Finite Group Databasebeta

← #10691833

Subgroups — #10691833

Order 768.

478 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 109 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#83216C211yesmaximal, characteristic
2#78491C211yesmaximal, characteristic
3C3×#1993C211yesmaximal, characteristic, complemented
4#71774C211yesmaximal, characteristic, complemented
5#69545C211yesmaximal, characteristic
6#67660C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9287—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#901—21no—
11#5477C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5110—21no—
15#5088C412yescomplemented
16#5084—22no—
17#5080C412yescomplemented
18#4968—22no—
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22#4706—22no—
23#4706—22no—
24C3×#340—21no—
25C3×#340—21no—
26#4546—22no—
27#4451—22no—
28#3227—31no—
29#2932—31no—
30#1993D311yescharacteristic, complemented
31#1548—31no—
32#1517—31no—
33#1286—31no—
34#1275—31no—
35C22×C3×Q8—21no—
36C2×#235—21no—
37C22×C3×C8D411yescharacteristic
38C22×C3×C8D411yescharacteristic
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8C2×C412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47#930—21no—
48#930D412yes—
49C2×#208—21no—
50#901—42no—
51#901—42no—
52C2×ΓL(1,𝔽25)—22no—
53C2×D3×C8—22no—
54#875—21no—
55#874C2×C411yescharacteristic
56C4×D12—21no—
57C4×D12D412yes—
58#869C2×C411yescharacteristic
59C3×#97—21no—
60C3×#97—21no—
61C3×#95—21no—
62C3×#95—21no—
63#830—21no—
64#830—21no—
65(C2×D3)⋊C8—24no—
66#809—24no—
67#808—24no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D412yes—
72C4×C3⋊C8—21no—
73C4×C3⋊C8C2×C412yes—
74C2×#131—61no—
75#532—31no—
76C2×C4⋊Q8C2×D311yescharacteristic
77#498—32no—
78#486—32no—
79#481—62no—
80#476—62no—
81#435—62no—
82#435—62no—
83#431—31no—
84#431—31no—
85#423—31no—
86#408—61no—
87C2×#97C2×D311yescharacteristic
88#382C2×D311yescharacteristic
89#340—21no—
90#340—21no—
91#333—62no—
92#327—62no—
93C2×C3×Q8—22no—
94C2×C3×Q8—41no—
95C2×C3×Q8—82no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C2×#67—22no—
102#235—41no—
103#235—41no—
104#235—41no—
105#235—41no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12C22⋊C412yes—
109C2×C4×D3—22no—
110C2×Dic6—21no—
111C2×C3×C8—21no—
112C2×C3×C8—21no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8SD1612yes—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129C4⋊Dic3—21no—
130C4⋊Dic3C22⋊C412yes—
131#208—41no—
132#208—41no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—41no—
136ΓL(1,𝔽25)—44no—
137D3×C8—44no—
138C2×#41—61no—
139C22×Q8—21no—
140#131—122no—
141#131—122no—
142C2×M16—62no—
143C2×M16—61no—
144C22×C8—62no—
145C22×C8#6711yescharacteristic
146C22×C8#6711yescharacteristic
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—31no—
151C4⋊Q8C4×D312yes—
152#127—31no—
153#124—61no—
154#121—32no—
155#121—61no—
156C4×D4—32no—
157C4×D4—61no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C42C22×D311yescharacteristic
162#106—64no—
163C4⋊C8—32no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167#102—64no—
168C22⋊C8—64no—
169C22⋊C8—61no—
170C22⋊C8—61no—
171#97—21no—
172#97—21no—
173C4×C8—32no—
174C4×C8—61no—
175#95—21no—
176#95—21no—
177C23×C3#12111yescharacteristic
178C22×D3—22no—
179C3×Q8—82no—
180C3×Q8—82no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—82no—
185C2×C3×C4—81no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4C2×C22⋊C411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12311yescharacteristic
