Finite Group Databasebeta

← #10691486

Subgroups — #10691486

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79309C211yesmaximal, characteristic
2#73300C211yesmaximal, characteristic, complemented
3#70789C211yesmaximal, characteristic
4#70770C211yesmaximal, characteristic, complemented
5#69395C211yesmaximal, characteristic
6C3×#1276C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#9238—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C43C2211yescharacteristic
14C4×C4⋊Dic3C2211yescharacteristic
15#4933—22no—
16#4932—21no—
17C4×Dic3⋊C4—21no—
18#4929—21no—
19#4928—21no—
20C2×#805—21no—
21C2×#805—21no—
22#4924C2211yescharacteristic
23#4851C412yescomplemented
24#4848C412yescomplemented
25#4738—21no—
26#4735—21no—
27C4×#204—22no—
28#4470—22no—
29#4456—22no—
30#3103—31no—
31#2295—31no—
32#1962—31no—
33#1502—31no—
34#1487—31no—
35#1303—31no—
36#1276D311yescharacteristic, complemented
37C2×#235—21no—
38C22×Dic6D411yescharacteristic
39C2×C3×C42—21no—
40C2×C3×C42—21no—
41C2×C3×C42D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3D411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—22no—
50C2×C4×Dic3—21no—
51C2×C4×Dic3—21no—
52C2×#208—21no—
53C2×#204—22no—
54#875—21no—
55#874C2×C411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869C2×C411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C412yes—
62C4×Dic6D412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8C2×C412yes—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68C3×C4×C8—21no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95D412yes—
73#821—22no—
74#817—22no—
75#816—22no—
76#805—42no—
77#805—42no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80#532—31no—
81C2×C4×Q8—31no—
82#481—61no—
83#475—32no—
84#460—32no—
85#448—61no—
86C4×SD16—62no—
87#431C2×D311yescharacteristic
88#431C2×D311yescharacteristic
89#423—31no—
90C2×#104—61no—
91C2×#104—61no—
92#391—61no—
93#389—61no—
94#387—62no—
95C4×C4⋊C4—31no—
96C4×C4⋊C4—61no—
97#376—61no—
98C43C2×D311yescharacteristic
99#327—62no—
100Q8⋊C8—62no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C2×#67—41no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—22no—
110#235—41no—
111#235—41no—
112#235—41no—
113#235—41no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×C4×D3—41no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—42no—
121C2×Dic6C22⋊C412yes—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C3×C42—21no—
125C3×C42—21no—
126C3×C42—21no—
127C3×C42—42no—
128C3×C42—42no—
129C3×C42SD1612yescomplemented
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—42no—
140C4⋊Dic3C22⋊C412yes—
141Dic3⋊C4—22no—
142Dic3⋊C4—22no—
143Dic3⋊C4—44no—
144C4×Dic3—22no—
145C4×Dic3—22no—
146C4×Dic3—41no—
147C4×Dic3—41no—
148#208—41no—
149#208—41no—
150C2×C3⋊C8—41no—
151#204—44no—
152C2×#41—61no—
153C22×Q8—31no—
154C2×SD16—62no—
155C2×M16—61no—
156C4⋊Q8—31no—
157#127—31no—
158#124—61no—
159#121—61no—
160#121—61no—
161C4×Q8—32no—
162C4×Q8—32no—
163C4×Q8—61no—
164C4×Q8—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—62no—
170C2×C4⋊C4—61no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—61no—
174C2×C42—61no—
175C2×C42D1211yescharacteristic
176C2×C42D1211yescharacteristic
177C2×C42C22×D311yescharacteristic
178#106—62no—
179C4⋊C8—21no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8C4×D312yes—
183#104—122no—
184#104—122no—
185#103—62no—
186#102—62no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C4×C8—21no—
190C4×C8—21no—
191#95—32no—
192#95—62no—
193#95—62no—
194#95—62no—
195C23×C3#12011yescharacteristic
196C22×D3—41no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×C3×C4C2×SD1611yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×C3×C4#13711yescharacteristic
