Finite Group Databasebeta

← #10693577

Subgroups — #10693577

Order 768.

454 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#1770C211yesmaximal, characteristic, complemented
2C2×#4485C211yesmaximal, characteristic
3#69772C211yesmaximal, characteristic
4#69487C211yesmaximal, characteristic
5#69467C211yesmaximal, characteristic, complemented
6#67702C211yesmaximal, characteristic, complemented
7#67654C211yesmaximal, characteristic, complemented
8#9572—31nomaximal, Sylow
9C2×#874C2211yescharacteristic
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×#97—21no—
12C2×C3×C4×C8C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#821—21no—
15C2×#821—21no—
16C2×#809—21no—
17C2×#809—21no—
18C2×#808—21no—
19#4935C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×#336—21no—
23#4529—21no—
24#4485C414yescomplemented
25#1982—31no—
26#1957—31no—
27#1948—31no—
28#1770D311yescharacteristic, complemented
29C2×#331—31no—
30#1296—31no—
31#1277—31no—
32C22×D12—21no—
33C22×D12—21no—
34C22×D12—21no—
35C22×C3×C8—21no—
36C22×C3×C8D411yescharacteristic
37C22×C3×C8D411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C42—21no—
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C22×C3⋊C8—21no—
46C22×C3⋊C8—21no—
47#874—21no—
48#874—21no—
49#874—21no—
50#874C2×C412yes—
51C3×C4⋊C8—21no—
52C3×C4⋊C8C2×C412yes—
53C3×#97—42no—
54C3×C4×C8—21no—
55C3×C4×C8—21no—
56C3×#95—21no—
57#821—42no—
58#821—42no—
59#809—42no—
60#809—42no—
61#808—42no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)C2×C412yes—
66C2×#127—31no—
67C2×#106—61no—
68C2×C4⋊C8—31no—
69C2×C4⋊C8—31no—
70C2×C4⋊C8C2×D311yescharacteristic
71C2×#102—61no—
72C2×#102—61no—
73C2×#102—61no—
74C2×#102—61no—
75C2×#97—21no—
76C2×C4×C8C2×D311yescharacteristic
77#384—31no—
78C4×C4⋊C4C2×D311yescharacteristic
79#340—61no—
80#336—21no—
81#331—34no—
82C23×D3—41no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4111yescharacteristic
88C22×Dic3—41no—
89C2×D12—22no—
90C2×D12—22no—
91C2×D12—22no—
92C2×D12—41no—
93C2×D12—41no—
94C2×D12—41no—
95C2×D12—41no—
96C2×D12—41no—
97C2×D12—41no—
98C2×D12—41no—
99C2×D12—41no—
100C2×D12—41no—
101C2×D12—41no—
102C2×D12—41no—
103C2×D12—81no—
104C2×D12—81no—
105C2×C3×C8—21no—
106C2×C3×C8—21no—
107C2×C3×C8—22no—
108C2×C3×C8—21no—
109C2×C3×C8—21no—
110C2×C3×C8—21no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C2×C3×C8SD1612yes—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C42—41no—
119C3×C42—41no—
120C3×C42C22×C411yescharacteristic
121C3×C42#4111yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124C4⋊Dic3—41no—
125C4⋊Dic3—41no—
126C4⋊Dic3—41no—
127C4⋊Dic3—41no—
128C4⋊Dic3—81no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C22×D4—61no—
136C22×D4—61no—
137C22×D4—61no—
138C22×C8—21no—
139C22×C8—61no—
140C22×C8—61no—
141C22×C8#6711yescharacteristic
142C22×C8#6711yescharacteristic
143#127—32no—
144#127—61no—
145#127—61no—
146#127—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4D1211yescharacteristic
151C2×C4⋊C4D1211yescharacteristic
152C2×C42—21no—
153C2×C42C22×D311yescharacteristic
154#106—122no—
155C4⋊C8—21no—
156C4⋊C8—32no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8C4×D312yes—
161#102—122no—
162#102—122no—
163#102—122no—
164#102—122no—
165#97—42no—
166C4×C8—21no—
167C4×C8—21no—
168#95—21no—
169C23×C3#12111yescharacteristic
170C22×D3—42no—
171C22×D3—81no—
172C22×D3—81no—
173C22×D3—81no—
174C22×D3—81no—
175C22×D3—81no—
176C22×D3—81no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4#11711yescharacteristic
186C2×C3×C4C4×D411yescharacteristic
187C2×C3×C4C4×D411yescharacteristic
188C2×C3×C4#12311yescharacteristic
189C2×C3×C4#12411yescharacteristic
190C2×C3×C4#12611yescharacteristic
191C2×C3×C4C2×SD1611yescharacteristic
192C2×C3×C4#13511yescharacteristic
193C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
196C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
198C2×C3×C4#13711yescharacteristic
199C2×Dic3—81no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202D12—42no—
203D12—42no—
204D12—42no—
205D12—42no—
206D12—82no—
207D12—82no—
208D12—81no—
209D12—81no—
210D12—81no—
211D12—81no—
212C3×C8—22no—
213C3×C8—42no—
214C3×C8—41no—
215C3⋊C8—42no—
216C3⋊C8—81no—
217C24—121no—
