Finite Group Databasebeta

← #10698215

Subgroups — #10698215

Order 768.

500 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73725C211yesmaximal, characteristic, complemented
2#70457C211yesmaximal, characteristic
3#69772C211yesmaximal, characteristic
4#69487C211yesmaximal, characteristic
5C3×#1279C211yesmaximal, characteristic, complemented
6#67702C211yesmaximal, characteristic, complemented
7#67655C211yesmaximal, characteristic, complemented
8#10351—31nomaximal, Sylow
9C2×#874C2211yescharacteristic
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×C4×C8C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13C2×#821—21no—
14C2×#821—21no—
15C2×#821—22no—
16#4977—21no—
17#4972—21no—
18C2×#809—21no—
19C2×#809—21no—
20#4936—21no—
21#4935C2211yescharacteristic
22C4×C4⋊Dic3C2211yescharacteristic
23#4928—21no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4660—22no—
26#2571—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—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3D411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×C4⋊Dic3D412yes—
47C2×Dic3⋊C4—22no—
48C2×C4×Dic3—21no—
49C22×C3⋊C8—21no—
50#874—21no—
51#874—21no—
52#874—21no—
53#874Q812yes—
54C3×C4⋊C8—21no—
55C3×C4⋊C8—21no—
56C3×C4×C8—21no—
57C3×C4×C8—21no—
58C3×#95—21no—
59C3⋊#95—22no—
60C3⋊#95—22no—
61C3⋊#95—22no—
62C3⋊#95—21no—
63#821—42no—
64#821—42no—
65#821—44no—
66#809—42no—
67#809—42no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C2×#127—31no—
71C2×C4⋊C8—31no—
72C2×C4⋊C8C2×D311yescharacteristic
73C2×#102—62no—
74C2×#102—61no—
75C2×#102—61no—
76C2×#102—61no—
77C2×#102—61no—
78C2×C4×C8C2×D311yescharacteristic
79#400—62no—
80#389—61no—
81#389—61no—
82#384—31no—
83#382—61no—
84#382—61no—
85C4×C4⋊C4—31no—
86C4×C4⋊C4C2×D311yescharacteristic
87C23×D3—41no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×Dic3—21no—
94C22×Dic3—21no—
95C22×Dic3—22no—
96C2×D12—22no—
97C2×D12—22no—
98C2×D12—22no—
99C2×D12—42no—
100C2×D12—42no—
101C2×D12—41no—
102C2×D12—41no—
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×C3×C8—22no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C2×C3×C8—41no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C42—41no—
123C3×C42—41no—
124C3×C42C2×Q811yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127C3×C42#4111yescharacteristic
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—42no—
137Dic3⋊C4—82no—
138C4×Dic3—41no—
139C4×Dic3—41no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C22×D4—61no—
144C22×D4—61no—
145C22×D4—61no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—61no—
149#127—32no—
150#127—61no—
151#127—61no—
152#127—61no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4D1211yescharacteristic
160C2×C4⋊C4D1211yescharacteristic
161C2×C42—21no—
162C2×C42—61no—
163C2×C42C22×D311yescharacteristic
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168#102—122no—
169#102—122no—
170#102—122no—
171#102—122no—
172#102—124no—
173C4×C8—21no—
174C4×C8—21no—
175#95—21no—
176#95—62no—
177#95—62no—
178#95—62no—
179#95—61no—
180C23×C3#12011yescharacteristic
181C22×D3—42no—
182C22×D3—81no—
183C22×D3—81no—
184C22×D3—81no—
185C22×D3—81no—
186C22×D3—81no—
187C22×D3—81no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4C22⋊Q811yescharacteristic
197C2×C3×C4C22⋊Q811yescharacteristic
198C2×C3×C4C22⋊Q811yescharacteristic
199C2×C3×C4#12411yescharacteristic
200C2×C3×C4#12411yescharacteristic
201C2×C3×C4#12411yescharacteristic
202C2×C3×C4#13511yescharacteristic
203C2×C3×C4#13511yescharacteristic
204C2×C3×C4#13511yescharacteristic
205C2×C3×C4#13511yescharacteristic
206C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
207C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
208C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
209C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219D12—44no—
220D12—42no—
221D12—42no—
222D12—42no—
223D12—42no—
224D12—82no—
