Finite Group Databasebeta

← #10697986

Subgroups — #10697986

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#487C211yesmaximal, characteristic, complemented
2C2×#5085C211yesmaximal, characteristic
3C2×#5088C211yesmaximal, characteristic
4#71787C211yesmaximal, characteristic, complemented
5#71787C211yesmaximal, characteristic, complemented
6#69772C211yesmaximal, characteristic
7C3⋊#1273C211yesmaximal, characteristic, complemented
8#10310—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C22×#215—22no—
12C2×#874C2211yescharacteristic
13C3×#487—22no—
14C3×#487—22no—
15C2×C3×#103—22no—
16C2×C3×C4×C8C2211yescharacteristic
17C2×#821—22no—
18#5088—22no—
19#5088—22no—
20#5085—22no—
21#5085—22no—
22C2×#809—22no—
23#4935C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4529—22no—
27#4529—22no—
28C2×#492—31no—
29C2×#487D311yescharacteristic, complemented
30C2×#486—31no—
31#1990—31no—
32#1990—31no—
33#1957—31no—
34#1273—31no—
35C22×C3×Q8—21no—
36C22×C3×Q8—21no—
37C22×D12—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×C3×C8D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C3×C4⋊Q8D412yes—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#215—28no—
55C2×#215—44no—
56C2×#215—44no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C22×C3⋊C8D412yes—
61C22×C3⋊C8D412yes—
62#874—21no—
63#874—21no—
64#874—21no—
65#874D412yes—
66C3×#103—28no—
67C3×C4×C8—21no—
68C3×C4×C8—21no—
69#821—44no—
70#809—28no—
71C4×C3⋊C8—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8—21no—
74C4×C3⋊C8—21no—
75C22×SD16—62no—
76C2×C4⋊Q8C2×D311yescharacteristic
77C2×C4⋊Q8C2×D311yescharacteristic
78C2×#127—31no—
79#492—62no—
80#492—62no—
81#487—22no—
82#487—22no—
83#486—62no—
84#486—62no—
85C2×#103—22no—
86C2×#102—62no—
87C2×#102—62no—
88C2×C4×C8—31no—
89C2×C4×C8—31no—
90C2×C4×C8C2×D311yescharacteristic
91#384—31no—
92#340—62no—
93#340—62no—
94C23×D3—41no—
95C2×C3×Q8—22no—
96C2×C3×Q8—22no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C2×C3×Q8—42no—
100C2×C3×Q8—42no—
101C2×C3×Q8—42no—
102C2×C3×Q8—42no—
103C22×C3×C4—41no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—41no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—42no—
113C2×D12—42no—
114C2×D12—42no—
115C2×D12—42no—
116C2×D12—42no—
117C2×D12—42no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#215—416no—
137C4⋊Dic3—42no—
138C4⋊Dic3—42no—
139C4⋊Dic3—42no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8SD1614yes—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×D4—61no—
149C22×D4—61no—
150C22×D4—61no—
151C2×SD16—68no—
152C2×SD16—124no—
153C2×SD16—124no—
154C22×C8—32no—
155C22×C8—32no—
156C22×C8#6712yes—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8#6712yes—
164C4⋊Q8#6712yes—
165#127—32no—
166#127—61no—
167#127—61no—
168#127—61no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C42C22×D311yescharacteristic
177#103—28no—
178#102—68no—
179#102—124no—
180C4×C8—21no—
181C4×C8—21no—
182C4×C8—61no—
183C4×C8—61no—
184C4×C8—61no—
185C4×C8—61no—
186C23×C3#12711yescharacteristic
187C22×D3—42no—
188C22×D3—81no—
189C22×D3—81no—
190C22×D3—81no—
191C22×D3—81no—
192C22×D3—81no—
193C22×D3—81no—
194C3×Q8—44no—
195C3×Q8—44no—
196C3×Q8—82no—
197C3×Q8—82no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—81no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4C2×SD1612yes—
211C2×C3×C4#13512yes—
212C2×C3×C4#13512yes—
213C2×C3×C4AGL(1,ℤ/8)12yes—
214C2×C3×C4AGL(1,ℤ/8)12yes—
215C2×C3×C4#13712yes—
216C2×Dic3—81no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219D12—44no—
