Finite Group Databasebeta

← #10691693

Subgroups — #10691693

Order 768.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83213C211yesmaximal, characteristic
2#78522C211yesmaximal, characteristic
3C3×#1997C211yesmaximal, characteristic, complemented
4C2×#4546C211yesmaximal, characteristic, complemented
5#70503C211yesmaximal, characteristic
6#67714C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#9267—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#907—21no—
11#5477C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5227C414yes—
15#5129—21no—
16#5085—21no—
17#5083—22no—
18#5079—21no—
19#4925C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22#4706—22no—
23#4688—21no—
24C8×D12—21no—
25C3×#343—21no—
26C3×#343—21no—
27#4546—21no—
28#4546—21no—
29#4485—21no—
30#4482—21no—
31#3206—31no—
32#2935—31no—
33#1997D311yescharacteristic, complemented
34#1547—31no—
35C2×#333—31no—
36#1301—31no—
37#1283—31no—
38C22×C3×Q8C2×C412yes—
39C2×#235—21no—
40C2×C3×M16—21no—
41C2×C3×M16D411yescharacteristic
42C2×C3×M16D411yescharacteristic
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—24no—
51#930—21no—
52#930C2×C412yes—
53C22×C3⋊C8—21no—
54C22×C3⋊C8—21no—
55#907—42no—
56#907—42no—
57C2×ΓL(1,𝔽25)—22no—
58C2×D3×C8—22no—
59#875—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12D412yes—
63#869D411yescharacteristic
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×#97—21no—
67C3×C4×C8—21no—
68#835—24no—
69#830C2×C412yes—
70(C2×D3)⋊C8—22no—
71(C2×D3)⋊C8—22no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D411yescharacteristic
77C4⋊(C3⋊C8)D411yescharacteristic
78C4×C3⋊C8—21no—
79C2×#131—61no—
80#532—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82#492—61no—
83#487—61no—
84#461—62no—
85#448—34no—
86#436—61no—
87#435—62no—
88C8×D4—61no—
89#431—31no—
90#423C2×D311yescharacteristic
91C2×C4⋊C8—31no—
92C2×C4⋊C8—31no—
93#408—61no—
94C4×M16C2×D311yescharacteristic
95#343—21no—
96#343—21no—
97#333—61no—
98#333—61no—
99#332—61no—
100#331—61no—
101C2×C3×Q8—22no—
102C2×C3×Q8—22no—
103C2×C3×Q8—82no—
104C22×C3×C4—22no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C2×#67—22no—
109#235—41no—
110#235—41no—
111#235—41no—
112#235—41no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12C22⋊C412yes—
116C2×C4×D3—22no—
117C2×Dic6—21no—
118C3×M16—21no—
119C3×M16—21no—
120C3×M16—21no—
121C3×M16—41no—
122C3×M16—41no—
123C3×M16SD1612yescomplemented
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135#215—84no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3C22⋊C412yes—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—81no—
147ΓL(1,𝔽25)—44no—
148D3×C8—44no—
149C2×#41—61no—
150C22×Q8C4×D312yes—
151C2×SD16—64no—
152#131—122no—
153#131—122no—
154C2×M16—21no—
155C2×M16—62no—
156C2×M16#6711yescharacteristic
157C2×M16#6711yescharacteristic
158C22×C8—62no—
159C22×C8—61no—
160C22×C8—61no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—31no—
166#127—31no—
167#124—61no—
168#121—32no—
169#121—61no—
170C4×D4—32no—
171C4×D4—61no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C42C22×D311yescharacteristic
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—31no—
178C4⋊C8—31no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183#103—64no—
184C22⋊C8—32no—
185C22⋊C8—62no—
186C22⋊C8—62no—
187#97—21no—
188C4×C8—21no—
189C4×C8—61no—
190C23×C3C4×D411yescharacteristic
191C22×D3—22no—
192C3×Q8—44no—
193C3×Q8—82no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—82no—
199C2×C3×C4C2×C22⋊C411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12311yescharacteristic
205C2×C3×C4C2×SD1611yescharacteristic
