Finite Group Databasebeta

← #10713348

Subgroups — #10713348

Order 768.

450 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
1#83786C211yesmaximal, characteristic
2C2×C3×#350C211yesmaximal, characteristic, complemented
3C2×#4561C211yesmaximal, characteristic, complemented
4C2×#4515C211yesmaximal, characteristic, complemented
5#71420C211yesmaximal, characteristic, complemented
6C2×#4508C211yesmaximal, characteristic, complemented
7C2×Dic3×C16C211yesmaximal, characteristic, complemented
8#13191—31nomaximal, Sylow
9C2×C3×#131C2211yescharacteristic
10#5832C411yescharacteristic
11C2×#901C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14#5234C411yescharacteristic
15#5129C411yescharacteristic
16#5128—22no—
17#5118C411yescharacteristic
18C8×#67—22no—
19C2×C8×Dic3C2211yescharacteristic
20C2×#812C2211yescharacteristic
21C22×C3⋊C16C2211yescharacteristic
22C3×#350—22no—
23#4561—22no—
24#4515—22no—
25#4508—22no—
26Dic3⋊C16—22no—
27Dic3×C16—22no—
28#2931—31no—
29#2150—31no—
30C2×#350—31no—
31C2×#350—31no—
32C2×#350—31no—
33C2×#350D311yescharacteristic, complemented
34C2×C4×C16—31no—
35C2×C3×#41C2×C411yescharacteristic
36C2×#235C2×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#947C811yescharacteristic
46#944C811yescharacteristic
47#940C811yescharacteristic
48#939—22no—
49#936C811yescharacteristic
50C4×#67—22no—
51C2×C4×Dic3C2×C411yescharacteristic
52C2×#208C2×C411yescharacteristic
53C22×C3⋊C8C2×C411yescharacteristic
54#901—21no—
55#901—21no—
56#901—21no—
57#901—21no—
58C2×ΓL(1,𝔽25)—21no—
59C2×D3×C8—21no—
60C3×M32—21no—
61C3×M32—21no—
62C2×C3×C16—21no—
63C2×C3×C16—21no—
64C2×C3×C16—21no—
65C2×C3×C16D412yescomplemented
66#830—22no—
67(C2×D3)⋊C8—22no—
68Dic3⋊C8—22no—
69C8×Dic3D412yes—
70C8×Dic3D412yes—
71#812—21no—
72#812—21no—
73C2×C3⋊C16—21no—
74C2×C3⋊C16—21no—
75C2×C3⋊C16—21no—
76C2×C3⋊C16D412yescomplemented
77C2×#131—31no—
78C2×#131C2×D311yescharacteristic
79#532—31no—
80C2×M32—31no—
81C2×M32C2×D311yescharacteristic
82C22×C16—31no—
83C22×C16C2×D311yescharacteristic
84#436—62no—
85C8×D4—62no—
86#423—31no—
87#408—31no—
88#408—31no—
89C2×C4×C8—31no—
90C4⋊C16—62no—
91#350—22no—
92#350—62no—
93#350—62no—
94#350—62no—
95C4×C16—62no—
96C3×#41—21no—
97C3×#41—21no—
98C3×#41—21no—
99C3×#41—21no—
100C2×C3×Q8C2×C811yescharacteristic
101C2×C3×D4—21no—
102C2×C3×D4C2×C811yescharacteristic
103C22×C3×C4—21no—
104C22×C3×C4C22×C411yescharacteristic
105C2×#67—21no—
106C2×#67—22no—
107C22×Dic3M1612yes—
108#235—21no—
109#235—21no—
110#235—21no—
111#235—21no—
112C2×D12C2×C811yescharacteristic
113C2×C4×D3—21no—
114C2×Dic6C2×C811yescharacteristic
115C3×M16—21no—
116C3×M16—21no—
117C3×M16—41no—
118C3×M16—41no—
119C2×C3×C8—21no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C2×C3×C8C2×D411yescharacteristic
123C2×C3×C8C2×D411yescharacteristic
124C2×C3×C8C2×D411yescharacteristic
125C2×C3×C8C2×D411yescharacteristic
126C2×C3×C8#4111yescharacteristic
127C2×C3×C8#4111yescharacteristic
128#217—22no—
129(C2×D3)⋊C4—22no—
130Dic3⋊C4—22no—
131C4×Dic3C22⋊C412yes—
132C4×Dic3C2×C811yescharacteristic
133C4×Dic3C2×C811yescharacteristic
134#208—21no—
135#208—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8C22⋊C412yes—
140ΓL(1,𝔽25)—41no—
141ΓL(1,𝔽25)—41no—
142D3×C8—41no—
143D3×C8—41no—
144C3×C16—22no—
145C3×C16—41no—
146C3⋊C16—22no—
147C3⋊C16—41no—
148C2×#41—31no—
149C2×#41C4×D311yescharacteristic
150#131—21no—
151#131—21no—
152#131—21no—
153#131—21no—
154#131—61no—
155#131—61no—
156#131—61no—
157#131—61no—
158C2×M16—21no—
159C2×M16—31no—
160C2×M16—61no—
161C2×M16C4×D311yescharacteristic
162C22×C8—21no—
163C22×C8—31no—
164C22×C8—61no—
165C22×C8C22×D311yescharacteristic
166C4⋊Q8—31no—
167#127—31no—
168#124—31no—
169#124—31no—
170#121—62no—
171C4×D4—62no—
172C2×C42—31no—
173M32—21no—
174M32—21no—
175M32—61no—
176M32—61no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—32no—
181C2×C16—61no—
182C2×C16—61no—
183C2×C16—61no—
184C2×C16#6712yescomplemented
185C4⋊C8—62no—
186C22⋊C8—62no—
187C22⋊C8—62no—
