Finite Group Databasebeta

← #10694588

Subgroups — #10694588

Order 768.

440 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78468C211yesmaximal, characteristic
2C3×#1726C211yesmaximal, characteristic, complemented
3#69801C211yesmaximal, characteristic, complemented
4#69798C211yesmaximal, characteristic
5#69732C211yesmaximal, characteristic, complemented
6#69433C211yesmaximal, characteristic, complemented
7#69432C211yesmaximal, characteristic
8#9761—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×#97—21no—
12C2×C3×#97C2211yescharacteristic
13C2×C3×C4×C8—21no—
14C2×C3×C4×C8C2211yescharacteristic
15C3×#376C2211yescharacteristic
16#5114—21no—
17#5114—21no—
18#4926C2211yescharacteristic
19#4926C2211yescharacteristic
20#4694C411yescharacteristic, complemented
21#4693C411yescharacteristic, complemented
22C4×#204—21no—
23C4×#204—21no—
24#4681C411yescharacteristic, complemented
25#4680C411yescharacteristic, complemented
26#4465—22no—
27#4464—22no—
28#4458—21no—
29#4457—21no—
30#4452—22no—
31#4451—22no—
32#3167—31no—
33#1726D311yescharacteristic, complemented
34#1568—31no—
35#1531—31no—
36#1530—31no—
37#1506—31no—
38#1505—31no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C22×C3×C8D411yescharacteristic
42C22×C3×C8D411yescharacteristic
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45#930—21no—
46#930—21no—
47#926—21no—
48#924—21no—
49#924—21no—
50C2×#204—21no—
51C2×#204—21no—
52#875—21no—
53#874C2×C411yescharacteristic
54C4×D12—21no—
55C4×D12—21no—
56#870—21no—
57#869C2×C411yescharacteristic
58#869C2×C411yescharacteristic
59#869C2×C411yescharacteristic
60C4×Dic6—21no—
61C4×Dic6—21no—
62C3×#97—24no—
63C3×#97—21no—
64C3×#97D412yes—
65C3×C4×C8—24no—
66C3×C4×C8C2×C411yescharacteristic
67C3×C4×C8C2×C411yescharacteristic
68C3×C4×C8D411yescharacteristic
69C3×C4×C8D411yescharacteristic
70C3×#95—21no—
71C3×#95D412yes—
72#830—21no—
73#830—21no—
74#821—21no—
75#821—21no—
76#817—21no—
77#817—21no—
78#816—21no—
79#816—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#803—21no—
84#533—31no—
85#532—31no—
86#498—31no—
87#492—31no—
88#487—31no—
89#486—31no—
90#465—61no—
91#465—61no—
92C4×SD16—61no—
93C4×SD16—61no—
94#432—31no—
95#432—31no—
96C2×#97—21no—
97C2×#97C2×D311yescharacteristic
98C2×C4×C8—21no—
99C2×C4×C8C2×D311yescharacteristic
100#376C2×D311yescharacteristic
101#335—61no—
102#334—61no—
103#330—62no—
104#329—62no—
105#328—62no—
106#327—62no—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×#67—41no—
112#235—82no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×C4×D3—41no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×C3×C8—22no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8C22⋊C411yescharacteristic
126C2×C3×C8C22⋊C411yescharacteristic
127C2×C3×C8C22⋊C411yescharacteristic
128C2×C3×C8C22⋊C411yescharacteristic
129C3×C42—22no—
130C3×C42—22no—
131C3×C42C22×C411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#217—41no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149#204—42no—
150#204—42no—
151C2×#41—61no—
152C2×SD16—61no—
153C2×SD16—61no—
154C22×C8—21no—
155C22×C8#6711yescharacteristic
156C22×C8#6711yescharacteristic
157C4⋊Q8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160#127—31no—
161#125—61no—
162#124—61no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165#121—61no—
166#121—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169C4×D4—61no—
170C4×D4—61no—
171#117—61no—
172C2×C42—21no—
173C2×C42C22×D311yescharacteristic
174#106—61no—
175#106—61no—
176C4⋊C8—61no—
177C4⋊C8—61no—
178#103—61no—
179#103—61no—