215#67—81no—
216#67—81no—
217C2×Dic3—22no—
218C2×Dic3—22no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225D12—42no—
226D12—41no—
227D12—41no—
228D12—81no—
229C4×D3—81no—
230C4×D3—81no—
231Dic6—22no—
232Dic6—22no—
233Dic6—42no—
234Dic6—41no—
235Dic6—41no—
236C3×C8—42no—
237C3×C8—81no—
238C3⋊C8—81no—
239#41—121no—
240#41—121no—
241#41—121no—
242#41—121no—
243C2×Q8—32no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—122no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—121no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—62no—
256C22×C4—121no—
257C22×C4C2×D1211yescharacteristic
258C22×C4D3×D411yescharacteristic
259C22×C4D3×D411yescharacteristic
260SD16—124no—
261M16—121no—
262M16—121no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—121no—
266C4⋊C4—32no—
267C4⋊C4—62no—
268C4⋊C4—62no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—124no—
273C4⋊C4—122no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—42no—
280C42—42no—
281C42—62no—
282C42—62no—
283C42—121no—
284C42—121no—
285C42#20412yescomplemented
286C42C2×C4×D311yescharacteristic
287C42#23511yescharacteristic
288C42D3×D411yescharacteristic
289C42#23911yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#38611yescharacteristic
293C22×C3#44811yescharacteristic
294C22×C3#44911yescharacteristic
295C2×D3—81no—
296C2×D3—81no—
297C2×D3—81no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—42no—
302C3×C4—42no—
303C3×C4#42112yes—
304C3×C4C4×SD1611yescharacteristic
305C3×C4#44011yescharacteristic
306C3×C4#46011yescharacteristic
307C3×C4#46411yescharacteristic
308Dic3—42no—
309Dic3—42no—
310Dic3—82no—
311Dic3—81no—
312C23—121no—
313C23#88211yescharacteristic
314Q8—62no—
315Q8—62no—
316Q8—122no—
317Q8—121no—
318Q8—121no—
319D4—121no—
320D4—121no—
321D4—122no—
322D4—241no—
323D4—241no—
324D4—241no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—122no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4C4×D1211yescharacteristic
344C2×C4D3×C22⋊C411yescharacteristic
345C2×C4#88411yescharacteristic
346C2×C4#89311yescharacteristic
347C2×C4#89411yescharacteristic
348C2×C4#89511yescharacteristic
349C2×C4C2×#20411yescharacteristic
350C2×C4#90411yescharacteristic
351C2×C4#90811yescharacteristic
352C2×C4#90911yescharacteristic
353C8—42no—
354C8—81no—
355C8—241no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3#188011yescharacteristic
359C2×C3#189711yescharacteristic
360C2×C3#190611yescharacteristic
361D3—161no—
362C22—21no—
363C22—21no—
364C22—241no—
365C22—241no—
366C22—241no—
367C22#466511yescharacteristic
368C22#473411yescharacteristic
369C22#473811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—42no—
374C4—42no—
375C4—122no—
376C4—122no—
377C4—242no—
378C4—241no—
379C4C4×#20411yescharacteristic
380C4#471011yescharacteristic
381C4#482112yes—
382C4#484511yescharacteristic
383C4#484811yescharacteristic
384C3#923811yescharacteristic, complemented, Sylow
385C2—21no—
386C2—21no—
387C2—481no—
388C2#6868611yescharacteristic
389C2#6875711yescharacteristic
390C2#6877811yescharacteristic
subgroupG/H
centerC2×C4#895
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1276D3
hypocenterC3#9238
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #73300C2semidirect
4. #70770C2semidirect
6. C3×#1276C2semidirect
7. #67746C2semidirect
23. #4851C4semidirect
24. #4848C4semidirect
36. #1276D3semidirect
129. C3×C42SD16semidirect
285. C42#204semidirect
384. C3#9238semidirect

Maximal subgroups

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

subgroupindex
#793092
#733002
#707892
#707702
#693952
C3×#12762
#677462
#92383