Finite Group Databasebeta

← #10693522

Subgroups — #10693522

Order 768.

466 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
1#70457C211yesmaximal, characteristic
2#70454C211yesmaximal, characteristic, complemented
3C2×#4485C211yesmaximal, characteristic
4#69772C211yesmaximal, characteristic
5C8×C4⋊Dic3C211yesmaximal, characteristic, complemented
6C3×#1277C211yesmaximal, characteristic, complemented
7#67702C211yesmaximal, characteristic, complemented
8#9564—31nomaximal, Sylow
9C2×#874C2211yescharacteristic
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#821—21no—
14C2×#821—21no—
15C2×#821—21no—
16C2×#821—21no—
17C2×#817—21no—
18C2×C8×Dic3—21no—
19#4935C2211yescharacteristic
20C4×C4⋊Dic3C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22#4551—21no—
23#4550—21no—
24#4485C414yescomplemented
25#1982—31no—
26#1957—31no—
27#1948—31no—
28C8×C4⋊C4—31no—
29C2×#331—31no—
30#1296—31no—
31#1277D311yescharacteristic, complemented
32C22×D12—21no—
33C22×D12—21no—
34C22×D12—21no—
35C22×C3×C8—21no—
36C22×C3×C8—21no—
37C22×C3×C8D411yescharacteristic
38C22×C3×C8D411yescharacteristic
39C2×C3×C42C2311yescharacteristic
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3D411yescharacteristic
44C2×C4⋊Dic3D411yescharacteristic
45C2×C4×Dic3—21no—
46C22×C3⋊C8—21no—
47#874—21no—
48#874—21no—
49#874—21no—
50#874C2×C412yes—
51C3×C4⋊C8—21no—
52C3×C4⋊C8—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8C2×C412yes—
55C3×C4×C8—21no—
56C3×C4×C8—21no—
57C3⋊#95—21no—
58#821—42no—
59#821—42no—
60#821—42no—
61#821—42no—
62#817—42no—
63C8×Dic3—42no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)C2×C412yes—
66C2×#127—31no—
67C2×#106—61no—
68C2×C4⋊C8—31no—
69C2×C4⋊C8C2×D311yescharacteristic
70C2×C4⋊C8C2×D311yescharacteristic
71C2×#102—61no—
72C2×#102—61no—
73C2×#102—61no—
74C2×#102—61no—
75C2×C4×C8—61no—
76C2×C4×C8C2×D311yescharacteristic
77#384—31no—
78C4×C4⋊C4—31no—
79#340—61no—
80#336—61no—
81#331—34no—
82C23×D3—41no—
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4#4111yescharacteristic
86C22×Dic3—21no—
87C22×Dic3—21no—
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×C8—41no—
114C2×C3×C8—41no—
115C2×C3×C8—41no—
116C2×C3×C8SD1612yes—
117C3×C42C22×C411yescharacteristic
118C3×C42#4111yescharacteristic
119C3×C42#4111yescharacteristic
120C3×C42#4111yescharacteristic
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125C4⋊Dic3—41no—
126C4⋊Dic3—41no—
127C4⋊Dic3—81no—
128C4⋊Dic3D812yes—
129C4⋊Dic3SD1612yes—
130C4×Dic3—41no—
131C4×Dic3—41no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C22×D4—61no—
136C22×D4—61no—
137C22×D4—61no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—61no—
141C22×C8D1211yescharacteristic
142C22×C8D1211yescharacteristic
143#127—32no—
144#127—61no—
145#127—61no—
146#127—61no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C42—61no—
153C2×C42C22×D311yescharacteristic
154#106—122no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—32no—
159C4⋊C8—61no—
160C4⋊C8C4×D312yes—
161#102—122no—
162#102—122no—
163#102—122no—
164#102—122no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—122no—
168#95—61no—
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#11711yescharacteristic
180C2×C3×C4C4×D411yescharacteristic
181C2×C3×C4C4×D411yescharacteristic
182C2×C3×C4#12311yescharacteristic
183C2×C3×C4#12411yescharacteristic
184C2×C3×C4#12611yescharacteristic
185C2×C3×C4C2×D811yescharacteristic
186C2×C3×C4C2×SD1611yescharacteristic
187C2×C3×C4C2×SD1611yescharacteristic
188C2×C3×C4#13511yescharacteristic
189C2×C3×C4#13511yescharacteristic
190C2×C3×C4#13511yescharacteristic
191C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
192C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
193C2×Dic3—22no—
194C2×Dic3—22no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
