Finite Group Databasebeta

← #10945110

Subgroups — #10945110

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83252C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic
3C2×#4842C211yesmaximal, characteristic
4C4×#893C211yesmaximal, characteristic
5C3×#1981C211yesmaximal, characteristic, complemented
6#70443C211yesmaximal, characteristic, complemented
7#70276C214yesmaximal
8#70223C214yesmaximal, complemented
9#69489C211yesmaximal, characteristic, complemented
10#33026—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C22×#215C2212yes—
13C2×#893C2212yes—
14C2×C4×D12C2211yescharacteristic
15C2×C42×D3—21no—
16C2×#869C2211yescharacteristic
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#103C2214yes—
19C3×C4×C4⋊C4C2211yescharacteristic
20#5229C2214yes—
21C2×#835C2212yes—
22C2×#821C2212yes—
23C2×#817C2212yes—
24#5083—24no—
25#4977—21no—
26#4977—21no—
27Dic3×C4⋊C4C2212yes—
28#4973—22no—
29C4×(C2×D3)⋊C4—21no—
30C42×Dic3C2211yescharacteristic
31C2×C4⋊(C3⋊C8)C2211yescharacteristic
32#4842—24no—
33#4666—22no—
34#4550C2214yes—
35#4529C2214yes—
36C2×#476—31no—
37C2×#461—31no—
38#2862—31no—
39C42×D4—31no—
40#1987—31no—
41#1981D311yescharacteristic, complemented
42#1927—34no—
43#1877—34no—
44#1831—31no—
45C22×C3×Q8C2312yes—
46C22×D12C2311yescharacteristic
47C22×C4×D3—21no—
48C22×C4×D3—22no—
49C22×Dic6—21no—
50C22×C3×C8C2312yes—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—22no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4C2312yes—
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#215—24no—
59C2×#215—28no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3C2312yes—
64C2×C4⋊Dic3C2311yescharacteristic
65C2×C4×Dic3—22no—
66C2×C4×Dic3D411yescharacteristic
67C2×C4×Dic3D412yes—
68C2×C4×Dic3D411yescharacteristic
69C2×C4×Dic3C2312yes—
70C22×C3⋊C8C2312yes—
71#893—24no—
72#893—24no—
73C4×D12—22no—
74C4×D12D414yes—
75C42×D3—44no—
76#869—22no—
77#869—22no—
78C3×C4⋊C8—22no—
79C3×#103—28no—
80C3×#95—21no—
81#835—24no—
82C3⋊#95—22no—
83C3⋊#95—24no—
84C3⋊#95—24no—
85#821—24no—
86#817—24no—
87C4⋊(C3⋊C8)—22no—
88C22×SD16—32no—
89C2×C4⋊Q8—31no—
90C2×C4⋊Q8C2×D311yescharacteristic
91C2×C4×D4—32no—
92C2×C4×D4—31no—
93C22×C42—61no—
94#476—64no—
95#461—64no—
96C2×#106—32no—
97C2×C4⋊C8—31no—
98C2×C4⋊C8C2×D311yescharacteristic
99C2×#103—32no—
100C2×#103C2×D314yes—
101C2×#102—32no—
102#391—34no—
103#391—62no—
104#382—61no—
105#382—61no—
106C4×C4⋊C4—32no—
107C4×C4⋊C4C2×D311yescharacteristic
108C4×C22⋊C4—62no—
109C4×C22⋊C4—61no—
110C43—31no—
111#340—34no—
112#336—34no—
113C23×D3—21no—
114C2×C3×Q8—22no—
115C2×C3×Q8—24no—
116C2×C3×Q8—24no—
117C2×C3×Q8—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2411yescharacteristic
123C22×Dic3—21no—
124C22×Dic3—22no—
125C22×Dic3C2×D412yes—
126C22×Dic3C2×D412yes—
127C2×D12—21no—
128C2×D12—22no—
129C2×D12C2×D412yes—
130C2×D12#4114yes—
131C2×C4×D3—28no—
132C2×C4×D3—24no—
133C2×C4×D3—44no—
134C2×C4×D3—42no—
135C2×C4×D3—42no—
136C2×C4×D3—44no—
137C2×Dic6—44no—
138C2×Dic6—42no—
139C2×C3×C8—22no—
140C2×C3×C8—24no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—42no—
143C3×C4⋊C4—42no—
144C3×C4⋊C4#4114yes—
145C3×C42—42no—
146C3×C42C2×D412yes—
147C3×C42#4112yes—
148#215—48no—
149(C2×D3)⋊C4—28no—
150(C2×D3)⋊C4—24no—
151C4⋊Dic3—21no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—42no—
155C4⋊Dic3C2×D412yes—
156C4×Dic3—22no—
157C4×Dic3—22no—
158C4×Dic3—22no—
159C4×Dic3—42no—
160C4×Dic3—44no—
161C4×Dic3SD1614yescomplemented
162C2×C3⋊C8—22no—
163C2×C3⋊C8—24no—
164C22×Q8—61no—
165C22×Q8C22×D312yes—
166C22×D4—31no—
167C23×C4—62no—
168C23×C4—61no—
169C2×SD16—68no—
170C2×SD16—64no—
171C22×C8—32no—
172C22×C8C22×D312yes—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—62no—
176C4⋊Q8—62no—
177C4×D4—34no—
178C4×D4—64no—
179C4×D4—64no—
180C4×D4—62no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4C22×D312yes—
187C2×C22⋊C4—62no—
188C2×C22⋊C4—61no—
189C2×C42—21no—
190C2×C42—31no—
191C2×C42—31no—
192C2×C42—32no—
193C2×C42—32no—
194C2×C42—62no—
195C2×C42—124no—
196C2×C42C22×D311yescharacteristic
197#106—64no—
198C4⋊C8—22no—
