Finite Group Databasebeta

← #10698232

Subgroups — #10698232

Order 768.

536 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73717C211yesmaximal, characteristic, complemented
2#70457C211yesmaximal, characteristic
3#69772C211yesmaximal, characteristic
4#69487C211yesmaximal, characteristic
5C3×#1279C211yesmaximal, characteristic, complemented
6#67702C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#10354—31nomaximal, Sylow
9C2×#890—21no—
10C2×#874C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#821—21no—
15C2×#821—21no—
16C2×#821—22no—
17#4984—21no—
18C2×#809—21no—
19C2×#809—21no—
20#4935—21no—
21#4935C2211yescharacteristic
22C4×C4⋊Dic3C2211yescharacteristic
23#4928—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4661—22no—
26#2552—31no—
27#1982—31no—
28#1982—31no—
29#1957—31no—
30#1296—31no—
31#1279—31no—
32#1279D311yescharacteristic, complemented
33C22×D12—21no—
34C22×D12—21no—
35C22×D12—21no—
36C22×C3×C8—21no—
37C22×C3×C8—21no—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C42—21no—
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3D411yescharacteristic
47C2×C4⋊Dic3D411yescharacteristic
48C2×C4⋊Dic3D412yes—
49C2×Dic3⋊C4—22no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52#890—42no—
53#890—42no—
54#874—21no—
55#874—21no—
56#874—21no—
57#874Q812yes—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4×C8—21no—
61C3×C4×C8—21no—
62C3×#95—21no—
63C3⋊#95—22no—
64C3⋊#95—21no—
65#821—42no—
66#821—42no—
67#821—44no—
68#809—42no—
69#809—42no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C2×#127—31no—
73C2×#125—61no—
74C2×C4⋊C8—31no—
75C2×C4⋊C8C2×D311yescharacteristic
76C2×#102—62no—
77C2×#102—61no—
78C2×#102—61no—
79C2×#102—61no—
80C2×#102—61no—
81C2×C4×C8C2×D311yescharacteristic
82#398—62no—
83#389—61no—
84#389—61no—
85#384—31no—
86#384—61no—
87C4×C4⋊C4—31no—
88C4×C4⋊C4C2×D311yescharacteristic
89C23×D3—41no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×Dic3—21no—
96C22×Dic3—21no—
97C22×Dic3—22no—
98C2×D12—22no—
99C2×D12—22no—
100C2×D12—22no—
101C2×D12—42no—
102C2×D12—42no—
103C2×D12—41no—
104C2×D12—41no—
105C2×D12—41no—
106C2×D12—41no—
107C2×D12—41no—
108C2×D12—41no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×C3×C8—22no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C2×C3×C8—41no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4D812yes—
123C3×C4⋊C4SD1612yes—
124C3×C42—41no—
125C3×C42—41no—
126C3×C42C2×Q811yescharacteristic
127C3×C42#4111yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—44no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3—42no—
140C4⋊Dic3—82no—
141Dic3⋊C4—44no—
142C4×Dic3—41no—
143C4×Dic3—41no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—41no—
147C22×D4—61no—
148C22×D4—61no—
149C22×D4—61no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—61no—
153#127—32no—
154#127—61no—
155#127—61no—
156#127—61no—
157#125—122no—
158#125—122no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—32no—
162C2×C4⋊C4—62no—
163C2×C4⋊C4—62no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—62no—
167C2×C4⋊C4D1211yescharacteristic
168C2×C4⋊C4D1211yescharacteristic
169C2×C42—21no—
170C2×C42—61no—
171C2×C42C22×D311yescharacteristic
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176#102—122no—
177#102—122no—
178#102—124no—
179#102—122no—
180#102—122no—
181C4×C8—21no—
182C4×C8—21no—
183#95—21no—
184#95—62no—
185#95—61no—
186C23×C3#12011yescharacteristic
187C22×D3—42no—
188C22×D3—81no—
189C22×D3—81no—
190C22×D3—81no—
191C22×D3—81no—
192C22×D3—81no—
193C22×D3—81no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4C22⋊Q811yescharacteristic
203C2×C3×C4C22⋊Q811yescharacteristic
204C2×C3×C4C22⋊Q811yescharacteristic
205C2×C3×C4#12411yescharacteristic
206C2×C3×C4#12411yescharacteristic
207C2×C3×C4#12411yescharacteristic
208C2×C3×C4C2×D811yescharacteristic
209C2×C3×C4C2×SD1611yescharacteristic
210C2×C3×C4#13511yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
215C2×C3×C4#13711yescharacteristic
