Finite Group Databasebeta

← #10692769

Subgroups — #10692769

Order 768.

440 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 97 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#83199C211yesmaximal, characteristic
2#79289C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic, complemented
4#71773C211yesmaximal, characteristic, complemented
5#70505C211yesmaximal, characteristic
6#67743C211yesmaximal, characteristic, complemented
7#67710C211yesmaximal, characteristic, complemented
8#9433—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5093—21no—
14#5089C414yes—
15#5088—21no—
16#4966—22no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4838—21no—
21#4834—21no—
22#4727—22no—
23#4688—21no—
24C8×D12—21no—
25C3×#333—21no—
26C3×#333—21no—
27#4547—21no—
28#4547—21no—
29#4485—21no—
30#4481—21no—
31#3141—31no—
32#2979—31no—
33#1560—31no—
34#1548D311yescharacteristic, complemented
35#1546—31no—
36#1299—31no—
37#1299—31no—
38C22×C3×Q8—21no—
39C2×#235D412yes—
40C2×C3×M16—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8C2×C412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48#930—21no—
49#930—21no—
50C2×#208—21no—
51C2×#208—21no—
52C2×ΓL(1,𝔽25)—22no—
53C2×D3×C8—22no—
54#875C2×C412yes—
55#874D411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869D411yescharacteristic
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8D411yescharacteristic
62C3×C4⋊C8D411yescharacteristic
63C3×C22⋊C8—21no—
64C3×C4×C8—21no—
65#830—21no—
66(C2×D3)⋊C8—22no—
67(C2×D3)⋊C8—22no—
68#810—24no—
69#809—24no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)D411yescharacteristic
73C4⋊(C3⋊C8)D411yescharacteristic
74#803C2×C412yes—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#532—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79#489—34no—
80#487—61no—
81#486—61no—
82#436—61no—
83#436—61no—
84C8×D4—61no—
85C8×D4—61no—
86#431—31no—
87#431C2×D311yescharacteristic
88#423—31no—
89#423C2×D311yescharacteristic
90#417—62no—
91#408—62no—
92C4×M16—31no—
93#333—21no—
94#333—21no—
95#333—61no—
96#332—61no—
97#332—61no—
98#331—61no—
99C2×C3×Q8—22no—
100C2×C3×Q8—41no—
101C2×C3×Q8—81no—
102C2×C3×Q8—81no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×#67—22no—
108#235—42no—
109#235—42no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12C22⋊C412yes—
113C2×C4×D3—22no—
114C2×Dic6C22⋊C412yes—
115C3×M16—81no—
116C2×C3×C8—21no—
117C2×C3×C8—21no—
118C2×C3×C8—41no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131#208—41no—
132#208—41no—
133#208—81no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—41no—
138ΓL(1,𝔽25)—82no—
139D3×C8—82no—
140C2×#41—32no—
141C22×Q8—21no—
142C2×M16—21no—
143C2×M16—62no—
144C2×M16—61no—
145C2×M16—61no—
146C22×C8—62no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—31no—
151C4⋊Q8C4×D312yes—
152#127—31no—
153#124—32no—
154#121—61no—
155#121—61no—
156C4×D4—61no—
157C4×D4—61no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C42C22×D311yescharacteristic
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—31no—
164C4⋊C8—31no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8#6711yescharacteristic
168C4⋊C8#6711yescharacteristic
169#103—64no—
170#102—64no—
171C22⋊C8—21no—
172C22⋊C8—62no—
173C22⋊C8—62no—
174C22⋊C8—61no—
175#97—32no—
176C4×C8—21no—
177C4×C8—61no—
178C4×C8—61no—
179C23×C3#12311yescharacteristic
180C22×D3—22no—
181C3×Q8—82no—
182C3×Q8—81no—
183C3×Q8—81no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—42no—
191C2×C3×C4—82no—
192C2×C3×C4C2×C22⋊C411yescharacteristic
193C2×C3×C4C4×D411yescharacteristic
194C2×C3×C4C4×D411yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198#67—82no—
199C2×Dic3—22no—
200D12—42no—
201D12—41no—
202D12—41no—
203D12—41no—
204D12—41no—
205C4×D3—42no—
206C4×D3—42no—
207Dic6—42no—
208C3×C8—41no—
209C3×C8—41no—
210C3×C8—81no—
211C3⋊C8—42no—
212C3⋊C8—41no—
213C3⋊C8—41no—
214C3⋊C8—81no—
215#41—122no—
216#41—122no—
217C2×Q8—22no—
218C2×Q8—32no—
219C2×Q8—41no—
220C2×Q8—81no—
221C2×Q8—81no—
222C2×D4—32no—
223C2×D4—62no—
224C2×D4—61no—
225C2×D4—61no—
226C22×C4—41no—
227C22×C4—62no—
228C22×C4#23511yescharacteristic
229C22×C4D3×D411yescharacteristic
230C22×C4#23711yescharacteristic
231M16—81no—
232M16—121no—
233M16—121no—
234M16—242no—
235M16—241no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—62no—
240C2×C8—61no—
241C2×C8—61no—
242C2×C8—121no—
243C2×C8—242no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C22⋊C4—62no—
251C22⋊C4—62no—
252C42C2×C4×D311yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C42C2×#6711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3#38611yescharacteristic
260C2×D3—42no—
261C2×D3—42no—
262C2×D3—42no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—82no—
267C3×C4#45511yescharacteristic
268C3×C4#45511yescharacteristic
269C3×C4#47111yescharacteristic
270C3×C4#47111yescharacteristic
271C3×C4#47211yescharacteristic
272C3×C4#47311yescharacteristic
273Dic3—42no—
274C23—62no—
275C23#88311yescharacteristic
276Q8—82no—
277Q8—81no—
278Q8—81no—
279Q8—122no—
280D4—122no—
281D4—121no—
282D4—121no—
283D4—121no—
284D4—121no—
285D4—242no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—62no—
294C2×C4—82no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4D3×C22⋊C411yescharacteristic
298C2×C4#67⋊C411yescharacteristic
299C2×C4#88411yescharacteristic
300C2×C4C4×#6711yescharacteristic
301C2×C4#93711yescharacteristic
302C2×C4#93811yescharacteristic
303C8—41no—
304C8—41no—
305C8—81no—
306C8—122no—
307C8—121no—
308C8—121no—
309C8—241no—
310C2×C3—41no—
311C2×C3#188511yescharacteristic
312C2×C3#190711yescharacteristic
313C2×C3#190811yescharacteristic
314D3—82no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—122no—
319C22—122no—
320C22—122no—
321C22#495611yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—82no—
326C4—122no—
327C4#474811yescharacteristic
328C4#474911yescharacteristic
329C4#517711yescharacteristic
330C4#519211yescharacteristic
331C4#520211yescharacteristic
332C4#520211yescharacteristic
333C3#943311yescharacteristic, complemented, Sylow
334C2—41no—
335C2—242no—
336C2#7008811yescharacteristic
337C2#7028611yescharacteristic
338C2#7034311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9433
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C3×#1548C2semidirect
4. #71773C2semidirect
6. #67743C2semidirect
7. #67710C2semidirect
34. #1548D3semidirect
333. C3#9433semidirect

Maximal subgroups

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

subgroupindex
#831992
#792892
C3×#15482
#717732
#705052
#677432
#677102
#94333