Finite Group Databasebeta

← #10712075

Subgroups — #10712075

Order 768.

474 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82007C211yesmaximal, characteristic, complemented
2#78458C211yesmaximal, characteristic
3#71438C211yesmaximal, characteristic, complemented
4#71437C211yesmaximal, characteristic, complemented
5#71435C211yesmaximal, characteristic, complemented
6C3×#1376C211yesmaximal, characteristic, complemented
7#67658C211yesmaximal, characteristic, complemented
8#12982—31nomaximal, Sylow
9C2×#901C2211yescharacteristic
10C2×#901C2211yescharacteristic
11#5477C411yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15#5110C411yescharacteristic
16#5110C411yescharacteristic
17C2×#812C2211yescharacteristic
18C2×#812C2211yescharacteristic
19#4924C411yescharacteristic
20#4907—22no—
21D3×M32—22no—
22#4901—22no—
23#4688—22no—
24C8×D12—22no—
25#4515—22no—
26#4514—22no—
27#4508—22no—
28#3407—31no—
29#2931—31no—
30#2117—31no—
31#2117—31no—
32#2115—31no—
33#1376—31no—
34#1376D311yescharacteristic, complemented
35C2×#235C2×C411yescharacteristic
36C2×#235C2×C411yescharacteristic
37C22×C3×C8C2×C411yescharacteristic
38C22×C3×C8C2311yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40#930—22no—
41C2×#208C2×C411yescharacteristic
42C2×#208C2×C411yescharacteristic
43#901—21no—
44#901—21no—
45#901—21no—
46#901—21no—
47#901—21no—
48#901—21no—
49#901—21no—
50#901—21no—
51C2×ΓL(1,𝔽25)—21no—
52C2×ΓL(1,𝔽25)D412yes—
53C2×D3×C8—21no—
54C2×D3×C8D412yes—
55#875C811yescharacteristic
56#875C811yescharacteristic
57#874C811yescharacteristic
58C4×D12—22no—
59#869C811yescharacteristic
60C3×M32—21no—
61C3×M32—21no—
62C3×M32—21no—
63C3×M32D412yescomplemented
64C2×C3×C16—21no—
65C2×C3×C16—21no—
66C3×C4×C8—21no—
67C3×C4×C8—21no—
68(C2×D3)⋊C8—22no—
69(C2×D3)⋊C8—22no—
70#812—21no—
71#812—21no—
72#812—21no—
73#812D412yescomplemented
74C2×C3⋊C16—21no—
75C2×C3⋊C16—21no—
76C4⋊(C3⋊C8)—22no—
77#798—24no—
78D3×C16—24no—
79C2×#131—31no—
80C2×#131—31no—
81#532—31no—
82#504—62no—
83#504—62no—
84C2×M32—31no—
85C2×M32—31no—
86C2×M32—62no—
87C2×M32C2×D311yescharacteristic
88C2×M32C2×D311yescharacteristic
89#436—62no—
90C8×D4—62no—
91#423—31no—
92#408—31no—
93#408—31no—
94C2×C4×C8C2×D311yescharacteristic
95#350—62no—
96#350—62no—
97#349—62no—
98C22×C3×C4—21no—
99C22×C3×C4C22×C411yescharacteristic
100C2×#67—21no—
101C2×#67M1612yes—
102#235—21no—
103#235—21no—
104#235—21no—
105#235—21no—
106#235—21no—
107#235—21no—
108#235—21no—
109#235—21no—
110C2×D12—22no—
111C2×D12C2×C811yescharacteristic
112C2×D12C2×C811yescharacteristic
113C2×C4×D3—21no—
114C2×C4×D3C22⋊C412yes—
115C2×Dic6C2×C811yescharacteristic
116C2×Dic6C2×C811yescharacteristic
117C2×C3×C8—21no—
118C2×C3×C8—21no—
119C2×C3×C8—21no—
120C2×C3×C8C2×D411yescharacteristic
121C2×C3×C8C2×D411yescharacteristic
122C2×C3×C8C2×D411yescharacteristic
123C2×C3×C8C2×D411yescharacteristic
124C2×C3×C8#4111yescharacteristic
125C2×C3×C8#4111yescharacteristic
126C3×C42—21no—
127C3×C42C2×C811yescharacteristic
128C3×C42C2×C811yescharacteristic
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131C4⋊Dic3—22no—
132#208—21no—
133#208—21no—
134#208—21no—
135#208—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8C22⋊C412yes—
138ΓL(1,𝔽25)—22no—
139ΓL(1,𝔽25)—22no—
140ΓL(1,𝔽25)—41no—
141ΓL(1,𝔽25)—41no—
142D3×C8—22no—
143D3×C8—22no—
144D3×C8—41no—
145D3×C8—41no—
146C3×C16—22no—
147C3×C16—41no—
148C3⋊C16—22no—
149C3⋊C16—41no—
150C2×#41—31no—
151C2×#41—31no—
152#131—61no—
153#131—61no—
154#131—61no—
155#131—61no—
156#131—61no—
157#131—61no—
158#131—61no—
159#131—61no—
160C2×M16—31no—
161C2×M16—32no—
162C2×M16—31no—
163C2×M16—61no—
164C22×C8—32no—
165C22×C8—61no—
166C22×C8C4×D311yescharacteristic
167C22×C8C22×D311yescharacteristic
168C4⋊Q8—31no—
169#127—31no—
170#124—31no—
171#124—31no—
172#121—62no—
173C4×D4—62no—
174C2×C42C4×D311yescharacteristic
175M32—21no—
176M32—21no—
177M32—21no—
178M32—32no—
179M32—64no—
180M32—61no—
181M32—61no—
182M32—61no—
183M32D1212yescomplemented
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—64no—
187C2×C16—61no—
188C2×C16—61no—
189C4⋊C8—62no—
190C22⋊C8—62no—
191C22⋊C8—62no—
192C4×C8—21no—
193C4×C8—21no—
