Finite Group Databasebeta

← #10692416

Subgroups — #10692416

Order 768.

442 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 95 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#83307C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic, complemented
3C3×#1551C211yesmaximal, characteristic, complemented
4#70811C211yesmaximal, characteristic, complemented
5#70426C211yesmaximal, characteristic, complemented
6#67757C211yesmaximal, characteristic, complemented
7#67708C211yesmaximal, characteristic, complemented
8#9371—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13#5238—22no—
14#5234—22no—
15#5083—21no—
16#5072—21no—
17#5058C412yes—
18#5055C412yes—
19#5012—21no—
20#5011—22no—
21D4×C3⋊C8—21no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25C3×#333—21no—
26C3×#331—21no—
27#4483—21no—
28#4483—21no—
29#4464—22no—
30#3214—31no—
31#2981—31no—
32#1551D311yescharacteristic, complemented
33#1536—31no—
34#1480—31no—
35#1290—31no—
36#1287—31no—
37C2×C3×#41D412yes—
38C2×#235—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8D411yescharacteristic
41C3×#127D411yescharacteristic
42C3×#124C2×C412yes—
43C3×#121—21no—
44C3×#121—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#215—22no—
49C2×#213—22no—
50#930—21no—
51#930—21no—
52C2×#208—22no—
53C22×C3⋊C8—22no—
54#875—21no—
55#874C2×C411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869C2×C411yescharacteristic
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×#97—21no—
64#835—22no—
65#832—22no—
66#830—22no—
67#830—22no—
68#830—21no—
69#830—21no—
70#830—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803C2×C412yes—
75#803D412yes—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#532—31no—
79#532C2×D311yescharacteristic
80#496—32no—
81#489—32no—
82#461—61no—
83#459—61no—
84#449—62no—
85#436—61no—
86#435—62no—
87C8×D4—61no—
88#432—31no—
89#432—31no—
90#432C2×D311yescharacteristic
91#431—31no—
92C2×C4⋊C8C2×D311yescharacteristic
93#408—62no—
94#333—21no—
95#331—21no—
96#329—62no—
97D4⋊C8—61no—
98D4⋊C8—61no—
99C3×#41—82no—
100C2×C3×Q8C22⋊C412yes—
101C2×C3×D4—22no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4C22⋊C412yes—
105C22×C3×C4—22no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—41no—
110#235—81no—
111#235—81no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12—21no—
115C2×C4×D3—41no—
116C2×Dic6—21no—
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C2×C3×C8—81no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128#215—82no—
129#213—82no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—41no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134#208—82no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—82no—
140C2×#41—61no—
141C2×#41D1212yes—
142C2×SD16—62no—
143C2×D8—62no—
144C2×M16—62no—
145C22×C8—21no—
146C22×C8—62no—
147C4⋊Q8—31no—
148C4⋊Q8#6711yescharacteristic
149#127—31no—
150#127#6711yescharacteristic
151#124—61no—
152#124C4×D312yes—
153#121—21no—
154#121—21no—
155#121—61no—
156#121—61no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—61no—
160C4×D4—61no—
161C2×C42C22×D311yescharacteristic
162C4⋊C8—21no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168#103—62no—
169#102—62no—
170C22⋊C8—21no—
171C22⋊C8—62no—
172C22⋊C8—62no—
173C22⋊C8—61no—
174C22⋊C8—61no—
175C22⋊C8—61no—
176#97—21no—
177#97—32no—
178#97—32no—
179C4×C8—31no—
180C4×C8—31no—
181C23×C3—22no—
182C23×C3#12311yescharacteristic
183C22×D3—41no—
184C3×Q8—42no—
185C3×D4—42no—
186C3×D4—41no—
187C3×D4—41no—
188C3×D4—81no—
189C3×D4—82no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—22no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4C2×C22⋊C411yescharacteristic
198C2×C3×C4C4×D411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203#67—82no—
204C2×Dic3—41no—
205D12—41no—
206D12—41no—
207D12—42no—
208D12—81no—
209C4×D3—81no—
210C4×D3—81no—
211Dic6—81no—
212C3×C8—81no—
213C3×C8—81no—
214C3⋊C8—41no—
215C3⋊C8—42no—
216C3⋊C8—41no—
217C3⋊C8—42no—
218#41—82no—
219#41—241no—
220#41—241no—
221C2×Q8—61no—
222C2×Q8(C2×D3)⋊C412yes—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—121no—
230C2×D4(C2×D3)⋊C412yes—
231C22×C4—22no—
232C22×C4—121no—
233C22×C4C2×D1211yescharacteristic
234C22×C4#23711yescharacteristic
235C22×C4#23711yescharacteristic
236SD16—242no—
237D8—242no—
238M16—242no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—61no—
242C2×C8—61no—
243C2×C8—62no—
244C2×C8—62no—
245C2×C8—81no—
246C2×C8—242no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C42C2×C4×D311yescharacteristic
256C42D3×D411yescharacteristic
257C42D3×D411yescharacteristic
258C42C2×#6711yescharacteristic
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3#38611yescharacteristic
266C2×D3—81no—
267C2×D3—81no—
268C2×D3—81no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—42no—
275C3×C4#47111yescharacteristic
276C3×C4#47211yescharacteristic
277Dic3—81no—
278C23—22no—
279C23—121no—
280C23#88611yescharacteristic
281Q8—42no—
282Q8—241no—
283D4—42no—
284D4—41no—
285D4—41no—
286D4—81no—
287D4—82no—
288D4—121no—
289D4—121no—
290D4—122no—
291D4—241no—
292D4—242no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—121no—
301C2×C4—241no—
302C2×C4—241no—
303C2×C4#67⋊C411yescharacteristic
304C2×C4#67⋊C411yescharacteristic
305C2×C4#88211yescharacteristic
306C2×C4C2×(C2×D3)⋊C411yescharacteristic
307C2×C4#93911yescharacteristic
308C2×C4#93911yescharacteristic
309C8—81no—
310C8—81no—
311C8—121no—
312C8—122no—
313C8—121no—
314C8—122no—
315C2×C3—41no—
316C2×C3—82no—
317C2×C3#188111yescharacteristic
318C2×C3#190711yescharacteristic
319C2×C3#191111yescharacteristic
320D3—161no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22#495811yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—241no—
338C4#519911yescharacteristic
339C4#520311yescharacteristic
340C3#937111yescharacteristic, complemented, Sylow
341C2—41no—
342C2—82no—
343C2—481no—
344C2#7007111yescharacteristic
345C2#7029311yescharacteristic
346C2#7030311yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1551C2
socleC22×C3#386
prime radical#83307C2
cosocleC2×C3×C42C23
hypercenter#1551D3
hypocenterC3#9371
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1551

Complements

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

NH ≅ G/Ntype
2. #83280C2semidirect
3. C3×#1551C2semidirect
4. #70811C2semidirect
5. #70426C2semidirect
6. #67757C2semidirect
7. #67708C2semidirect
32. #1551D3semidirect
340. C3#9371semidirect

Maximal subgroups

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

subgroupindex
#833072
#832802
C3×#15512
#708112
#704262
#677572
#677082
#93713