Finite Group Databasebeta

← #10712080

Subgroups — #10712080

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C16C211yesmaximal, characteristic, complemented
2#78553C211yesmaximal, characteristic
3C2×#4515C211yesmaximal, characteristic, complemented
4#71431C211yesmaximal, characteristic, complemented
5C2×#4508C211yesmaximal, characteristic, complemented
6C3×#1379C211yesmaximal, characteristic, complemented
7#67671C211yesmaximal, characteristic, complemented
8#12983—31nomaximal, Sylow
9C2×#901C2211yescharacteristic
10C22×D3×C8C2211yescharacteristic
11#5524C411yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C3×#405C2211yescharacteristic
15#5129C411yescharacteristic
16#5126C411yescharacteristic
17C2×#812C2211yescharacteristic
18C22×C3⋊C16C2211yescharacteristic
19#4999C411yescharacteristic
20C2×D3×C16—22no—
21C2×D3×C16—22no—
22C2×D3×C16—22no—
23#4706—22no—
24C8×D12—22no—
25#4515—22no—
26#4508—22no—
27(C2×D3)⋊C16—22no—
28C23×C16—31no—
29#2929—31no—
30C2×#350—31no—
31C2×#350—31no—
32#2112—31no—
33#1379—31no—
34#1379D311yescharacteristic, complemented
35C2×#235C2×C411yescharacteristic
36C22×C4×D3C2×C411yescharacteristic
37C2×C3×M16C2×C411yescharacteristic
38C22×C3×C8C2311yescharacteristic
39C3×#117C2×C411yescharacteristic
40C2×#208C2×C411yescharacteristic
41C22×C3⋊C8C2×C411yescharacteristic
42#901—21no—
43#901—21no—
44#901—21no—
45#901—21no—
46C2×ΓL(1,𝔽25)—21no—
47C2×D3×C8—21no—
48C2×D3×C8—21no—
49C2×D3×C8—21no—
50C2×D3×C8—21no—
51C2×D3×C8—21no—
52C2×D3×C8D412yes—
53C2×D3×C8D412yes—
54#897C812yes—
55#895C812yes—
56#886—21no—
57#882—21no—
58C4×D12—22no—
59C3×M32—21no—
60C3×M32—21no—
61C2×C3×C16—21no—
62C2×C3×C16—21no—
63C2×C3×C16—21no—
64C2×C3×C16D412yescomplemented
65C3×#97—21no—
66C3×C4×C8—21no—
67(C2×D3)⋊C8—22no—
68(C2×D3)⋊C8—22no—
69#812—21no—
70#812—21no—
71C2×C3⋊C16—21no—
72C2×C3⋊C16—21no—
73C2×C3⋊C16—21no—
74C2×C3⋊C16D412yescomplemented
75C4⋊(C3⋊C8)—22no—
76D3×C16—28no—
77C2×#131—31no—
78C23×C8—31no—
79#525—31no—
80C2×M32—31no—
81C2×M32C2×D311yescharacteristic
82C22×C16—31no—
83C22×C16—62no—
84C22×C16—62no—
85C22×C16—62no—
86C22×C16C2×D311yescharacteristic
87#435—62no—
88C8×D4—62no—
89#424—31no—
90#408—31no—
91#407—31no—
92#405C2×D311yescharacteristic
93#350—62no—
94#350—62no—
95C22⋊C16—62no—
96C23×D3M1611yescharacteristic
97C22×C3×C4C22×C411yescharacteristic
98C2×#67—21no—
99C22×Dic3M1611yescharacteristic
100#235—21no—
101#235—21no—
102#235—21no—
103#235—21no—
104C2×D12—22no—
105C2×D12C2×C811yescharacteristic
106C2×C4×D3—21no—
107C2×C4×D3—21no—
108C2×C4×D3—21no—
109C2×C4×D3—21no—
110C2×C4×D3—21no—
111C2×C4×D3C22⋊C412yes—
112C2×C4×D3C2×C812yes—
113C2×Dic6C2×C811yescharacteristic
114C3×M16—21no—
115C3×M16—21no—
116C2×C3×C8—21no—
117C2×C3×C8C2×D411yescharacteristic
118C2×C3×C8C2×D411yescharacteristic
119C2×C3×C8C2×D411yescharacteristic
120C2×C3×C8C2×D411yescharacteristic
121C2×C3×C8#4111yescharacteristic
122C2×C3×C8#4111yescharacteristic
123C3×C4⋊C4C2×C812yes—
124C3×C22⋊C4—21no—
125C3×C42—21no—
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—22no—
128C4⋊Dic3—22no—
129#208—21no—
130#208—21no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8C22⋊C412yes—
135ΓL(1,𝔽25)—41no—
136ΓL(1,𝔽25)—41no—
137D3×C8—22no—
138D3×C8—22no—
139D3×C8—22no—
140D3×C8—22no—
141D3×C8—41no—
142D3×C8—41no—
143C3×C16—22no—
144C3×C16—41no—
145C3⋊C16—22no—
146C3⋊C16—41no—
147C2×#41—31no—
148C23×C4—31no—
149#131—61no—
150#131—61no—
151#131—61no—
152#131—61no—
153C2×M16—31no—
154C2×M16—61no—
155C2×M16C4×D311yescharacteristic
156C22×C8—31no—
157C22×C8—32no—
158C22×C8—32no—
159C22×C8—61no—
160C22×C8—61no—
161C22×C8—61no—
162C22×C8—61no—
163C22×C8—61no—
164C22×C8C22×D311yescharacteristic
165#123—61no—
166C22⋊Q8—32no—
167#121—32no—
168#120—61no—
169C4×D4—62no—
170#117C4×D311yescharacteristic
171M32—21no—
172M32—21no—
173M32—61no—
174M32—61no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—32no—
179C2×C16—68no—
180C2×C16—61no—
181C2×C16—61no—
182C2×C16—61no—
183C2×C16D1212yescomplemented
184C4⋊C8—62no—
185C22⋊C8—62no—
186C22⋊C8—62no—
187#97—21no—
188C4×C8—21no—
189C23×C3C2×M1611yescharacteristic
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—24no—
