Finite Group Databasebeta

← #10692783

Subgroups — #10692783

Order 768.

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

normal subgroups only

This group has 504 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#83203C211yesmaximal, characteristic
2#79289C211yesmaximal, characteristic
3C3×#1549C211yesmaximal, characteristic, complemented
4#71775C211yesmaximal, characteristic, complemented
5#70505C211yesmaximal, characteristic
6#67743C211yesmaximal, characteristic, complemented
7#67710C211yesmaximal, characteristic, complemented
8#9435—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#942—22no—
11#5477C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5092—21no—
15#5086C414yes—
16#5085—21no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4838—21no—
21#4834—21no—
22#4727—22no—
23#4688—21no—
24C8×D12—21no—
25C3×#332—21no—
26C3×#332—21no—
27#4547—21no—
28#4547—21no—
29#4485—21no—
30#4481—21no—
31#3152—31no—
32#2979—31no—
33#1560—31no—
34#1549—31no—
35#1549D311yescharacteristic, complemented
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
48C2×#216—24no—
49#942—44no—
50#942—44no—
51C2×#215—24no—
52#930—21no—
53#930—21no—
54C2×#208—21no—
55C2×#208—21no—
56C2×ΓL(1,𝔽25)—22no—
57C2×D3×C8—22no—
58#875C2×C412yes—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8D411yescharacteristic
66C3×C4⋊C8D411yescharacteristic
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69#830—21no—
70(C2×D3)⋊C8—22no—
71(C2×D3)⋊C8—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D411yescharacteristic
75C4⋊(C3⋊C8)D411yescharacteristic
76#803C2×C412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C2×#137—62no—
80#532—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82#497—34no—
83#494—61no—
84#492—61no—
85#436—61no—
86#436—61no—
87C8×D4—61no—
88C8×D4—61no—
89#431—31no—
90#431C2×D311yescharacteristic
91#423—31no—
92#423C2×D311yescharacteristic
93#408—62no—
94C4×M16—31no—
95#333—61no—
96#332—21no—
97#332—21no—
98#332—61no—
99#332—61no—
100#331—61no—
101C2×C3×Q8—22no—
102C2×C3×Q8—42no—
103C2×C3×Q8—42no—
104C2×C3×Q8—41no—
105C22×C3×C4—41no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—22no—
110#235—42no—
111#235—42no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12C22⋊C412yes—
115C2×C4×D3—22no—
116C2×Dic6C22⋊C412yes—
117C3×M16—81no—
118C2×C3×C8—21no—
119C2×C3×C8—21no—
120C2×C3×C8—41no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C42C22×C411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129#216—48no—
130#215—48no—
131(C2×D3)⋊C4—22no—
132(C2×D3)⋊C4—22no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135#208—22no—
136#208—22no—
137#208—81no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—41no—
142ΓL(1,𝔽25)—82no—
143D3×C8—82no—
144C2×#41—32no—
145C22×Q8—21no—
146#137—124no—
147#137—124no—
148C2×Q16—64no—
149C2×SD16—64no—
150C2×M16—21no—
151C2×M16—62no—
152C2×M16—61no—
153C2×M16—61no—
154C22×C8—62no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—31no—
159C4⋊Q8C4×D312yes—
160#127—31no—
161#124—32no—
162#121—61no—
163#121—61no—
164C4×D4—61no—
165C4×D4—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C42C22×D311yescharacteristic
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—31no—
172C4⋊C8—31no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175C4⋊C8#6711yescharacteristic
176C4⋊C8#6711yescharacteristic
177C22⋊C8—21no—
178C22⋊C8—62no—
179C22⋊C8—62no—
180C22⋊C8—61no—
181#97—32no—
182C4×C8—21no—
183C4×C8—61no—
184C4×C8—61no—
185C23×C3#12311yescharacteristic
186C22×D3—22no—
187C3×Q8—44no—
188C3×Q8—81no—
189C3×Q8—81no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—42no—
197C2×C3×C4—82no—
198C2×C3×C4C2×C22⋊C411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204#67—82no—
205C2×Dic3—22no—
206D12—42no—
207D12—41no—
208D12—41no—
209D12—41no—
210D12—41no—
211C4×D3—42no—
212C4×D3—42no—
213Dic6—42no—
214C3×C8—41no—
215C3×C8—41no—
216C3×C8—81no—
217C3⋊C8—24no—
218C3⋊C8—41no—
219C3⋊C8—41no—
220C3⋊C8—81no—
221#41—122no—
222#41—122no—
223C2×Q8—22no—
224C2×Q8—32no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×D4—32no—
229C2×D4—62no—
230C2×D4—61no—
231C2×D4—61no—
232C22×C4—41no—
233C22×C4—62no—
234C22×C4#23511yescharacteristic
235C22×C4D3×D411yescharacteristic
236C22×C4#23711yescharacteristic
237Q16—128no—
238SD16—128no—
239M16—62no—
240M16—62no—
241M16—81no—
242M16—242no—
243M16—241no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—41no—
247C2×C8—62no—
248C2×C8—61no—
249C2×C8—61no—
250C2×C8—121no—
251C2×C8—242no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C22⋊C4—62no—
259C22⋊C4—62no—
260C42C2×C4×D311yescharacteristic
261C42D3×D411yescharacteristic
262C42D3×D411yescharacteristic
263C42C2×#6711yescharacteristic
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3#38611yescharacteristic
268C2×D3—42no—
269C2×D3—42no—
270C2×D3—42no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—82no—
275C3×C4#45311yescharacteristic
276C3×C4#45311yescharacteristic
277C3×C4#47111yescharacteristic
278C3×C4#47111yescharacteristic
279C3×C4#47311yescharacteristic
280C3×C4#47311yescharacteristic
281Dic3—42no—
282C23—62no—
283C23#88311yescharacteristic
284Q8—44no—
285Q8—81no—
286Q8—81no—
287Q8—122no—
288D4—122no—
289D4—121no—
290D4—121no—
291D4—121no—
292D4—121no—
293D4—242no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—62no—
302C2×C4—82no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4D3×C22⋊C411yescharacteristic
306C2×C4#67⋊C411yescharacteristic
307C2×C4#88411yescharacteristic
308C2×C4C4×#6711yescharacteristic
309C2×C4#93711yescharacteristic
310C2×C4#93811yescharacteristic
311C8—41no—
312C8—41no—
313C8—64no—
314C8—81no—
315C8—121no—
316C8—121no—
317C8—241no—
318C2×C3—41no—
319C2×C3#188711yescharacteristic
320C2×C3#190811yescharacteristic
321C2×C3#190811yescharacteristic
322D3—82no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—122no—
327C22—122no—
328C22—122no—
329C22#495611yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—82no—
334C4—122no—
335C4#475411yescharacteristic
336C4#475511yescharacteristic
337C4#517711yescharacteristic
338C4#519211yescharacteristic
339C4#519811yescharacteristic
340C4#519811yescharacteristic
341C3#943511yescharacteristic, complemented, Sylow
342C2—41no—
343C2—242no—
344C2#7012311yescharacteristic
345C2#7028611yescharacteristic
346C2#7032311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1549D3
hypocenterC3#9435
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. C3×#1549C2semidirect
4. #71775C2semidirect
6. #67743C2semidirect
7. #67710C2semidirect
35. #1549D3semidirect
341. C3#9435semidirect

Maximal subgroups

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

subgroupindex
#832032
#792892
C3×#15492
#717752
#705052
#677432
#677102
#94353