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—81no—
216C3⋊C8—42no—
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—61no—
235C22×C4—61no—
236C22×C4—121no—
237C22×C4C2×D1211yescharacteristic
238C22×C4D3×D411yescharacteristic
239C22×C4#23911yescharacteristic
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—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—62no—
252C2×C8—121no—
253C2×C8—121no—
254C2×C8#20412yes—
255C4⋊C4—32no—
256C4⋊C4—32no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—241no—
264C42—121no—
265C42—121no—
266C42C2×C4×D311yescharacteristic
267C42#23511yescharacteristic
268C42#23711yescharacteristic
269C42#23711yescharacteristic
270C22×C3#38811yescharacteristic
271C22×C3#46011yescharacteristic
272C22×C3#46311yescharacteristic
273C22×C3#46511yescharacteristic
274C22×C3#46611yescharacteristic
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×C4C4×D811yescharacteristic
287C3×C4C4×SD1611yescharacteristic
288C3×C4C4×SD1611yescharacteristic
289C3×C4#44211yescharacteristic
290C3×C4#48111yescharacteristic
291C3×C4#48211yescharacteristic
292C3×C4#48611yescharacteristic
293C3×C4#48811yescharacteristic
294Dic3—42no—
295Dic3—42no—
296Dic3—161no—
297C23—122no—
298C23—241no—
299C23—241no—
300C23—241no—
301C23—241no—
302C23—241no—
303C23—241no—
304C23#89511yescharacteristic
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—62no—
318C2×C4—62no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—241no—
324C2×C4—241no—
325C2×C4—241no—
326C2×C4C4×D1211yescharacteristic
327C2×C4#67⋊C411yescharacteristic
328C2×C4#88511yescharacteristic
329C2×C4#88611yescharacteristic
330C2×C4#89211yescharacteristic
331C2×C4#89811yescharacteristic
332C2×C4C2×#20411yescharacteristic
333C2×C4#90411yescharacteristic
334C2×C4#90811yescharacteristic
335C2×C4#90811yescharacteristic
336C2×C4D3×D811yescharacteristic
337C2×C4D3×SD1611yescharacteristic
338C2×C4#91611yescharacteristic
339C2×C4#91911yescharacteristic
340C8—22no—
341C8—41no—
342C8—42no—
343C8—81no—
344C8—122no—
345C2×C3#166611yescharacteristic
346C2×C3#166811yescharacteristic
347C2×C3#167711yescharacteristic
348C2×C3#168111yescharacteristic
349C2×C3#192611yescharacteristic
350C2×C3#193111yescharacteristic
351C2×C3#193411yescharacteristic
352D3—162no—
353C22—242no—
354C22—242no—
355C22—242no—
356C22—481no—
357C22—481no—
358C22—481no—
359C22—481no—
360C22#467011yescharacteristic
361C22#484411yescharacteristic
362C22#484811yescharacteristic
363C22#486411yescharacteristic
364C22#488311yescharacteristic
365C22#489711yescharacteristic
366C22#489811yescharacteristic
367C4—21no—
368C4—21no—
369C4—122no—
370C4—122no—
371C4—481no—
372C4C4×#20411yescharacteristic
373C4#471111yescharacteristic
374C4#473511yescharacteristic
375C4#473711yescharacteristic
376C4#475611yescharacteristic
377C4#476011yescharacteristic
378C4#478811yescharacteristic
379C4#480011yescharacteristic
380C3#956411yescharacteristic, complemented, Sylow
381C2—482no—
382C2#6853811yescharacteristic
383C2#6854611yescharacteristic
384C2#6858011yescharacteristic
385C2#6858611yescharacteristic
386C2#6862611yescharacteristic
387C2#6885311yescharacteristic
388C2#6893111yescharacteristic
subgroupG/H
centerC23#895
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1277C2
socleC23×C3#121
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1277D3
hypocenterC3#9564
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
2. #70454C2semidirect
5. C8×C4⋊Dic3C2semidirect
6. C3×#1277C2semidirect
7. #67702C2semidirect
24. #4485C4semidirect
31. #1277D3semidirect
380. C3#9564semidirect

Maximal subgroups

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

subgroupindex
#704572
#704542
C2×#44852
#697722
C8×C4⋊Dic32
C3×#12772
#677022
#95643