194C23×C3C2×M1611yescharacteristic
195C22×D3—21no—
196C22×D3C22⋊C812yes—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4C2×C22⋊C411yescharacteristic
201C2×C3×C4C4×D411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4C22×C811yescharacteristic
204C2×C3×C4#13111yescharacteristic
205C2×C3×C4#13111yescharacteristic
206#67—22no—
207#67—22no—
208#67—41no—
209#67—41no—
210C2×Dic3—21no—
211C2×Dic3C22⋊C812yes—
212D12—21no—
213D12—21no—
214D12—21no—
215D12—21no—
216D12—44no—
217C4×D3—22no—
218C4×D3—22no—
219C4×D3—41no—
220C4×D3—41no—
221Dic6—21no—
222Dic6—21no—
223Dic6—21no—
224Dic6—21no—
225C3×C8—41no—
226C3×C8#12011yescharacteristic
227C3×C8#12111yescharacteristic
228C3×C8#12111yescharacteristic
229C3×C8#12311yescharacteristic
230C3⋊C8—22no—
231C3⋊C8—41no—
232#41—61no—
233#41—61no—
234#41—61no—
235#41—61no—
236#41—61no—
237#41—61no—
238#41—61no—
239#41—61no—
240C2×Q8—31no—
241C2×Q8—31no—
242C2×D4—32no—
243C2×D4—31no—
244C2×D4—31no—
245C2×D4—62no—
246C2×D4—61no—
247C22×C4—21no—
248C22×C4—32no—
249C22×C4—61no—
250C22×C4C2×C4×D311yescharacteristic
251M16—61no—
252M16—61no—
253M16—62no—
254M16—62no—
255M16—61no—
256M16—61no—
257M16—121no—
258M16—121no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—32no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—61no—
266C2×C8—121no—
267C2×C8—121no—
268C2×C8C2×D1211yescharacteristic
269C2×C8#23511yescharacteristic
270C2×C8D3×D411yescharacteristic
271C2×C8D3×D411yescharacteristic
272C2×C8D3×D411yescharacteristic
273C2×C8#23911yescharacteristic
274C4⋊C4—62no—
275C22⋊C4—62no—
276C22⋊C4—62no—
277C42—21no—
278C42D3×C811yescharacteristic
279C42D3×C811yescharacteristic
280C16—22no—
281C16—41no—
282C16—62no—
283C16—121no—
284C22×C3—41no—
285C22×C3C2×C22⋊C811yescharacteristic
286C22×C3#43511yescharacteristic
287C22×C3#43511yescharacteristic
288C2×D3—22no—
289C2×D3—42no—
290C2×D3—42no—
291C2×D3—41no—
292C3×C4—22no—
293C3×C4—41no—
294C3×C4#38611yescharacteristic
295C3×C4#40811yescharacteristic
296C3×C4C8×D411yescharacteristic
297C3×C4C8×D411yescharacteristic
298Dic3—22no—
299Dic3—41no—
300C23—32no—
301C23—61no—
302C23D3×M1611yescharacteristic
303Q8—61no—
304Q8—61no—
305Q8—61no—
306Q8—61no—
307D4—62no—
308D4—62no—
309D4—61no—
310D4—61no—
311D4—61no—
312D4—61no—
313D4—124no—
314D4—121no—
315D4—121no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—32no—
319C2×C4—42no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—61no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4C4×D1211yescharacteristic
326C2×C4D3×C22⋊C411yescharacteristic
327C2×C4#89311yescharacteristic
328C2×C4C2×D3×C811yescharacteristic
329C2×C4#90111yescharacteristic
330C2×C4#90711yescharacteristic
331C8—41no—
332C8—62no—
333C8—121no—
334C8#88211yescharacteristic
335C8#88411yescharacteristic
336C8#89411yescharacteristic
337C8#89511yescharacteristic
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#186911yescharacteristic
342D3—42no—
343D3—81no—
344C22—41no—
345C22—62no—
346C22—122no—
347C22—122no—
348C22—121no—
349C22#470611yescharacteristic
350C22D3×C22⋊C811yescharacteristic
351C22#483611yescharacteristic
352C4—22no—
353C4—41no—
354C4—62no—
355C4—121no—
356C4#466511yescharacteristic
357C4C8×D1211yescharacteristic
358C4#472711yescharacteristic
359C4#483411yescharacteristic
360C3#1298211yescharacteristic, complemented, Sylow
361C2—21no—
362C2—21no—
363C2—21no—
364C2—122no—
365C2—241no—
366C2#6866811yescharacteristic
subgroupG/H
centerC8#882
commutatorC2×C3×C4C22×C8
FrattiniC22×C8C22×D3
FittingC3×#1376C2
socleC2×C3#1869
prime radical#5477C4
cosocleC22×C3×C8C23
hypercenter#1376D3
hypocenterC3#12982
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #1376

Complements

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

NH ≅ G/Ntype
1. #82007C2semidirect
3. #71438C2semidirect
4. #71437C2semidirect
5. #71435C2semidirect
6. C3×#1376C2semidirect
7. #67658C2semidirect
34. #1376D3semidirect
63. C3×M32D4semidirect
73. #812D4semidirect
183. M32D12semidirect
360. C3#12982semidirect

Maximal subgroups

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

subgroupindex
#820072
#784582
#714382
#714372
#714352
C3×#13762
#676582
#129823