193C22×D3—21no—
194C22×D3C22⋊C812yes—
195C2×C3×C4—22no—
196C2×C3×C4C2×C22⋊C411yescharacteristic
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4C4×D411yescharacteristic
199C2×C3×C4C22×C811yescharacteristic
200C2×C3×C4#13111yescharacteristic
201C2×C3×C4#13111yescharacteristic
202#67—41no—
203#67—41no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3C22⋊C812yes—
208D12—21no—
209D12—21no—
210D12—44no—
211C4×D3—22no—
212C4×D3—22no—
213C4×D3—22no—
214C4×D3—22no—
215C4×D3—41no—
216C4×D3—41no—
217Dic6—21no—
218Dic6—21no—
219C3×C8—41no—
220C3×C8#12011yescharacteristic
221C3×C8#12111yescharacteristic
222C3×C8#12111yescharacteristic
223C3×C8#12311yescharacteristic
224C3⋊C8—22no—
225C3⋊C8—41no—
226C24—31no—
227#41—61no—
228#41—61no—
229#41—61no—
230#41—61no—
231C2×Q8—31no—
232C2×D4—31no—
233C2×D4—62no—
234C2×D4—61no—
235C22×C4—31no—
236C22×C4—32no—
237C22×C4—32no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4C2×C4×D311yescharacteristic
244M16—21no—
245M16—21no—
246M16—61no—
247M16—61no—
248M16—121no—
249M16—121no—
250C2×C8—21no—
251C2×C8—32no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—62no—
255C2×C8—62no—
256C2×C8—62no—
257C2×C8—62no—
258C2×C8—61no—
259C2×C8—121no—
260C2×C8—121no—
261C2×C8C2×D1211yescharacteristic
262C2×C8#23511yescharacteristic
263C2×C8D3×D411yescharacteristic
264C2×C8D3×D411yescharacteristic
265C2×C8D3×D411yescharacteristic
266C2×C8#23911yescharacteristic
267C4⋊C4—62no—
268C4⋊C4D3×C812yes—
269C22⋊C4—21no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C42—21no—
273C16—22no—
274C16—41no—
275C16—62no—
276C16—121no—
277C22×C3—22no—
278C22×C3C2×C22⋊C811yescharacteristic
279C22×C3#43511yescharacteristic
280C22×C3#43511yescharacteristic
281C2×D3—24no—
282C2×D3—24no—
283C2×D3—22no—
284C2×D3—22no—
285C2×D3—42no—
286C2×D3—41no—
287C2×D3#35014yescomplemented
288C3×C4—42no—
289C3×C4#38611yescharacteristic
290C3×C4#40811yescharacteristic
291C3×C4C8×D411yescharacteristic
292C3×C4C8×D411yescharacteristic
293Dic3—22no—
294Dic3—41no—
295C23—32no—
296C23—64no—
297C23—61no—
298C23—61no—
299C23—61no—
300C23D3×M1611yescharacteristic
301Q8—61no—
302Q8—61no—
303D4—61no—
304D4—61no—
305D4—124no—
306D4—121no—
307D4—121no—
308C2×C4—22no—
309C2×C4—32no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—61no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4C4×D1211yescharacteristic
320C2×C4D3×C22⋊C411yescharacteristic
321C2×C4#89311yescharacteristic
322C2×C4C2×D3×C811yescharacteristic
323C2×C4#90111yescharacteristic
324C2×C4#90711yescharacteristic
325C8—41no—
326C8—62no—
327C8—121no—
328C8#88211yescharacteristic
329C8#88411yescharacteristic
330C8#89411yescharacteristic
331C8#89511yescharacteristic
332C2×C3—21no—
333C2×C3—21no—
334C2×C3#186911yescharacteristic
335C2×C3C2×#35012yes—
336D3—24no—
337D3—81no—
338C22—22no—
339C22—34no—
340C22—64no—
341C22—64no—
342C22—62no—
343C22—62no—
344C22—122no—
345C22—121no—
346C22#470611yescharacteristic
347C22D3×C22⋊C811yescharacteristic
348C22#483611yescharacteristic
349C4—42no—
350C4—62no—
351C4—121no—
352C4#466511yescharacteristic
353C4C8×D1211yescharacteristic
354C4#472711yescharacteristic
355C4#483411yescharacteristic
356C3#1298311yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—64no—
360C2—241no—
361C2#6866811yescharacteristic
362C2D3×#35012yes—
subgroupG/H
centerC2×C8D3×D4
commutatorC2×C3×C4C22×C8
FrattiniC22×C8C22×D3
FittingC3×#1379C2
socleC22×C3C2×C22⋊C8
prime radical#5524C4
cosocleC22×C3×C8C23
hypercenter#1379D3
hypocenterC3#12983
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ #1379

Complements

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

NH ≅ G/Ntype
1. C22×D3×C16C2semidirect
3. C2×#4515C2semidirect
4. #71431C2semidirect
5. C2×#4508C2semidirect
6. C3×#1379C2semidirect
7. #67671C2semidirect
34. #1379D3semidirect
64. C2×C3×C16D4semidirect
74. C2×C3⋊C16D4semidirect
183. C2×C16D12semidirect
287. C2×D3#350semidirect
356. C3#12983semidirect

Maximal subgroups

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

subgroupindex
C22×D3×C162
#785532
C2×#45152
#714312
C2×#45082
C3×#13792
#676712
#129833