199C4⋊C8—62no—
200#103—28no—
201#103—64no—
202#102—64no—
203#95—21no—
204#95—62no—
205#95—64no—
206#95—64no—
207C23×C3C22×D411yescharacteristic
208C22×D3—22no—
209C22×D3—24no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—42no—
213C3×Q8—42no—
214C3×Q8—44no—
215C3×Q8—42no—
216C2×C3×C4—22no—
217C2×C3×C4—24no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—44no—
222C2×C3×C4#12112yes—
223C2×C3×C4#12112yes—
224C2×C3×C4C2×SD1612yes—
225C2×C3×C4#13712yes—
226C2×C3×C4C22×D411yescharacteristic
227C2×C3×C4C2×#4111yescharacteristic
228C2×C3×C4C2×#4112yes—
229C2×C3×C4C2×#4112yes—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239C2×Dic3—44no—
240C2×Dic3C2×SD1614yes—
241D12—24no—
242D12—22no—
243D12#12414yes—
244C4×D3—416no—
245C4×D3—48no—
246Dic6—84no—
247C3×C8—42no—
248C3⋊C8—42no—
249C24—61no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—124no—
255C2×Q8—122no—
256C2×D4—32no—
257C2×D4—34no—
258C2×D4—61no—
259C2×D4—62no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—32no—
263C22×C4—32no—
264C22×C4—68no—
265C22×C4—61no—
266C22×C4—64no—
267C22×C4—62no—
268C22×C4—124no—
269C22×C4—124no—
270C22×C4—122no—
271C22×C4—122no—
272C22×C4D3×D411yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4C23×D311yescharacteristic
275SD16—128no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—62no—
279C2×C8—64no—
280C4⋊C4—22no—
281C4⋊C4—32no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—61no—
285C4⋊C4—62no—
286C4⋊C4—62no—
287C4⋊C4—122no—
288C4⋊C4#23714yes—
289C22⋊C4—68no—
290C22⋊C4—64no—
291C42—34no—
292C42—42no—
293C42—62no—
294C42—62no—
295C42—62no—
296C42—122no—
297C42—124no—
298C42#23512yes—
299C42D3×D412yes—
300C22×C3C2×#12111yescharacteristic
301C22×C3#52512yes—
302C22×C3C22×SD1611yescharacteristic
303C22×C3C2×#13711yescharacteristic
304C22×C3#57512yes—
305C2×D3—24no—
306C2×D3—44no—
307C2×D3—44no—
308C2×D3—42no—
309C2×D3—42no—
310C3×C4—22no—
311C3×C4—44no—
312C3×C4—82no—
313C3×C4#46114yes—
314C3×C4C2×#12411yescharacteristic
315C3×C4#52912yes—
316C3×C4#53211yescharacteristic
317Dic3—24no—
318Dic3—42no—
319Dic3—42no—
320Dic3—82no—
321C23—62no—
322C23—64no—
323C23—121no—
324C23—121no—
325C23—122no—
326C23C2×D3×D411yescharacteristic
327Q8—42no—
328Q8—42no—
329Q8—44no—
330Q8—244no—
331D4—34no—
332D4—62no—
333D4—64no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—34no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—62no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—1216no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—128no—
351C2×C4—122no—
352C2×C4—124no—
353C2×C4#88412yes—
354C2×C4#88412yes—
355C2×C4D3×SD1612yes—
356C2×C4#91812yes—
357C2×C4C2×#23511yescharacteristic
358C2×C4C2×D3×D411yescharacteristic
359C2×C4C2×#23712yes—
360C2×C4D3×#4112yes—
361C8—42no—
362C8—122no—
363C2×C3#285011yescharacteristic
364C2×C3C2×#46111yescharacteristic
365C2×C3#304011yescharacteristic
366C2×C3#310112yes—
367C2×C3#311612yes—
368D3—44no—
369C22—64no—
370C22—124no—
371C22—124no—
372C22—122no—
373C22—122no—
374C22C2×#88411yescharacteristic
375C22#560612yes—
376C22C2×D3×SD1611yescharacteristic
377C22C2×#91811yescharacteristic
378C22#577712yes—
379C4—22no—
380C4—44no—
381C4—64no—
382C4—82no—
383C4—122no—
384C4—122no—
385C4—242no—
386C4#481814yes—
387C4#555712yes—
388C4D3×#12411yescharacteristic
389C4#571611yescharacteristic
390C3#3302611yescharacteristic, complemented, Sylow
391C2—124no—
392C2#7647411yescharacteristic
393C2C2×#481811yescharacteristic
394C2#7897511yescharacteristic
395C2#8040812yes—
396C2#8045912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1981C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1981D3
hypocenterC3#33026
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
1. #83252C2semidirect
5. C3×#1981C2semidirect
6. #70443C2semidirect
8. #70223C2semidirect
9. #69489C2semidirect
41. #1981D3semidirect
161. C4×Dic3SD16semidirect
390. C3#33026semidirect

Maximal subgroups

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

subgroupindex
#832522
C2×#50832
C2×#48422
C4×#8932
C3×#19812
#704432
#702762
#702232
#694892
#330263