225D12—81no—
226D12—81no—
227D12—81no—
228D12—81no—
229C3×C8—42no—
230C3×C8—81no—
231C3⋊C8—81no—
232C24—121no—
233C2×D4—62no—
234C2×D4—62no—
235C2×D4—62no—
236C2×D4—122no—
237C2×D4—122no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—121no—
244C2×D4—121no—
245C2×D4—121no—
246C2×D4—121no—
247C2×D4—121no—
248C2×D4—121no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—62no—
254C22×C4C2×D1211yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—121no—
264C2×C8—121no—
265C2×C8—121no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—122no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C4⋊C4—242no—
280C42—41no—
281C42—41no—
282C42—121no—
283C42—121no—
284C42#23511yescharacteristic
285C42#23511yescharacteristic
286C42#23711yescharacteristic
287C42D3×Q811yescharacteristic
288C22×C3#39311yescharacteristic
289C22×C3#44911yescharacteristic
290C22×C3#44911yescharacteristic
291C22×C3#44911yescharacteristic
292C22×C3#44911yescharacteristic
293C22×C3#45311yescharacteristic
294C22×C3#45411yescharacteristic
295C2×D3—82no—
296C2×D3—82no—
297C2×D3—82no—
298C2×D3—161no—
299C2×D3—161no—
300C2×D3—161no—
301C2×D3—161no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—81no—
305C3×C4—81no—
306C3×C4#47811yescharacteristic
307C3×C4#47811yescharacteristic
308C3×C4#47811yescharacteristic
309C3×C4#47811yescharacteristic
310C3×C4#48811yescharacteristic
311C3×C4#48811yescharacteristic
312C3×C4#49011yescharacteristic
313C3×C4#49011yescharacteristic
314Dic3—82no—
315Dic3—81no—
316Dic3—81no—
317C23—122no—
318C23—241no—
319C23—241no—
320C23—241no—
321C23—241no—
322C23—241no—
323C23—241no—
324C23#88211yescharacteristic
325D4—124no—
326D4—122no—
327D4—122no—
328D4—122no—
329D4—122no—
330D4—242no—
331D4—241no—
332D4—241no—
333D4—241no—
334D4—241no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4#87511yescharacteristic
353C2×C4#88511yescharacteristic
354C2×C4#88511yescharacteristic
355C2×C4#89611yescharacteristic
356C2×C4#89611yescharacteristic
357C2×C4#89711yescharacteristic
358C2×C4#90411yescharacteristic
359C2×C4#90411yescharacteristic
360C2×C4#90811yescharacteristic
361C2×C4#90811yescharacteristic
362C2×C4#91111yescharacteristic
363C2×C4#91411yescharacteristic
364C2×C4#91611yescharacteristic
365C2×C4#91911yescharacteristic
366C8—42no—
367C8—81no—
368C8—241no—
369C2×C3#162011yescharacteristic
370C2×C3#162411yescharacteristic
371C2×C3#162811yescharacteristic
372C2×C3#162811yescharacteristic
373C2×C3#202511yescharacteristic
374C2×C3#202511yescharacteristic
375C2×C3#202711yescharacteristic
376D3—162no—
377C22—242no—
378C22—242no—
379C22—242no—
380C22—481no—
381C22—481no—
382C22—481no—
383C22—481no—
384C22#467311yescharacteristic
385C22#473411yescharacteristic
386C22#473411yescharacteristic
387C22#478111yescharacteristic
388C22#481011yescharacteristic
389C22#482211yescharacteristic
390C22#482711yescharacteristic
391C4—21no—
392C4—21no—
393C4—81no—
394C4—81no—
395C4—241no—
396C4—241no—
397C4—242no—
398C4#469711yescharacteristic
399C4#471311yescharacteristic
400C4#476411yescharacteristic
401C4#480011yescharacteristic
402C4#484711yescharacteristic
403C4#484711yescharacteristic
404C4#487111yescharacteristic
405C4#489111yescharacteristic
406C3#1035111yescharacteristic, complemented, Sylow
407C2—482no—
408C2#6819911yescharacteristic
409C2#6820111yescharacteristic
410C2#6824511yescharacteristic
411C2#6825811yescharacteristic
412C2#6867911yescharacteristic
413C2#6876311yescharacteristic
414C2#6889911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenter#1279D3
hypocenterC3#10351
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. #73725C2semidirect
5. C3×#1279C2semidirect
6. #67702C2semidirect
7. #67655C2semidirect
32. #1279D3semidirect
406. C3#10351semidirect

Maximal subgroups

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

subgroupindex
#737252
#704572
#697722
#694872
C3×#12792
#677022
#676552
#103513