Finite Group Databasebeta

← #10713391

Subgroups — #10713391

Order 768.

438 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#83794C211yesmaximal, characteristic
2C3×#2117C211yesmaximal, characteristic, complemented
3#72006C211yesmaximal, characteristic, complemented
4#71431C211yesmaximal, characteristic, complemented
5#71431C211yesmaximal, characteristic, complemented
6#71425C211yesmaximal, characteristic, complemented
7#71423C211yesmaximal, characteristic, complemented
8#13200—31nomaximal, Sylow
9C2×C3×#131C2211yescharacteristic
10#5828C411yescharacteristic
11C22×D3×C8C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14#5234C411yescharacteristic
15#5127—22no—
16#5126C411yescharacteristic
17#5119C411yescharacteristic
18C8×#67—22no—
19#5101C2211yescharacteristic
20C2×#812C2211yescharacteristic
21C2×#812C2211yescharacteristic
22C3×#350—22no—
23#4561—22no—
24#4512—22no—
25#4511—22no—
26(C2×D3)⋊C16—22no—
27(C2×D3)⋊C16—22no—
28#2929—31no—
29#2153—31no—
30#2117—31no—
31#2117D311yescharacteristic, complemented
32#2112—31no—
33#2112—31no—
34#2110—31no—
35C2×C3×#41C2×C411yescharacteristic
36C22×C4×D3C2×C411yescharacteristic
37C3×#131—21no—
38C3×#131—21no—
39C3×#131—21no—
40C3×#131—21no—
41C2×C3×M16—21no—
42C2×C3×M16C2×C411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8C2311yescharacteristic
45#946—21no—
46#938—21no—
47#937—21no—
48#935—21no—
49C4×#67—22no—
50#926C2×C411yescharacteristic
51C2×#208C2×C411yescharacteristic
52C22×C3⋊C8C2×C411yescharacteristic
53C2×D3×C8—21no—
54C2×D3×C8—21no—
55C2×D3×C8—21no—
56C2×D3×C8—21no—
57C2×D3×C8—21no—
58C2×D3×C8—21no—
59C3×M32—21no—
60C3×M32—21no—
61C3×M32—21no—
62C3×M32—21no—
63C2×C3×C16—21no—
64C2×C3×C16D412yescomplemented
65#830—22no—
66(C2×D3)⋊C8—22no—
67#815D412yes—
68Dic3⋊C8—22no—
69C8×Dic3D412yes—
70#812—21no—
71#812—21no—
72#812—21no—
73#812—21no—
74C2×C3⋊C16—21no—
75C2×C3⋊C16D412yescomplemented
76C2×#131C2×D311yescharacteristic
77C23×C8—31no—
78#525—31no—
79C2×M32—31no—
80C2×M32—31no—
81C2×M32C2×D311yescharacteristic
82C2×M32C2×D311yescharacteristic
83#435—62no—
84C8×D4—62no—
85#424—31no—
86#408—31no—
87#407—31no—
88#405—31no—
89#364—62no—
90#350—22no—
91#350—62no—
92C22⋊C16—62no—
93C22⋊C16—62no—
94#347—62no—
95C23×D3M1611yescharacteristic
96C3×#41—21no—
97C3×#41—21no—
98C3×#41—21no—
99C3×#41—21no—
100C2×C3×Q8M1611yescharacteristic
101C2×C3×D4—21no—
102C2×C3×D4M1611yescharacteristic
103C22×C3×C4—21no—
104C22×C3×C4C22×C411yescharacteristic
105C2×#67—22no—
106C22×Dic3M1611yescharacteristic
107C2×C4×D3—21no—
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—21no—
113C3×M16—21no—
114C3×M16—21no—
115C3×M16—41no—
116C3×M16—41no—
117C2×C3×C8—21no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C2×C3×C8C2×D411yescharacteristic
121C2×C3×C8C2×D411yescharacteristic
122C2×C3×C8C2×D411yescharacteristic
123C2×C3×C8C2×D411yescharacteristic
124C2×C3×C8#4111yescharacteristic
125C2×C3×C8#4111yescharacteristic
126#217—22no—
127#217—21no—
128(C2×D3)⋊C4—22no—
129C4⋊Dic3—21no—
130Dic3⋊C4—22no—
131C4×Dic3C22⋊C412yes—
132#208—21no—
133#208—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8C22⋊C412yes—
138D3×C8—41no—
139D3×C8—41no—
140D3×C8—41no—
141D3×C8—41no—
142C3×C16—22no—
143C3×C16—41no—
144C3⋊C16—22no—
145C3⋊C16—41no—
146C2×#41C4×D311yescharacteristic
147C23×C4—31no—
148#131—21no—
149#131—21no—
150#131—21no—
151#131—21no—
152C2×M16—21no—
153C2×M16—31no—
154C2×M16C4×D311yescharacteristic
155C22×C8—21no—
156C22×C8—31no—
157C22×C8—61no—
158C22×C8—61no—
159C22×C8—61no—
160C22×C8—61no—
161C22×C8—61no—
162C22×C8—61no—
163C22×C8C22×D311yescharacteristic
164#123—61no—
165C22⋊Q8—61no—
166#121—61no—
167#120—61no—
168C4×D4—62no—
169#117—31no—
170M32—21no—
171M32—21no—
172M32—21no—
173M32—21no—
174M32—61no—
175M32—61no—
176M32—61no—
177M32—61no—
178C2×C16—21no—
179C2×C16—32no—
180C2×C16—61no—
181C2×C16#6712yescomplemented
182C4⋊C8—62no—
183C22⋊C8—62no—
184C22⋊C8—62no—
185#97—32no—
186C4×C8—32no—
187C23×C3—21no—
