137#125—21no—
138#124—61no—
139C22⋊Q8—21no—
140C22⋊Q8#6712yes—
141#121—61no—
142#121—61no—
143C4×Q8—21no—
144C4×Q8D1212yes—
145C4×D4—61no—
146C4×D4—61no—
147#117—21no—
148C2×C42C22×D311yescharacteristic
149#107—62no—
150C4⋊C8—21no—
151C4⋊C8—64no—
152C4⋊C8—61no—
153C4⋊C8—61no—
154#103—62no—
155#102—62no—
156C22⋊C8—21no—
157C22⋊C8—32no—
158C22⋊C8—61no—
159#97—21no—
160#97—32no—
161#97—62no—
162#97—61no—
163#97—61no—
164C4×C8—31no—
165C4×C8—31no—
166C4×C8—62no—
167#95—21no—
168#95—21no—
169#95—21no—
170C23×C3#12111yescharacteristic
171C22×D3—41no—
172C3×Q8—42no—
173C3×Q8—81no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—81no—
180C2×C3×C4—81no—
181C2×C3×C4—81no—
182C2×C3×C4C2×C22⋊C411yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4C4×D411yescharacteristic
185C2×C3×C4#12011yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12311yescharacteristic
188#67—82no—
189C2×Dic3—41no—
190D12—42no—
191D12—81no—
192D12—81no—
193C4×D3—82no—
194Dic6—81no—
195C3×C8—81no—
196C3⋊C8—22no—
197C3⋊C8—42no—
198C3⋊C8—42no—
199C3⋊C8—41no—
200#41—242no—
201C2×Q8—21no—
202C2×Q8—61no—
203C2×Q8(C2×D3)⋊C412yes—
204C2×D4—61no—
205C2×D4—61no—
206C2×D4—61no—
207C2×D4—121no—
208C22×C4—41no—
209C22×C4—121no—
210C22×C4D3×D411yescharacteristic
211C22×C4#23711yescharacteristic
212C22×C4C2×#6711yescharacteristic
213SD16—124no—
214M16—122no—
215C2×C8—41no—
216C2×C8—61no—
217C2×C8—62no—
218C2×C8—62no—
219C2×C8—61no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4(C2×D3)⋊C412yes—
228C22⋊C4—22no—
229C22⋊C4—121no—
230C22⋊C4—121no—
231C42—22no—
232C42C2×C4×D311yescharacteristic
233C42C2×D1211yescharacteristic
234C42D3×D411yescharacteristic
235C42#23711yescharacteristic
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3#38611yescharacteristic
240C2×D3—81no—
241C2×D3—81no—
242C2×D3—81no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—42no—
250C3×C4—42no—
251C3×C4—161no—
252Dic3—81no—
253C23—121no—
254C23#93911yescharacteristic
255Q8—42no—
256Q8—81no—
257Q8—241no—
258D4—122no—
259D4—241no—
260D4—241no—
261D4—242no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—121no—
271C2×C4—242no—
272C2×C4#67⋊C411yescharacteristic
273C2×C4#67⋊C411yescharacteristic
274C2×C4#88211yescharacteristic
275C2×C4#88611yescharacteristic
276C2×C4C2×(C2×D3)⋊C411yescharacteristic
277C2×C4#93911yescharacteristic
278C8—62no—
279C8—81no—
280C8—121no—
281C8—122no—
282C8—122no—
283C2×C3—41no—
284C2×C3#188411yescharacteristic
285C2×C3#190311yescharacteristic
286C2×C3#190511yescharacteristic
287D3—161no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—241no—
292C22—241no—
293C22—241no—
294C22#495811yescharacteristic
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—161no—
304C4—241no—
305C3#941811yescharacteristic, complemented, Sylow
306C2—41no—
307C2—481no—
308C2#7004511yescharacteristic
309C2#7033411yescharacteristic
310C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1515C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1515D3
hypocenterC3#9418
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
2. #83333C2semidirect
3. C3×#1515C2semidirect
6. #67753C2semidirect
7. #67633C2semidirect
13. #5088C4semidirect
14. #5085C4semidirect
15. #5079C4semidirect
16. #5078C4semidirect
31. #1515D3semidirect
305. C3#9418semidirect

Maximal subgroups

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

subgroupindex
#833472
#833332
C3×#15152
#708112
#695642
#677532
#676332
#94183