Finite Group Databasebeta

← #10692051

Subgroups — #10692051

Order 768.

458 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83212C211yesmaximal, characteristic
2#79292C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic, complemented
4#71779C211yesmaximal, characteristic, complemented
5#70529C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9319—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C4×#235C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×#825—21no—
14#5094C412yescomplemented
15#5091—21no—
16#5087C412yescomplemented
17#5084—21no—
18#4966—21no—
19#4964—21no—
20C4×C4⋊Dic3C2211yescharacteristic
21#4924C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23#4837—21no—
24#4835—21no—
25#4730—22no—
26#4688—21no—
27#4679—21no—
28C3×#333—21no—
29C3×#333—21no—
30#4555—21no—
31#4529—22no—
32#3079—31no—
33#2989—31no—
34#1987—31no—
35#1806—31no—
36#1548D311yescharacteristic, complemented
37#1303—31no—
38#1292—31no—
39C22×C3×Q8—21no—
40C2×#235D411yescharacteristic
41C2×C3×M16—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8C2×C412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C4×#67—21no—
50C4×#67—21no—
51#926D411yescharacteristic
52C2×C4⋊Dic3—21no—
53C2×C4×Dic3—21no—
54C2×#208—21no—
55C22×C3⋊C8D412yes—
56C2×ΓL(1,𝔽25)—22no—
57C4×D12C2×C411yescharacteristic
58C4×D12D411yescharacteristic
59#872—21no—
60C42×D3—21no—
61C4×Dic6C2×C411yescharacteristic
62C4×Dic6D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8D411yescharacteristic
66C3×C4⋊C8D411yescharacteristic
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69C3⋊#95—21no—
70#825—42no—
71(C2×D3)⋊C8—22no—
72#815—22no—
73Dic3⋊C8—22no—
74#810—22no—
75#809—22no—
76#808—22no—
77#807—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)C2×C412yes—
82C2×C4⋊Q8C2×D311yescharacteristic
83C4×#41—31no—
84#477—61no—
85#476—61no—
86#475—32no—
87#474—32no—
88#446—61no—
89#436—61no—
90#435—62no—
91#432—61no—
92#431C2×D311yescharacteristic
93#428—61no—
94#423—31no—
95#423—61no—
96#423C2×D311yescharacteristic
97C2×C4⋊C8—31no—
98C2×#104—61no—
99#417—61no—
100C4×C4⋊C4—31no—
101#340—62no—
102#339—61no—
103#333—21no—
104#333—21no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—82no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—21no—
113C22×Dic3—41no—
114#235—21no—
115#235—21no—
116#235—21no—
117#235—21no—
118C2×D12C22⋊C411yescharacteristic
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×Dic6C22⋊C411yescharacteristic
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8—41no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#217—21no—
137(C2×D3)⋊C4—21no—
138(C2×D3)⋊C4—21no—
139C4⋊Dic3—22no—
140C4⋊Dic3—41no—
141C4⋊Dic3C22⋊C411yescharacteristic
142C4⋊Dic3C22⋊C411yescharacteristic
143Dic3⋊C4—21no—
144Dic3⋊C4—21no—
145C4×Dic3—21no—
146C4×Dic3—21no—
147C4×Dic3—21no—
148C4×Dic3—41no—
149C4×Dic3—41no—
150#208—41no—
151#208—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155ΓL(1,𝔽25)—82no—
156C2×#41—31no—
157C22×Q8—21no—
158C2×M16—21no—
159C2×M16—62no—
160C2×M16—61no—
161C22×C8—32no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8C4×D312yes—
166C4×Q8—31no—
167C4×Q8—31no—
168C4×D4—31no—
169C4×D4—31no—
170C4×D4—61no—
171C4×D4—61no—
172#117—31no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—61no—
177C2×C42—61no—
178C2×C42—61no—
179C2×C42C22×D311yescharacteristic
180#107—62no—
181#106—62no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—32no—
185C4⋊C8—62no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8#6711yescharacteristic
190C4⋊C8#6711yescharacteristic
191#104—122no—
192#103—62no—
193#102—62no—
194C22⋊C8—21no—
195C22⋊C8—62no—
196#97—62no—
197C4×C8—21no—
198#95—61no—
199C23×C3#12111yescharacteristic
200C22×D3—21no—
201C3×Q8—82no—
202C3×Q8—82no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—82no—
