Finite Group Databasebeta

← #10695106

Subgroups — #10695106

Order 768.

498 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 113 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#83291C211yesmaximal, characteristic
2C3×#1962C211yesmaximal, characteristic, complemented
3#71802C211yesmaximal, characteristic, complemented
4#70498C211yesmaximal, characteristic, complemented
5#70490C211yesmaximal, characteristic, complemented
6#69395C211yesmaximal, characteristic, complemented
7#69388C211yesmaximal, characteristic, complemented
8#9852—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×#104—22no—
13C3×C43C2211yescharacteristic
14#5088C411yescharacteristic, complemented
15#5078C411yescharacteristic, complemented
16#5073C411yescharacteristic, complemented
17#5070C411yescharacteristic, complemented
18#5038—22no—
19C4×#213—22no—
20C2×#805—22no—
21#4924C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24D4⋊(C3⋊C8)—22no—
25#4524—22no—
26#4483—22no—
27#4480—22no—
28#3169—31no—
29#1962—31no—
30#1962D311yescharacteristic, complemented
31#1744—31no—
32#1566—31no—
33#1492—31no—
34#1492—31no—
35C2×C3×#41—21no—
36C2×#235—21no—
37C2×C3×M16—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#127C2×C411yescharacteristic
40C3×#124—21no—
41C3×#121—21no—
42C3×#121—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C2×C3×C42—21no—
46C2×C3×C42—21no—
47C2×C3×C42D412yes—
48C2×C3×C42C2311yescharacteristic
49C2×#213—22no—
50#930—21no—
51#930—21no—
52C2×#208—21no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8D412yes—
55#875—21no—
56#874C2×C411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#869C2×C411yescharacteristic
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×#104—44no—
63#832—22no—
64#809—22no—
65#807—22no—
66#805—44no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4×C3⋊C8—21no—
70C4×C3⋊C8C2×C411yescharacteristic
71C4×C3⋊C8C2×C411yescharacteristic
72C4×C3⋊C8D412yes—
73C4×C3⋊C8D412yes—
74#532—31no—
75#532C2×D311yescharacteristic
76#500—31no—
77#493—31no—
78#486—31no—
79#486—31no—
80#466—62no—
81C4×D8—62no—
82#423—31no—
83#423C2×D311yescharacteristic
84C2×#104—22no—
85C2×#104—62no—
86C2×C4×C8—31no—
87C2×C4×C8—31no—
88C43C2×D311yescharacteristic
89#336—62no—
90#334—62no—
91D4⋊C8—62no—
92D4⋊C8—62no—
93C3×#41—42no—
94C3×#41—42no—
95C2×C3×Q8—21no—
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—41no—
100C22×C3×C4—22no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×#67—41no—
106#235—42no—
107#235—42no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×C4×D3—41no—
112C2×Dic6—21no—
113C3×M16—42no—
114C2×C3×C8—41no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C42—24no—
120C3×C42—22no—
121C3×C42—22no—
122C3×C42—42no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127#213—44no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—41no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132#208—42no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—42no—
135C2×C3⋊C8—42no—
136C2×C3⋊C8—41no—
137C2×C3⋊C8C22⋊C412yes—
138C2×C3⋊C8C22⋊C412yes—
139C2×#41—21no—
140C2×#41—61no—
141C2×D8—62no—
142C2×M16—21no—
143C2×M16—61no—
144C22×C8—32no—
145C22×C8—61no—
146C4⋊Q8—31no—
147C4⋊Q8C4×D311yescharacteristic
148#127—31no—
149#127C4×D311yescharacteristic
150#124—21no—
151#124—61no—
152#121—21no—
153#121—21no—
154#121—61no—
155#121—61no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—61no—
159C4×D4—61no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42#6712yes—
163C2×C42C22×D311yescharacteristic
164#107—62no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—61no—
168C4⋊C8—61no—
169#104—44no—
170#104—124no—
171#102—62no—
172#102—62no—
173C4×C8—31no—
174C4×C8—31no—