188C4×C8—32no—
189C4×C8—32no—
190C23×C3—21no—
191C23×C3C2×M1611yescharacteristic
192C22×D3—21no—
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×C4C22×C811yescharacteristic
206C2×C3×C4#13111yescharacteristic
207C2×C3×C4#13111yescharacteristic
208#67—41no—
209#67—41no—
210#67—44no—
211C2×Dic3—24no—
212C2×Dic3—21no—
213C2×Dic3C22⋊C812yes—
214C2×Dic3C22⋊C812yes—
215D12—21no—
216D12—21no—
217C4×D3—41no—
218C4×D3—41no—
219Dic6—21no—
220Dic6—21no—
221C3×C8—41no—
222C3×C8#12111yescharacteristic
223C3×C8#12111yescharacteristic
224C3×C8#12411yescharacteristic
225C3×C8#12711yescharacteristic
226C3⋊C8—22no—
227C3⋊C8—41no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—61no—
233#41—61no—
234#41—61no—
235#41—61no—
236C2×Q8—31no—
237C2×Q8D3×C811yescharacteristic
238C2×D4—21no—
239C2×D4—31no—
240C2×D4—62no—
241C2×D4—61no—
242C2×D4D3×C811yescharacteristic
243C22×C4—21no—
244C22×C4—32no—
245C22×C4—61no—
246C22×C4C2×C4×D311yescharacteristic
247M16—21no—
248M16—21no—
249M16—41no—
250M16—41no—
251M16—61no—
252M16—61no—
253M16—121no—
254M16—121no—
255C2×C8—21no—
256C2×C8—32no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—121no—
263C2×C8—121no—
264C2×C8#23511yescharacteristic
265C2×C8D3×D411yescharacteristic
266C2×C8D3×D411yescharacteristic
267C2×C8D3×D411yescharacteristic
268C2×C8#23711yescharacteristic
269C2×C8C2×#6711yescharacteristic
270C4⋊C4—62no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C42—32no—
274C42—31no—
275C42—31no—
276C16—22no—
277C16—41no—
278C16—62no—
279C16—121no—
280C22×C3—22no—
281C22×C3—42no—
282C22×C3—41no—
283C22×C3C2×C22⋊C811yescharacteristic
284C22×C3#43611yescharacteristic
285C22×C3#43611yescharacteristic
286C2×D3—42no—
287C2×D3—41no—
288C3×C4—41no—
289C3×C4#39011yescharacteristic
290C3×C4#40811yescharacteristic
291C3×C4C8×D411yescharacteristic
292C3×C4C8×D411yescharacteristic
293Dic3—22no—
294Dic3—41no—
295Dic3#35014yescomplemented
296C23—21no—
297C23—61no—
298C23D3×M1611yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—61no—
302Q8—61no—
303D4—21no—
304D4—21no—
305D4—41no—
306D4—41no—
307D4—61no—
308D4—61no—
309D4—124no—
310D4—121no—
311D4—121no—
312C2×C4—21no—
313C2×C4—32no—
314C2×C4—32no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—64no—
318C2×C4—61no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4D3×C22⋊C411yescharacteristic
322C2×C4#67⋊C411yescharacteristic
323C2×C4C2×D3×C811yescharacteristic
324C2×C4#90111yescharacteristic
325C2×C4#90711yescharacteristic
326C2×C4C4×#6711yescharacteristic
327C8—41no—
328C8—62no—
329C8—121no—
330C8#88411yescharacteristic
331C8#88511yescharacteristic
332C8#93911yescharacteristic
333C8#94011yescharacteristic
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—81no—
337C2×C3#195411yescharacteristic
338C2×C3C2×#35012yes—
339D3—81no—
340C22—22no—
341C22—42no—
342C22—41no—
343C22—122no—
344C22—121no—
345C22D3×C22⋊C811yescharacteristic
346C22#472911yescharacteristic
347C22#512811yescharacteristic
348C4—34no—
349C4—41no—
350C4—62no—
351C4—121no—
352C4#67⋊C811yescharacteristic
353C4#472711yescharacteristic
354C4#495711yescharacteristic
355C4C8×#6711yescharacteristic
356C3#1319111yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—81no—
360C2—241no—
361C2D3×#35012yes—
362C2#6991611yescharacteristic
subgroupG/H
centerC2×C8D3×D4
commutatorC2×C3×C4C22×C8
FrattiniC22×C8C22×D3
FittingC2×C3×#350C2
socleC22×C3C2×C22⋊C8
prime radical#5832C4
cosocleC22×C3×C8C23
hypercenterC2×#350D3
hypocenterC3#13191
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. C2×C3×#350C2semidirect
3. C2×#4561C2semidirect
4. C2×#4515C2semidirect
5. #71420C2semidirect
6. C2×#4508C2semidirect
7. C2×Dic3×C16C2semidirect
33. C2×#350D3semidirect
65. C2×C3×C16D4semidirect
76. C2×C3⋊C16D4semidirect
184. C2×C16#67semidirect
295. Dic3#350semidirect
356. C3#13191semidirect

Maximal subgroups

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

subgroupindex
#837862
C2×C3×#3502
C2×#45612
C2×#45152
#714202
C2×#45082
C2×Dic3×C162
#131913