Finite Group Databasebeta

← #10713389

Subgroups — #10713389

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#83734C211yesmaximal, characteristic
2C3×#2112C211yesmaximal, characteristic, complemented
3#71963C211yesmaximal, characteristic, complemented
4#71438C211yesmaximal, characteristic, complemented
5#71437C211yesmaximal, characteristic, complemented
6#71425C211yesmaximal, characteristic, complemented
7#71423C211yesmaximal, characteristic, complemented
8#13200—31nomaximal, Sylow
9C23×C3×C8C2211yescharacteristic
10#5791C411yescharacteristic
11C2×#901C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14#5207C411yescharacteristic
15#5111—22no—
16#5110C411yescharacteristic
17C8×#67—22no—
18#5102C411yescharacteristic
19#5101C2211yescharacteristic
20C2×#812C2211yescharacteristic
21C2×#812C2211yescharacteristic
22C3×C22⋊C16—22no—
23#4549—22no—
24#4515—22no—
25#4512—22no—
26#4511—22no—
27#4508—22no—
28#2929—31no—
29#2153—31no—
30#2117—31no—
31#2117—31no—
32#2112—31no—
33#2112D311yescharacteristic, complemented
34#2110—31no—
35C23×C3×C4C2×C411yescharacteristic
36C2×#235C2×C411yescharacteristic
37C22×C3×C8—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C22×C3×C8C2×C411yescharacteristic
44C22×C3×C8C2311yescharacteristic
45#953—21no—
46#930—21no—
47#929—21no—
48C4×#67—22no—
49#926C2×C411yescharacteristic
50#924—21no—
51C2×#208C2×C411yescharacteristic
52C2×#208C2×C411yescharacteristic
53#901—21no—
54#901—21no—
55#901—21no—
56#901—21no—
57C2×ΓL(1,𝔽25)—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×#131—31no—
77C23×C8C2×D311yescharacteristic
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—62no—
91#350—62no—
92C22⋊C16—22no—
93C22⋊C16—62no—
94#347—62no—
95C24×C3M1611yescharacteristic
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4M1611yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C2×#67—22no—
105C2×#67—21no—
106#235—21no—
107#235—21no—
108#235—21no—
109#235—21no—
110C2×D12M1611yescharacteristic
111C2×C4×D3—21no—
112C2×Dic6M1611yescharacteristic
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8—41no—
117C2×C3×C8—41no—
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—
134#208—21no—
135#208—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8C22⋊C412yes—
138ΓL(1,𝔽25)—41no—
139ΓL(1,𝔽25)—41no—
140D3×C8—41no—
141D3×C8—41no—
142C3×C16—22no—
143C3×C16—41no—
144C3⋊C16—22no—
145C3⋊C16—41no—
146C2×#41—31no—
147C23×C4C4×D311yescharacteristic
148#131—61no—
149#131—61no—
150#131—61no—
151#131—61no—
152C2×M16—31no—
153C2×M16—31no—
154C2×M16—61no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—61no—
162C22×C8C4×D311yescharacteristic
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×C3—21no—
189C23×C3—21no—
190C23×C3—42no—
191C23×C3C2×M1611yescharacteristic
192C22×D3—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4C2×C22⋊C411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4C2×M1611yescharacteristic
204C2×C3×C4#13111yescharacteristic
205C2×C3×C4#13111yescharacteristic
206#67—44no—
207#67—41no—
208#67—41no—
209C2×Dic3—22no—
210C2×Dic3—21no—
211D12—21no—
212D12—21no—
213C4×D3—41no—
214C4×D3—41no—
215Dic6—21no—
216Dic6—21no—
217C3×C8—41no—
218C3×C8#12111yescharacteristic
219C3×C8#12111yescharacteristic
220C3×C8#12411yescharacteristic
221C3×C8#12711yescharacteristic
222C3⋊C8—22no—
223C3⋊C8—41no—
224C24ΓL(1,𝔽25)11yescharacteristic
225#41—61no—
226#41—61no—
227#41—61no—
228#41—61no—
229C2×Q8—31no—
230C2×D4—31no—
231C2×D4—62no—
232C2×D4—61no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—61no—
240C22×C4ΓL(1,𝔽25)11yescharacteristic
241C22×C4C2×C4×D311yescharacteristic
242M16—61no—
243M16—61no—
244M16—61no—
245M16—61no—
246M16—121no—
247M16—121no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—32no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—61no—
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—22no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3#40711yescharacteristic
284C22×C3#43511yescharacteristic
285C22×C3#43511yescharacteristic
286C2×D3—42no—
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—21no—
297C23—21no—
298C23—42no—
299C23—61no—
300C23C2×ΓL(1,𝔽25)11yescharacteristic
301Q8—61no—
302Q8—61no—
303D4—61no—
304D4—61no—
305D4—124no—
306D4—121no—
307D4—121no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—62no—
316C2×C4—61no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4D3×C22⋊C411yescharacteristic
320C2×C4#67⋊C411yescharacteristic
321C2×C4#90111yescharacteristic
322C2×C4D3×M1611yescharacteristic
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—42no—
336C2×C3#195511yescharacteristic
337D3—81no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—122no—
347C22—121no—
348C22#472611yescharacteristic
349C22#473011yescharacteristic
350C22#511111yescharacteristic
351C4—41no—
352C4—122no—
353C4—121no—
354C4#472711yescharacteristic
355C4#472811yescharacteristic
356C4#495711yescharacteristic
357C4#512711yescharacteristic
358C3#1320011yescharacteristic, complemented, Sylow
359C2—21no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—241no—
364C2#6991711yescharacteristic
subgroupG/H
centerC8#940
commutatorC22×C3×C4C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2112C2
socleC2×C3#1955
prime radical#5791C4
cosocleC22×C3×C8C23
hypercenter#2112D3
hypocenterC3#13200
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. C3×#2112C2semidirect
3. #71963C2semidirect
4. #71438C2semidirect
5. #71437C2semidirect
6. #71425C2semidirect
7. #71423C2semidirect
33. #2112D3semidirect
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
#837342
C3×#21122
#719632
#714382
#714372
#714252
#714232
#132003