Finite Group Databasebeta

← #10691487

Subgroups — #10691487

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
1C3×#2295C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic
3#71622C211yesmaximal, characteristic, complemented
4#69469C211yesmaximal, characteristic
5#69395C211yesmaximal, characteristic
6#67620C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#9238—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#391—21no—
12C3×#389—21no—
13C3×#387—22no—
14C3×C4×C4⋊C4—21no—
15C3×C4×C4⋊C4C2211yescharacteristic
16C3×#376—21no—
17C3×C43C2211yescharacteristic
18#5227—21no—
19#5027C412yescomplemented
20C4×#215—22no—
21#5020C412yescomplemented
22#4968—21no—
23C2×#805—21no—
24C2×#805—21no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4535—22no—
29#4451—22no—
30#3103—31no—
31#2295D311yescharacteristic, complemented
32#1962—31no—
33#1502—31no—
34#1487—31no—
35#1303—31no—
36#1276—31no—
37C22×C3×Q8D411yescharacteristic
38C2×#235—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8C2×C412yes—
42C3×C4×Q8D412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C42—21no—
48C2×C3×C42—21no—
49C2×C3×C42—21no—
50C2×C3×C42—21no—
51C2×C3×C42D411yescharacteristic
52C2×C3×C42D411yescharacteristic
53C2×C3×C42C2311yescharacteristic
54C2×#215—22no—
55#930—21no—
56#930—21no—
57C2×#208—21no—
58#875—21no—
59#874C2×C411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869C2×C411yescharacteristic
63C3×#95—22no—
64C3×#95—22no—
65C3×#95—22no—
66C3×#95D412yes—
67#835—22no—
68#830—21no—
69#830—21no—
70#809—22no—
71#808—22no—
72#805—42no—
73#805—42no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yes—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80#532—31no—
81C2×C4×Q8C2×D311yescharacteristic
82#481—61no—
83#475—32no—
84#460—32no—
85#448—61no—
86C4×SD16—62no—
87#431—31no—
88#431—31no—
89#423—31no—
90C2×#104—61no—
91C2×#104—61no—
92#391—21no—
93#389—21no—
94#387—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4C2×D311yescharacteristic
97#376—21no—
98C43C2×D311yescharacteristic
99#327—62no—
100Q8⋊C8—62no—
101C2×C3×Q8—21no—
102C2×C3×Q8—42no—
103C2×C3×Q8C22⋊C412yes—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—41no—
113#235—41no—
114#235—41no—
115#235—41no—
116#235—41no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—21no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—44no—
127C3×C4⋊C4C22⋊C412yes—
128C3×C42—22no—
129C3×C42—22no—
130C3×C42—21no—
131C3×C42—21no—
132C3×C42—21no—
133C3×C42—42no—
134C3×C42—42no—
135C3×C42—41no—
136C3×C42—41no—
137C3×C42SD1612yescomplemented
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142#215—44no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147#208—41no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C2×#41—61no—
153C22×Q8#6711yescharacteristic
154C2×SD16—62no—
155C2×M16—61no—
156C4⋊Q8—31no—
157#127—31no—
158#124—61no—
159#121—61no—
160#121—61no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8C4×D312yes—
164C4×Q8D1212yes—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4#6711yescharacteristic
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42#6711yescharacteristic
176C2×C42#6711yescharacteristic
177C2×C42C22×D311yescharacteristic
178#106—62no—
179C4⋊C8—32no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183#104—122no—
184#104—122no—
185#103—62no—
186#102—62no—
187C22⋊C8—61no—
188C22⋊C8—61no—
189C4×C8—61no—
190C4×C8—61no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95#6712yes—
195C23×C3#12011yescharacteristic
196C22×D3—41no—
197C3×Q8—22no—
198C3×Q8—22no—
199C3×Q8—42no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4C2×C22⋊C411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12311yescharacteristic
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4#13711yescharacteristic
225#67—81no—
226#67—81no—
227C2×Dic3—41no—
228D12—42no—
229D12—41no—
230D12—41no—
231D12—81no—
232C4×D3—81no—
233C4×D3—81no—
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—21no—
244C2×Q8—42no—
245C2×Q8—61no—
246C2×Q8(C2×D3)⋊C412yes—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—121no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—121no—
257C22×C4C2×#6711yescharacteristic
258C22×C4C2×#6711yescharacteristic
259C22×C4C2×#6711yescharacteristic
260SD16—124no—
261M16—121no—
262M16—121no—
263C2×C8—62no—
264C2×C8—121no—
265C2×C8—121no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4(C2×D3)⋊C412yes—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C42—22no—
277C42—22no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—42no—
282C42—42no—
283C42—41no—
284C42—41no—
285C42#21512yescomplemented
286C42C2×C4×D311yescharacteristic
287C42C2×D1211yescharacteristic
288C42#23511yescharacteristic
289C42#23511yescharacteristic
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—42no—
304C3×C4—42no—
305C3×C4—82no—
306C3×C4#42112yes—
307C3×C4C4×SD1611yescharacteristic
308C3×C4#44011yescharacteristic
309C3×C4#46011yescharacteristic
310C3×C4#46411yescharacteristic
311Dic3—81no—
312C23—121no—
313C23#95311yescharacteristic
314Q8—22no—
315Q8—22no—
316Q8—42no—
317Q8—121no—
318Q8—121no—
319D4—122no—
320D4—121no—
321D4—121no—
322D4—241no—
323D4—241no—
324D4—241no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—121no—
341C2×C4—241no—
342C2×C4—241no—
343C2×C4C2×(C2×D3)⋊C411yescharacteristic
344C2×C4C4×#6711yescharacteristic
345C2×C4C4×#6711yescharacteristic
346C2×C4#92911yescharacteristic
347C2×C4#93011yescharacteristic
348C2×C4#93011yescharacteristic
349C2×C4C2×#21511yescharacteristic
350C2×C4#94211yescharacteristic
351C2×C4#94911yescharacteristic
352C2×C4#95011yescharacteristic
353C8—122no—
354C8—241no—
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#521311yescharacteristic
368C22#522711yescharacteristic
369C22#523811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—82no—
378C4—241no—
379C4#500512yes—
380C4C4×#21511yescharacteristic
381C4#502611yescharacteristic
382C4#502711yescharacteristic
383C4#502811yescharacteristic
384C3#923811yescharacteristic, complemented, Sylow
385C2—21no—
386C2—21no—
387C2—481no—
388C2#7159111yescharacteristic
389C2#7163311yescharacteristic
390C2#7164511yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2295C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2295D3
hypocenterC3#9238
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2295

Complements

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

NH ≅ G/Ntype
1. C3×#2295C2semidirect
3. #71622C2semidirect
6. #67620C2semidirect
7. #67611C2semidirect
19. #5027C4semidirect
21. #5020C4semidirect
31. #2295D3semidirect
137. C3×C42SD16semidirect
285. C42#215semidirect
384. C3#9238semidirect

Maximal subgroups

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

subgroupindex
C3×#22952
#824422
#716222
#694692
#693952
#676202
#676112
#92383