206C2×C3×C4#13511yescharacteristic
207C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
208C2×C3×C4#13711yescharacteristic
209#67—42no—
210#67—42no—
211C2×Dic3—22no—
212D12—42no—
213D12—41no—
214D12—41no—
215D12—41no—
216D12—41no—
217C4×D3—42no—
218C4×D3—42no—
219Dic6—41no—
220Dic6—41no—
221C3×C8—22no—
222C3×C8—41no—
223C3×C8—81no—
224C3⋊C8—42no—
225C3⋊C8—41no—
226C3⋊C8—41no—
227C3⋊C8—81no—
228#41—121no—
229#41—121no—
230#41—121no—
231#41—121no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—61no—
235C2×Q8—82no—
236C2×D4—32no—
237C2×D4—62no—
238C2×D4—61no—
239C2×D4—61no—
240C22×C4—22no—
241C22×C4—62no—
242C22×C4C2×C4×D311yescharacteristic
243C22×C4#23511yescharacteristic
244C22×C4C2×#6711yescharacteristic
245SD16—244no—
246M16—21no—
247M16—21no—
248M16—21no—
249M16—41no—
250M16—41no—
251M16—124no—
252M16#21512yescomplemented
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—62no—
257C2×C8—61no—
258C2×C8—61no—
259C2×C8—124no—
260C2×C8—121no—
261C2×C8—121no—
262C2×C8—121no—
263C2×C8—241no—
264C4⋊C4—32no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—61no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
274C42D3×D411yescharacteristic
275C42#23711yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#38611yescharacteristic
279C22×C3C4×SD1611yescharacteristic
280C22×C3#44011yescharacteristic
281C2×D3—42no—
282C2×D3—42no—
283C2×D3—42no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—82no—
288C3×C4#45011yescharacteristic
289C3×C4#45211yescharacteristic
290C3×C4#46111yescharacteristic
291C3×C4#46311yescharacteristic
292C3×C4#47111yescharacteristic
293C3×C4#47311yescharacteristic
294Dic3—42no—
295C23—62no—
296C23C4×#6711yescharacteristic
297Q8—44no—
298Q8—82no—
299Q8—121no—
300Q8—121no—
301D4—122no—
302D4—122no—
303D4—122no—
304D4—121no—
305D4—121no—
306D4—121no—
307D4—121no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—62no—
313C2×C4—82no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4D3×C22⋊C411yescharacteristic
317C2×C4#67⋊C411yescharacteristic
318C2×C4#88311yescharacteristic
319C2×C4#88411yescharacteristic
320C2×C4#93711yescharacteristic
321C2×C4#93811yescharacteristic
322C2×C4C2×#21511yescharacteristic
323C2×C4#94211yescharacteristic
324C2×C4#94911yescharacteristic
325C2×C4#95011yescharacteristic
326C8—22no—
327C8—41no—
328C8—81no—
329C8—122no—
330C8—121no—
331C8—121no—
332C8—241no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3#187711yescharacteristic
336C2×C3#190811yescharacteristic
337C2×C3#191411yescharacteristic
338D3—82no—
339C22—21no—
340C22—21no—
341C22—122no—
342C22—122no—
343C22—122no—
344C22#495611yescharacteristic
345C22C4×#21511yescharacteristic
346C22#502611yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—82no—
351C4—122no—
352C4#475411yescharacteristic
353C4#475511yescharacteristic
354C4#504611yescharacteristic
355C4#504711yescharacteristic
356C4#505711yescharacteristic
357C4#508311yescharacteristic
358C3#926711yescharacteristic, complemented, Sylow
359C2—21no—
360C2—21no—
361C2—242no—
362C2#6968711yescharacteristic
363C2#7008911yescharacteristic
364C2#7012311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1997D3
hypocenterC3#9267
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
3. C3×#1997C2semidirect
4. C2×#4546C2semidirect
6. #67714C2semidirect
7. #67668C2semidirect
33. #1997D3semidirect
123. C3×M16SD16semidirect
252. M16#215semidirect
358. C3#9267semidirect

Maximal subgroups

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

subgroupindex
#832132
#785222
C3×#19972
C2×#45462
#705032
#677142
#676682
#92673