Finite Group Databasebeta

← #10691691

Subgroups — #10691691

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#79312C211yesmaximal, characteristic
2#79004C211yesmaximal, characteristic
3#70524C211yesmaximal, characteristic, complemented
4#70503C211yesmaximal, characteristic
5C2×#4481C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#9267—31nomaximal, Sylow
9C2×#901—21no—
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5110—21no—
16C2×C4⋊(C3⋊C8)C2211yescharacteristic
17C4×#208C2211yescharacteristic
18#4841—22no—
19#4838—21no—
20#4836—22no—
21#4834—21no—
22#4738C414yes—
23#4693—21no—
24#4681—21no—
25#4554—21no—
26#4554—21no—
27#4485—21no—
28#4482—21no—
29#4481—21no—
30#4481—21no—
31#3206—31no—
32#2935—31no—
33#1997—31no—
34#1547—31no—
35C2×#333—31no—
36#1301D311yescharacteristic, complemented
37#1283—31no—
38C2×#235—21no—
39C22×Dic6C2×C412yes—
40C2×C3×M16—21no—
41C22×C3×C8—21no—
42C2×C3×C42C2311yescharacteristic
43#930—21no—
44#930C2×C412yes—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×#208D411yescharacteristic
48C2×#208D411yescharacteristic
49C22×C3⋊C8—21no—
50C2×#204—24no—
51#901—42no—
52#901—42no—
53C2×ΓL(1,𝔽25)—22no—
54C2×D3×C8—22no—
55#875—21no—
56#874D411yescharacteristic
57C4×D12—21no—
58C4×D12D412yes—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869D411yescharacteristic
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D411yescharacteristic
69C3×C4⋊C8D411yescharacteristic
70C3×C22⋊C8C2×C412yes—
71C3×C4×C8—21no—
72(C2×D3)⋊C8—22no—
73(C2×D3)⋊C8—22no—
74#816—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77#803—21no—
78C4×C3⋊C8—21no—
79C2×#131—61no—
80#532—31no—
81C2×C4⋊Q8—31no—
82#492—61no—
83#487—61no—
84#461—62no—
85#448—34no—
86#436—61no—
87#435—62no—
88C8×D4—61no—
89#431C2×D311yescharacteristic
90#423C2×D311yescharacteristic
91C2×C4⋊C8—31no—
92C2×C4⋊C8C2×D311yescharacteristic
93#408—61no—
94C4×M16—31no—
95#343—61no—
96#343—61no—
97#333—61no—
98#333—61no—
99#332—61no—
100#331—61no—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C2×#67—22no—
105C22×Dic3—22no—
106#235—41no—
107#235—41no—
108#235—41no—
109#235—41no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12C22⋊C412yes—
113C2×C4×D3—22no—
114C2×Dic6—21no—
115C2×Dic6—22no—
116C2×Dic6—22no—
117C2×Dic6—82no—
118C3×M16—41no—
119C3×M16—41no—
120C2×C3×C8—22no—
121C2×C3×C8—21no—
122C2×C3×C8—21no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—22no—
132C4⋊Dic3—21no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3C22⋊C412yes—
138#208—21no—
139#208—21no—
140#208—21no—
141#208SD1612yescomplemented
142C2×C3⋊C8—21no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8—81no—
146#204—84no—
147ΓL(1,𝔽25)—44no—
148D3×C8—44no—
149C2×#41—61no—
150C22×Q8—32no—
151C2×SD16—64no—
152#131—122no—
153#131—122no—
154C2×M16—21no—
155C2×M16—31no—
156C2×M16—31no—
157C2×M16—62no—
158C22×C8—21no—
159C22×C8—62no—
160C22×C8—61no—
161C4⋊Q8—31no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166#127—31no—
167#124—61no—
168#121—32no—
169#121—61no—
170C4×D4—32no—
171C4×D4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C42C22×D311yescharacteristic
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8D1211yescharacteristic
182C4⋊C8D1211yescharacteristic
183#103—64no—
184C22⋊C8—62no—
185C22⋊C8—62no—
186C22⋊C8C4×D312yes—
187#97—61no—