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225D12—44no—
226D12—42no—
227D12—42no—
228D12—42no—
229D12—42no—
230D12—82no—
231D12—81no—
232D12—81no—
233D12—81no—
234D12—81no—
235C3×C8—42no—
236C3×C8—81no—
237C3⋊C8—81no—
238C24—121no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—121no—
245C2×D4—121no—
246C2×D4—121no—
247C2×D4—121no—
248C2×D4—121no—
249C2×D4—121no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—62no—
260C22×C4C2×D1211yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—121no—
270C2×C8—121no—
271C2×C8—121no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—122no—
279C4⋊C4—124no—
280C4⋊C4—124no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—242no—
286C4⋊C4#20412yes—
287C4⋊C4D2412yes—
288C42—41no—
289C42—41no—
290C42—121no—
291C42—121no—
292C42#23511yescharacteristic
293C42#23511yescharacteristic
294C42#23711yescharacteristic
295C42D3×Q811yescharacteristic
296C22×C3#39311yescharacteristic
297C22×C3#44711yescharacteristic
298C22×C3#44811yescharacteristic
299C22×C3#44911yescharacteristic
300C22×C3#44911yescharacteristic
301C22×C3#45311yescharacteristic
302C22×C3#45411yescharacteristic
303C2×D3—82no—
304C2×D3—82no—
305C2×D3—82no—
306C2×D3—161no—
307C2×D3—161no—
308C2×D3—161no—
309C2×D3—161no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—42no—
313C3×C4—42no—
314C3×C4#47411yescharacteristic
315C3×C4#47611yescharacteristic
316C3×C4#47811yescharacteristic
317C3×C4#47811yescharacteristic
318C3×C4#48611yescharacteristic
319C3×C4#48811yescharacteristic
320C3×C4#48911yescharacteristic
321C3×C4#49011yescharacteristic
322Dic3—81no—
323Dic3—82no—
324Dic3—81no—
325C23—122no—
326C23—241no—
327C23—241no—
328C23—241no—
329C23—241no—
330C23—241no—
331C23—241no—
332C23#88211yescharacteristic
333D4—124no—
334D4—122no—
335D4—122no—
336D4—122no—
337D4—122no—
338D4—242no—
339D4—241no—
340D4—241no—
341D4—241no—
342D4—241no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4#87511yescharacteristic
361C2×C4#88511yescharacteristic
362C2×C4#88511yescharacteristic
363C2×C4#89611yescharacteristic
364C2×C4#89611yescharacteristic
365C2×C4#89711yescharacteristic
366C2×C4C2×#20411yescharacteristic
367C2×C4C2×D2411yescharacteristic
368C2×C4#90811yescharacteristic
369C2×C4#90911yescharacteristic
370C2×C4#91111yescharacteristic
371C2×C4#91411yescharacteristic
372C2×C4#91611yescharacteristic
373C2×C4#91911yescharacteristic
374C8—42no—
375C8—81no—
376C8—241no—
377C2×C3#162111yescharacteristic
378C2×C3#162211yescharacteristic
379C2×C3#162511yescharacteristic
380C2×C3#162611yescharacteristic
381C2×C3#202311yescharacteristic
382C2×C3#202511yescharacteristic
383C2×C3#202711yescharacteristic
384D3—162no—
385C22—242no—
386C22—242no—
387C22—242no—
388C22—481no—
389C22—481no—
390C22—481no—
391C22—481no—
392C22#467311yescharacteristic
393C22#473211yescharacteristic
394C22#473811yescharacteristic
395C22#478111yescharacteristic
396C22#481011yescharacteristic
397C22#482211yescharacteristic
398C22#482711yescharacteristic
399C4—21no—
400C4—21no—
401C4—42no—
402C4—42no—
403C4—241no—
404C4—241no—
405C4—242no—
406C4#469411yescharacteristic
407C4#471411yescharacteristic
408C4#476411yescharacteristic
409C4#480011yescharacteristic
410C4#484211yescharacteristic
411C4#484611yescharacteristic
412C4#487111yescharacteristic
413C4#489111yescharacteristic
414C3#1035411yescharacteristic, complemented, Sylow
415C2—482no—
416C2#6819811yescharacteristic
417C2#6820011yescharacteristic
418C2#6825211yescharacteristic
419C2#6825911yescharacteristic
420C2#6867811yescharacteristic
421C2#6876311yescharacteristic
422C2#6889911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenter#1279D3
hypocenterC3#10354
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. #73717C2semidirect
5. C3×#1279C2semidirect
6. #67702C2semidirect
7. #67655C2semidirect
32. #1279D3semidirect
414. C3#10354semidirect

Maximal subgroups

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

subgroupindex
#737172
#704572
#697722
#694872
C3×#12792
#677022
#676552
#103543