208C2×C3×C4C2×C22⋊C411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
215C2×C3×C4#13711yescharacteristic
216C2×C3×C4#13711yescharacteristic
217C2×C3×C4#13711yescharacteristic
218#67—41no—
219#67—41no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—42no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226D12—21no—
227D12—21no—
228C4×D3—41no—
229C4×D3—41no—
230C4×D3—41no—
231C4×D3—41no—
232C4×D3—81no—
233Dic6—21no—
234Dic6—21no—
235C3×C8—41no—
236C3×C8—41no—
237C3×C8—81no—
238C3⋊C8—42no—
239C3⋊C8—42no—
240#41—61no—
241#41—61no—
242#41—61no—
243#41—61no—
244C2×Q8—22no—
245C2×Q8—31no—
246C2×Q8—41no—
247C2×Q8—82no—
248C2×D4—31no—
249C2×D4—61no—
250C22×C4—41no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—121no—
255C22×C4#23511yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4D3×D411yescharacteristic
258M16—41no—
259M16—41no—
260M16—121no—
261M16—121no—
262M16—242no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—41no—
266C2×C8—62no—
267C2×C8—62no—
268C2×C8—122no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—31no—
272C4⋊C4—31no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—62no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—121no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C42—61no—
283C42—61no—
284C42—61no—
285C42—121no—
286C42—121no—
287C42C2×C4×D311yescharacteristic
288C42D3×D411yescharacteristic
289C42#23711yescharacteristic
290C42C2×#6711yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3#38611yescharacteristic
294C22×C3#46811yescharacteristic
295C22×C3#47011yescharacteristic
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—41no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—82no—
303C3×C4#41711yescharacteristic
304C3×C4#41911yescharacteristic
305C3×C4#45511yescharacteristic
306C3×C4#45611yescharacteristic
307C3×C4#46811yescharacteristic
308C3×C4#47011yescharacteristic
309Dic3—41no—
310Dic3—41no—
311Dic3—41no—
312Dic3—82no—
313C23—61no—
314C23#88411yescharacteristic
315Q8—61no—
316Q8—61no—
317Q8—82no—
318Q8—82no—
319D4—61no—
320D4—61no—
321D4—121no—
322D4—121no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—82no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—122no—
336C2×C4—241no—
337C2×C4—241no—
338C2×C4—241no—
339C2×C4D3×C22⋊C411yescharacteristic
340C2×C4#67⋊C411yescharacteristic
341C2×C4#88311yescharacteristic
342C2×C4#91411yescharacteristic
343C2×C4#91511yescharacteristic
344C2×C4#91811yescharacteristic
345C2×C4#91811yescharacteristic
346C2×C4C4×#6711yescharacteristic
347C2×C4#93711yescharacteristic
348C2×C4#93811yescharacteristic
349C8—41no—
350C8—41no—
351C8—81no—
352C8—122no—
353C8—122no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3#188211yescharacteristic
357C2×C3#189411yescharacteristic
358C2×C3#190311yescharacteristic
359D3—81no—
360C22—21no—
361C22—21no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22#481211yescharacteristic
366C22#481611yescharacteristic
367C22#495611yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—121no—
373C4—121no—
374C4—121no—
375C4—242no—
376C4#480211yescharacteristic
377C4#480311yescharacteristic
378C4#517411yescharacteristic
379C4#518911yescharacteristic
380C4#520211yescharacteristic
381C4#520511yescharacteristic
382C3#931911yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—241no—
386C2#7008711yescharacteristic
387C2#7027111yescharacteristic
388C2#7034811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9319
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C3×#1548C2semidirect
4. #71779C2semidirect
6. #67746C2semidirect
7. #67711C2semidirect
14. #5094C4semidirect
16. #5087C4semidirect
36. #1548D3semidirect
382. C3#9319semidirect

Maximal subgroups

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

subgroupindex
#832122
#792922
C3×#15482
#717792
#705292
#677462
#677112
#93193