188C4×C8—21no—
189C4×C8—61no—
190C23×C3C4×D411yescharacteristic
191C22×D3—22no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—41no—
195C2×C3×C4C2×C22⋊C411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C2×C3×C4C2×SD1611yescharacteristic
202C2×C3×C4#13511yescharacteristic
203C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
204C2×C3×C4#13711yescharacteristic
205#67—42no—
206#67—42no—
207C2×Dic3—22no—
208C2×Dic3—42no—
209C2×Dic3—82no—
210D12—42no—
211D12—41no—
212D12—41no—
213D12—41no—
214D12—41no—
215C4×D3—42no—
216C4×D3—42no—
217Dic6—41no—
218Dic6—41no—
219Dic6—44no—
220Dic6—82no—
221C3×C8—42no—
222C3×C8—41no—
223C3×C8—41no—
224C3×C8—81no—
225C3⋊C8—22no—
226C3⋊C8—41no—
227C3⋊C8—81no—
228#41—121no—
229#41—121no—
230#41—121no—
231#41—121no—
232C2×Q8—61no—
233C2×Q8—62no—
234C2×Q8—62no—
235C2×Q8—242no—
236C2×D4—32no—
237C2×D4—62no—
238C2×D4—61no—
239C2×D4—61no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4C2×C4×D311yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4#23911yescharacteristic
245SD16—244no—
246M16—32no—
247M16—41no—
248M16—41no—
249M16—61no—
250M16—61no—
251M16—61no—
252M16—124no—
253C2×C8—22no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—124no—
262C2×C8—121no—
263C2×C8—241no—
264C4⋊C4—32no—
265C4⋊C4—61no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C42C2×D1211yescharacteristic
273C42#23511yescharacteristic
274C42D3×D411yescharacteristic
275C42D3×D411yescharacteristic
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#45011yescharacteristic
288C3×C4#45211yescharacteristic
289C3×C4#46111yescharacteristic
290C3×C4#46311yescharacteristic
291C3×C4#47111yescharacteristic
292C3×C4#47311yescharacteristic
293Dic3—42no—
294Dic3—82no—
295C23—62no—
296C23#89311yescharacteristic
297Q8—124no—
298Q8—121no—
299Q8—121no—
300Q8—242no—
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—62no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—242no—
316C2×C4C4×D1211yescharacteristic
317C2×C4D3×C22⋊C411yescharacteristic
318C2×C4#88211yescharacteristic
319C2×C4#88411yescharacteristic
320C2×C4#89411yescharacteristic
321C2×C4#89511yescharacteristic
322C2×C4#91111yescharacteristic
323C2×C4D3×SD1611yescharacteristic
324C2×C4#91611yescharacteristic
325C2×C4#91811yescharacteristic
326C8—42no—
327C8—41no—
328C8—41no—
329C8—62no—
330C8—81no—
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#466511yescharacteristic
345C22#485211yescharacteristic
346C22#488911yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—122no—
351C4—242no—
352C4#477611yescharacteristic
353C4#478711yescharacteristic
354C4#480811yescharacteristic
355C4#481811yescharacteristic
356C4#489611yescharacteristic
357C4#489811yescharacteristic
358C3#926711yescharacteristic, complemented, Sylow
359C2—21no—
360C2—21no—
361C2—242no—
362C2#6889611yescharacteristic
363C2#6894911yescharacteristic
364C2#6904011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9267
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
3. #70524C2semidirect
5. C2×#4481C2semidirect
6. C3×#1301C2semidirect
7. #67744C2semidirect
36. #1301D3semidirect
141. #208SD16semidirect
358. C3#9267semidirect

Maximal subgroups

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

subgroupindex
#793122
#790042
#705242
#705032
C2×#44812
C3×#13012
#677442
#92673