175C4×C8—32no—
176C4×C8—32no—
177C4×C8—61no—
178C23×C3—41no—
179C23×C3#12111yescharacteristic
180C22×D3—41no—
181C3×Q8—42no—
182C3×D4—42no—
183C3×D4—42no—
184C3×D4—42no—
185C3×D4—82no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—24no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×C3×C4C2×C22⋊C411yescharacteristic
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12411yescharacteristic
199C2×C3×C4#12711yescharacteristic
200C2×C3×C4AGL(1,ℤ/8)12yes—
201C2×C3×C4AGL(1,ℤ/8)12yes—
202#67—82no—
203C2×Dic3—41no—
204D12—42no—
205D12—42no—
206D12—42no—
207C4×D3—81no—
208C4×D3—81no—
209Dic6—42no—
210C3×C8—81no—
211C3⋊C8—22no—
212C3⋊C8—22no—
213C3⋊C8—42no—
214C3⋊C8—81no—
215#41—42no—
216#41—42no—
217#41—122no—
218#41—122no—
219C2×Q8—21no—
220C2×Q8—61no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—41no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—121no—
229C22×C4—22no—
230C22×C4—41no—
231C22×C4—121no—
232C22×C4D3×D411yescharacteristic
233C22×C4#23711yescharacteristic
234C22×C4C2×#6711yescharacteristic
235D8—124no—
236M16—42no—
237M16—122no—
238C2×C8—32no—
239C2×C8—32no—
240C2×C8—41no—
241C2×C8—62no—
242C2×C8—122no—
243C2×C8—122no—
244C2×C8—121no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—121no—
252C22⋊C4—121no—
253C42—24no—
254C42—22no—
255C42—22no—
256C42—42no—
257C42C2×C4×D311yescharacteristic
258C42#23511yescharacteristic
259C42D3×D411yescharacteristic
260C42D3×D411yescharacteristic
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—81no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3#39011yescharacteristic
267C22×C3#46912yes—
268C2×D3—81no—
269C2×D3—81no—
270C2×D3—81no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—44no—
274C3×C4—81no—
275C3×C4#42112yes—
276C3×C4#44212yes—
277C3×C4#44312yes—
278C3×C4#49611yescharacteristic
279C3×C4#49611yescharacteristic
280Dic3—81no—
281C23—41no—
282C23—121no—
283C23#93911yescharacteristic
284Q8—42no—
285Q8—122no—
286D4—42no—
287D4—42no—
288D4—42no—
289D4—82no—
290D4—122no—
291D4—122no—
292D4—122no—
293D4—242no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—121no—
303C2×C4—241no—
304C2×C4—241no—
305C2×C4D3×C22⋊C411yescharacteristic
306C2×C4#67⋊C411yescharacteristic
307C2×C4#88411yescharacteristic
308C2×C4#88511yescharacteristic
309C2×C4C4×#6711yescharacteristic
310C2×C4#93212yes—
311C2×C4#94011yescharacteristic
312C2×C4#94912yes—
313C8—62no—
314C8—62no—
315C8—81no—
316C8—122no—
317C8—241no—
318C2×C3—22no—
319C2×C3—161no—
320C2×C3#196311yescharacteristic
321C2×C3#197211yescharacteristic
322C2×C3#197411yescharacteristic
323D3—161no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—241no—
330C22—241no—
331C22—241no—
332C22#495711yescharacteristic
333C22#503912yes—
334C4—21no—
335C4—21no—
336C4—44no—
337C4—81no—
338C4—241no—
339C4#482112yes—
340C4#482412yes—
341C4#501412yes—
342C4#505811yescharacteristic
343C4#507411yescharacteristic
344C3#985211yescharacteristic, complemented, Sylow
345C2—22no—
346C2—161no—
347C2—481no—
348C2#6964811yescharacteristic
349C2#7035711yescharacteristic
350C2#7037311yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#390
prime radical#83291C2
cosocleC2×C3×C42C23
hypercenter#1962D3
hypocenterC3#9852
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. C3×#1962C2semidirect
3. #71802C2semidirect
4. #70498C2semidirect
5. #70490C2semidirect
6. #69395C2semidirect
7. #69388C2semidirect
14. #5088C4semidirect
15. #5078C4semidirect
16. #5073C4semidirect
17. #5070C4semidirect
30. #1962D3semidirect
344. C3#9852semidirect

Maximal subgroups

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

subgroupindex
#832912
C3×#19622
#718022
#704982
#704902
#693952
#693882
#98523