220D12—44no—
221D12—44no—
222D12—82no—
223D12—82no—
224D12—82no—
225C3×C8—42no—
226C3⋊C8—24no—
227C3⋊C8—42no—
228C24—121no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×D4—62no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—122no—
241C2×D4—122no—
242C2×D4—122no—
243C2×D4—122no—
244C2×D4—121no—
245C2×D4—121no—
246C2×D4—122no—
247C2×D4—122no—
248C2×D4—121no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—121no—
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4C2×#6711yescharacteristic
255SD16—1216no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—34no—
260C2×C8—62no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—62no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—122no—
274C4⋊C4—122no—
275C4⋊C4—122no—
276C42D3×D411yescharacteristic
277C42#23711yescharacteristic
278C42C2×#6711yescharacteristic
279C42C2×#6711yescharacteristic
280C22×C3#39211yescharacteristic
281C22×C3#49211yescharacteristic
282C22×C3#49511yescharacteristic
283C22×C3#49511yescharacteristic
284C22×C3#49611yescharacteristic
285C22×C3#49611yescharacteristic
286C22×C3#49711yescharacteristic
287C2×D3—82no—
288C2×D3—82no—
289C2×D3—82no—
290C2×D3—161no—
291C2×D3—161no—
292C2×D3—161no—
293C2×D3—161no—
294C3×C4—82no—
295C3×C4—82no—
296C3×C4#44912yes—
297C3×C4#45012yes—
298C3×C4#45212yes—
299C3×C4#46012yes—
300C3×C4#46312yes—
301C3×C4#46412yes—
302Dic3—161no—
303C23—122no—
304C23—241no—
305C23—241no—
306C23—241no—
307C23—241no—
308C23—241no—
309C23—241no—
310C23#94011yescharacteristic
311Q8—44no—
312Q8—44no—
313Q8—82no—
314Q8—82no—
315D4—124no—
316D4—124no—
317D4—124no—
318D4—242no—
319D4—242no—
320D4—242no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4—241no—
330C2×C4D3×SD1612yes—
331C2×C4#91412yes—
332C2×C4#91812yes—
333C2×C4#91912yes—
334C2×C4#93711yescharacteristic
335C2×C4#93711yescharacteristic
336C2×C4#93811yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4#93911yescharacteristic
339C2×C4#94912yes—
340C2×C4#95012yes—
341C2×C4#95311yescharacteristic
342C8—42no—
343C8—64no—
344C8—122no—
345C2×C3#170811yescharacteristic
346C2×C3#171411yescharacteristic
347C2×C3#171711yescharacteristic
348C2×C3#171911yescharacteristic
349C2×C3#201211yescharacteristic
350C2×C3#201411yescharacteristic
351C2×C3#201511yescharacteristic
352D3—162no—
353C22—242no—
354C22—242no—
355C22—242no—
356C22—481no—
357C22—481no—
358C22—481no—
359C22—481no—
360C22#505811yescharacteristic
361C22#505911yescharacteristic
362C22#517511yescharacteristic
363C22#517611yescharacteristic
364C22#519011yescharacteristic
365C22#519111yescharacteristic
366C22#522311yescharacteristic
367C4—82no—
368C4—82no—
369C4—481no—
370C4#505712yes—
371C4#516312yes—
372C4#516912yes—
373C4#518512yes—
374C4#518712yes—
375C4#523812yes—
376C3#1031011yescharacteristic, complemented, Sylow
377C2—482no—
378C2#7121311yescharacteristic
379C2#7121611yescharacteristic
380C2#7121811yescharacteristic
381C2#7122411yescharacteristic
382C2#7172811yescharacteristic
383C2#7205611yescharacteristic
384C2#7207211yescharacteristic
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC23×C3#127
prime radicalC2×#874C22
hypercenterC2×#487D3
hypocenterC3#10310
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
1. C2×C3×#487C2semidirect
4. #71787C2semidirect
5. #71787C2semidirect
7. C3⋊#1273C2semidirect
29. C2×#487D3semidirect
376. C3#10310semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4872
C2×#50852
C2×#50882
#717872
#697722
C3⋊#12732
#103103