188C23×C3C2×M1611yescharacteristic
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—42no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—41no—
198C3×D4—41no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4C2×C22⋊C411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4C2×M1611yescharacteristic
206C2×C3×C4#13111yescharacteristic
207C2×C3×C4#13111yescharacteristic
208#67—44no—
209C2×Dic3—22no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C4×D3—41no—
214C4×D3—41no—
215C4×D3—41no—
216C4×D3—41no—
217C3×C8—41no—
218C3×C8#12111yescharacteristic
219C3×C8#12111yescharacteristic
220C3×C8#12411yescharacteristic
221C3×C8#12711yescharacteristic
222C3⋊C8—22no—
223C3⋊C8—41no—
224C24—31no—
225#41—21no—
226#41—21no—
227#41—21no—
228#41—21no—
229C2×Q8ΓL(1,𝔽25)11yescharacteristic
230C2×D4—21no—
231C2×D4—62no—
232C2×D4ΓL(1,𝔽25)11yescharacteristic
233C22×C4—21no—
234C22×C4—31no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4C2×C4×D311yescharacteristic
242M16—21no—
243M16—21no—
244M16—41no—
245M16—41no—
246M16—61no—
247M16—61no—
248C2×C8—21no—
249C2×C8—32no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—121no—
256C2×C8—121no—
257C2×C8—121no—
258C2×C8—121no—
259C2×C8#23511yescharacteristic
260C2×C8D3×D411yescharacteristic
261C2×C8D3×D411yescharacteristic
262C2×C8D3×D411yescharacteristic
263C2×C8#23711yescharacteristic
264C2×C8C2×#6711yescharacteristic
265C4⋊C4—62no—
266C4⋊C4—61no—
267C22⋊C4—62no—
268C22⋊C4—62no—
269C22⋊C4—61no—
270C42—32no—
271C16—22no—
272C16—41no—
273C16—62no—
274C16—121no—
275C22×C3—41no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3#40711yescharacteristic
279C22×C3#43511yescharacteristic
280C22×C3#43511yescharacteristic
281C2×D3—22no—
282C2×D3—22no—
283C2×D3—22no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C3×C4—41no—
289C3×C4#39011yescharacteristic
290C3×C4#40811yescharacteristic
291C3×C4#43511yescharacteristic
292C3×C4#43511yescharacteristic
293Dic3—42no—
294Dic3—41no—
295C23—21no—
296C23—61no—
297C23—61no—
298C23—61no—
299C23—122no—
300C23D3×M1611yescharacteristic
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—21no—
305D4—41no—
306D4—41no—
307D4—124no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—61no—
312C2×C4—62no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4D3×C22⋊C411yescharacteristic
320C2×C4#67⋊C411yescharacteristic
321C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
322C2×C4#90111yescharacteristic
323C2×C4#90711yescharacteristic
324C2×C4C4×#6711yescharacteristic
325C8—41no—
326C8—62no—
327C8—121no—
328C8#88411yescharacteristic
329C8#88511yescharacteristic
330C8#93911yescharacteristic
331C8#94011yescharacteristic
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—81no—
336C2×C3#195511yescharacteristic
337D3—42no—
338C22—41no—
339C22—42no—
340C22—41no—
341C22—62no—
342C22—62no—
343C22—62no—
344C22—121no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22#472611yescharacteristic
349C22#472811yescharacteristic
350C22#512711yescharacteristic
351C4—41no—
352C4—122no—
353C4—121no—
354C4#472711yescharacteristic
355C4#473011yescharacteristic
356C4#495711yescharacteristic
357C4#511111yescharacteristic
358C3#1320011yescharacteristic, complemented, Sylow
359C2—21no—
360C2—21no—
361C2—21no—
362C2—81no—
363C2—122no—
364C2#6991711yescharacteristic
subgroupG/H
centerC8#940
commutatorC22×C3×C4C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2117C2
socleC2×C3#1955
prime radical#5828C4
cosocleC22×C3×C8C23
hypercenter#2117D3
hypocenterC3#13200
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. C3×#2117C2semidirect
3. #72006C2semidirect
4. #71431C2semidirect
5. #71431C2semidirect
6. #71425C2semidirect
7. #71423C2semidirect
31. #2117D3semidirect
64. C2×C3×C16D4semidirect
75. C2×C3⋊C16D4semidirect
181. C2×C16#67semidirect
358. C3#13200semidirect

Maximal subgroups

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

subgroupindex
#837942
C3×#21172
#720062
#714312
#714252
#714232
#132003