218C2×D4—62no—
219C2×D4—62no—
220C2×D4—62no—
221C2×D4—121no—
222C2×D4—121no—
223C2×D4—121no—
224C2×D4—121no—
225C2×D4—121no—
226C2×D4—121no—
227C2×D4—121no—
228C2×D4—121no—
229C2×D4—121no—
230C2×D4—121no—
231C2×D4—121no—
232C2×D4—241no—
233C2×D4—241no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—121no—
237C22×C4C2×D1211yescharacteristic
238C22×C4#23511yescharacteristic
239C22×C4C2×#6711yescharacteristic
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—22no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—62no—
249C2×C8—121no—
250C2×C8—121no—
251C2×C8—121no—
252C2×C8—121no—
253C2×C8—121no—
254C2×C8#21512yes—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—241no—
264C42—41no—
265C42—41no—
266C42C2×C4×D311yescharacteristic
267C42#23511yescharacteristic
268C42#23511yescharacteristic
269C42#23511yescharacteristic
270C22×C3#38811yescharacteristic
271C22×C3#46011yescharacteristic
272C22×C3#46311yescharacteristic
273C22×C3#46811yescharacteristic
274C22×C3#46911yescharacteristic
275C22×C3#47111yescharacteristic
276C22×C3#47211yescharacteristic
277C2×D3—82no—
278C2×D3—82no—
279C2×D3—82no—
280C2×D3—161no—
281C2×D3—161no—
282C2×D3—161no—
283C2×D3—161no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—81no—
287C3×C4—81no—
288C3×C4C4×SD1611yescharacteristic
289C3×C4#44011yescharacteristic
290C3×C4#44211yescharacteristic
291C3×C4#44211yescharacteristic
292C3×C4#48111yescharacteristic
293C3×C4#48211yescharacteristic
294C3×C4#48911yescharacteristic
295C3×C4#49011yescharacteristic
296Dic3—161no—
297C23—122no—
298C23—241no—
299C23—241no—
300C23—241no—
301C23—241no—
302C23—241no—
303C23—241no—
304C23#93011yescharacteristic
305D4—122no—
306D4—122no—
307D4—122no—
308D4—122no—
309D4—242no—
310D4—242no—
311D4—241no—
312D4—241no—
313D4—241no—
314D4—241no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—241no—
324C2×C4—241no—
325C2×C4—241no—
326C2×C4#87211yescharacteristic
327C2×C4C4×D1211yescharacteristic
328C2×C4#87511yescharacteristic
329C2×C4#87611yescharacteristic
330C2×C4#90811yescharacteristic
331C2×C4#90811yescharacteristic
332C2×C4#90811yescharacteristic
333C2×C4#90911yescharacteristic
334C2×C4C4×#6711yescharacteristic
335C2×C4#92911yescharacteristic
336C2×C4#93211yescharacteristic
337C2×C4C2×#21511yescharacteristic
338C2×C4#94911yescharacteristic
339C2×C4#95011yescharacteristic
340C8—22no—
341C8—41no—
342C8—42no—
343C8—122no—
344C8—241no—
345C2×C3#168611yescharacteristic
346C2×C3#168811yescharacteristic
347C2×C3#169711yescharacteristic
348C2×C3#170111yescharacteristic
349C2×C3#192611yescharacteristic
350C2×C3#193111yescharacteristic
351C2×C3#193611yescharacteristic
352D3—162no—
353C22—242no—
354C22—242no—
355C22—242no—
356C22—481no—
357C22—481no—
358C22—481no—
359C22—481no—
360C22#494011yescharacteristic
361C22#501611yescharacteristic
362C22#502711yescharacteristic
363C22#513411yescharacteristic
364C22#513511yescharacteristic
365C22#514111yescharacteristic
366C22#514211yescharacteristic
367C4—21no—
368C4—21no—
369C4—81no—
370C4—81no—
371C4—481no—
372C4#471011yescharacteristic
373C4#471111yescharacteristic
374C4#471311yescharacteristic
375C4#471411yescharacteristic
376C4#496811yescharacteristic
377C4#500311yescharacteristic
378C4#501411yescharacteristic
379C4C4×#21511yescharacteristic
380C3#957211yescharacteristic, complemented, Sylow
381C2—482no—
382C2#6946311yescharacteristic
383C2#6981811yescharacteristic
384C2#6989211yescharacteristic
385C2#7086911yescharacteristic
386C2#7087011yescharacteristic
387C2#7095811yescharacteristic
388C2#7095911yescharacteristic
subgroupG/H
centerC23#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1770C2
socleC23×C3#121
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1770D3
hypocenterC3#9572
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1770

Complements

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

NH ≅ G/Ntype
1. C3×#1770C2semidirect
5. #69467C2semidirect
6. #67702C2semidirect
7. #67654C2semidirect
24. #4485C4semidirect
28. #1770D3semidirect
380. C3#9572semidirect

Maximal subgroups

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

subgroupindex
C3×#17702
C2×#44852
#697722
#694872
#694672
#677022
#676542
#95723