180#102—61no—
181#102—61no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184#97—24no—
185#97—21no—
186#97—61no—
187#97—61no—
188#97D1212yes—
189C4×C8—24no—
190C4×C8C4×D311yescharacteristic
191C4×C8C4×D311yescharacteristic
192C4×C8D1211yescharacteristic
193C4×C8D1211yescharacteristic
194#95—21no—
195#95D1212yes—
196C23×C3#12111yescharacteristic
197C22×D3—41no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—24no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12411yescharacteristic
210C2×C3×C4#12711yescharacteristic
211#67—82no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215D12—41no—
216D12—41no—
217D12—81no—
218D12—81no—
219C4×D3—82no—
220Dic6—41no—
221Dic6—41no—
222Dic6—81no—
223Dic6—81no—
224C3×C8—24no—
225C3×C8—21no—
226C3×C8—21no—
227C3×C8#10412yes—
228C3×C8#10412yes—
229C3⋊C8—81no—
230C3⋊C8—81no—
231#41—242no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—121no—
239C22×C4—22no—
240C22×C4—121no—
241C22×C4C2×D1211yescharacteristic
242C22×C4#23511yescharacteristic
243C22×C4C2×#6711yescharacteristic
244SD16—122no—
245SD16—122no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—121no—
252C2×C8—121no—
253C2×C8(C2×D3)⋊C411yescharacteristic
254C2×C8(C2×D3)⋊C411yescharacteristic
255C2×C8(C2×D3)⋊C411yescharacteristic
256C2×C8(C2×D3)⋊C411yescharacteristic
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C42—22no—
269C42—22no—
270C42—121no—
271C42C2×C4×D311yescharacteristic
272C42C2×D1211yescharacteristic
273C42C2×D1211yescharacteristic
274C42#23511yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3#39011yescharacteristic
278C22×C3#47112yes—
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—44no—
288C3×C4C2×#10411yescharacteristic
289C3×C4C2×#10411yescharacteristic
290Dic3—81no—
291Dic3—81no—
292Dic3—81no—
293C23—121no—
294C23#93011yescharacteristic
295Q8—121no—
296Q8—121no—
297Q8—241no—
298Q8—241no—
299D4—121no—
300D4—121no—
301D4—241no—
302D4—241no—
303D4—242no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—242no—
315C2×C4C4×D1211yescharacteristic
316C2×C4C4×D1211yescharacteristic
317C2×C4#87411yescharacteristic
318C2×C4#87511yescharacteristic
319C2×C4C2×(C2×D3)⋊C411yescharacteristic
320C2×C4#93011yescharacteristic
321C8—24no—
322C8—21no—
323C8—21no—
324C8—241no—
325C8—241no—
326C8#80512yes—
327C8#80512yes—
328C2×C3—22no—
329C2×C3#196211yescharacteristic
330C2×C3#197611yescharacteristic
331C2×C3#197711yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—241no—
336C22—241no—
337C22—241no—
338C22#493911yescharacteristic
339C22#514112yes—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—44no—
346C4—241no—
347C4—241no—
348C4—241no—
349C4C2×#80511yescharacteristic
350C4C2×#80511yescharacteristic
351C3#976111yescharacteristic, complemented, Sylow
352C2—22no—
353C2—481no—
354C2#6939511yescharacteristic
355C2#6989711yescharacteristic
356C2#6990211yescharacteristic
subgroupG/H
centerC22#4939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1726C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1726D3
hypocenterC3#9761
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1726

Complements

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

NH ≅ G/Ntype
2. C3×#1726C2semidirect
3. #69801C2semidirect
5. #69732C2semidirect
6. #69433C2semidirect
20. #4694C4semidirect
21. #4693C4semidirect
24. #4681C4semidirect
25. #4680C4semidirect
33. #1726D3semidirect
351. C3#9761semidirect

Maximal subgroups

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

subgroupindex
#784682
C3×#17262
#698012
#697982
#697322
#694332
#694322
#97613