194C2×C3×C4C2×SD1611yescharacteristic
195C2×C3×C4#13511yescharacteristic
196C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
197C2×C3×C4#13711yescharacteristic
198#67—42no—
199#67—42no—
200C2×Dic3—22no—
201D12—42no—
202D12—41no—
203D12—41no—
204D12—81no—
205C4×D3—42no—
206C4×D3—42no—
207Dic6—41no—
208Dic6—41no—
209C3×C8—22no—
210C3×C8—41no—
211C3⋊C8—42no—
212C3⋊C8—42no—
213C3⋊C8—81no—
214#41—121no—
215#41—121no—
216#41—121no—
217#41—121no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—61no—
221C2×Q8—82no—
222C2×D4—32no—
223C2×D4—62no—
224C2×D4—61no—
225C2×D4—61no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—62no—
229C22×C4D3×D411yescharacteristic
230C22×C4#23711yescharacteristic
231C22×C4C2×#6711yescharacteristic
232M16—124no—
233M16—121no—
234M16—121no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—62no—
241C2×C8—62no—
242C2×C8—124no—
243C2×C8—121no—
244C2×C8#21512yes—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—32no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—61no—
253C22⋊C4—62no—
254C22⋊C4—62no—
255C42C2×C4×D311yescharacteristic
256C42#23511yescharacteristic
257C42D3×D411yescharacteristic
258C42D3×D411yescharacteristic
259C22×C3—21no—
260C22×C3—21no—
261C22×C3#38611yescharacteristic
262C22×C3#46111yescharacteristic
263C22×C3#46311yescharacteristic
264C2×D3—42no—
265C2×D3—42no—
266C2×D3—42no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—82no—
272C3×C4—161no—
273C3×C4C4×SD1611yescharacteristic
274C3×C4#44011yescharacteristic
275C3×C4#45011yescharacteristic
276C3×C4#45211yescharacteristic
277Dic3—42no—
278C23—62no—
279C23#93811yescharacteristic
280Q8—82no—
281Q8—82no—
282Q8—121no—
283Q8—121no—
284D4—122no—
285D4—122no—
286D4—122no—
287D4—121no—
288D4—121no—
289D4—241no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—62no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4D3×C22⋊C411yescharacteristic
301C2×C4#67⋊C411yescharacteristic
302C2×C4#88311yescharacteristic
303C2×C4#88411yescharacteristic
304C2×C4C4×#6711yescharacteristic
305C2×C4#93711yescharacteristic
306C2×C4C2×#21511yescharacteristic
307C2×C4#94211yescharacteristic
308C2×C4#94911yescharacteristic
309C2×C4#95011yescharacteristic
310C8—22no—
311C8—41no—
312C8—122no—
313C8—122no—
314C8—241no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3#187711yescharacteristic
318C2×C3#190311yescharacteristic
319C2×C3#190911yescharacteristic
320D3—82no—
321C22—21no—
322C22—21no—
323C22—122no—
324C22—122no—
325C22—122no—
326C22#495611yescharacteristic
327C22#505711yescharacteristic
328C22#508311yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—82no—
334C4—122no—
335C4—161no—
336C4C4×#21511yescharacteristic
337C4#502611yescharacteristic
338C4#504611yescharacteristic
339C4#504711yescharacteristic
340C3#928711yescharacteristic, complemented, Sylow
341C2—21no—
342C2—21no—
343C2—242no—
344C2#6968711yescharacteristic
345C2#7032911yescharacteristic
346C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1993C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1993D3
hypocenterC3#9287
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
3. C3×#1993C2semidirect
4. #71774C2semidirect
6. #67660C2semidirect
7. #67631C2semidirect
15. #5088C4semidirect
17. #5080C4semidirect
30. #1993D3semidirect
340. C3#9287semidirect

Maximal subgroups

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

subgroupindex
#832162
#784912
C3×#19932
#717742
#695452
#676602
#676312
#92873