Finite Group Databasebeta

← #10697993

Subgroups — #10697993

Order 768.

622 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#487C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3C2×#5088C211yesmaximal, characteristic
4#71787C211yesmaximal, characteristic, complemented
5#71785C211yesmaximal, characteristic, complemented
6#69772C211yesmaximal, characteristic
7C3⋊#1273C211yesmaximal, characteristic, complemented
8#10311—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#874C2211yescharacteristic
12C3×#487—22no—
13C3×#487—22no—
14C2×C3×#103—22no—
15C2×C3×C4×C8C2211yescharacteristic
16C2×#835—22no—
17C2×#821—22no—
18#5088—22no—
19#5088—22no—
20#5088—22no—
21#5088—22no—
22C2×#809—22no—
23C2×#809—22no—
24#4935C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4529—22no—
28C2×#487D311yescharacteristic, complemented
29C2×#486—31no—
30C2×#486—31no—
31#1990—31no—
32#1957—31no—
33#1956—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×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C22×C3⋊C8D412yes—
58C22×C3⋊C8D412yes—
59#874—21no—
60#874—21no—
61#874—21no—
62#874D412yes—
63C3×#103—28no—
64C3×C4×C8—21no—
65C3×C4×C8—21no—
66#835—44no—
67#821—44no—
68#809—28no—
69#809—28no—
70C4×C3⋊C8—21no—
71C4×C3⋊C8—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8—21no—
74C2×C4⋊Q8C2×D311yescharacteristic
75C2×C4⋊Q8C2×D311yescharacteristic
76C2×#127—31no—
77#487—22no—
78#487—22no—
79#486—62no—
80#486—62no—
81#486—62no—
82#486—62no—
83C2×#103—22no—
84C2×#103—62no—
85C2×#102—62no—
86C2×#102—62no—
87C2×#102—62no—
88C2×C4×C8—31no—
89C2×C4×C8—31no—
90C2×C4×C8C2×D311yescharacteristic
91#384—31no—
92#340—62no—
93C23×D3—41no—
94C2×C3×Q8—22no—
95C2×C3×Q8—22no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8—42no—
99C2×C3×Q8—42no—
100C2×C3×Q8—42no—
101C2×C3×Q8—42no—
102C22×C3×C4—41no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—41no—
108C2×D12—22no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—42no—
112C2×D12—42no—
113C2×D12—42no—
114C2×D12—42no—
115C2×D12—42no—
116C2×D12—42no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×C3×C8—22no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135C4⋊Dic3—42no—
136C4⋊Dic3—42no—
137C4⋊Dic3—42no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×D4—61no—
147C22×D4—61no—
148C22×D4—61no—
149C22×C8—32no—
150C22×C8—32no—
151C22×C8#6712yes—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8#6712yes—
159C4⋊Q8#6712yes—
160#127—32no—
161#127—61no—
162#127—61no—
163#127—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C42C22×D311yescharacteristic
172#103—28no—
173#103—124no—
174#102—68no—
175#102—68no—
176#102—124no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8—61no—
180C4×C8—61no—
181C4×C8—61no—
182C4×C8—61no—
183C23×C3#12711yescharacteristic
184C22×D3—42no—
185C22×D3—81no—
186C22×D3—81no—
187C22×D3—81no—
188C22×D3—81no—
189C22×D3—81no—
190C22×D3—81no—
191C3×Q8—44no—
192C3×Q8—44no—
193C3×Q8—82no—
194C3×Q8—82no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#13512yes—
208C2×C3×C4#13512yes—
209C2×C3×C4#13512yes—
210C2×C3×C4AGL(1,ℤ/8)12yes—
211C2×C3×C4AGL(1,ℤ/8)12yes—
212C2×C3×C4#13712yes—
213C2×Dic3—81no—
214C2×Dic3—81no—
215C2×Dic3—81no—
216D12—44no—
217D12—44no—
218D12—44no—
219D12—82no—
220D12—82no—
221D12—82no—
222C3×C8—42no—
223C3⋊C8—42no—
224C3⋊C8—42no—
225C24—121no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×D4—62no—
235C2×D4—62no—
236C2×D4—62no—
237C2×D4—122no—
238C2×D4—122no—
239C2×D4—122no—
240C2×D4—122no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—122no—
244C2×D4—122no—
245C2×D4—121no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—121no—
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251C22×C4C2×#6711yescharacteristic
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—62no—
256C2×C8—62no—
257C2×C8—62no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—62no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—122no—
270C4⋊C4—122no—
271C4⋊C4—122no—
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C42C2×#6711yescharacteristic
275C42C2×#6711yescharacteristic
276C22×C3#39211yescharacteristic
277C22×C3#49511yescharacteristic
278C22×C3#49511yescharacteristic
279C22×C3#49511yescharacteristic
280C22×C3#49611yescharacteristic
281C22×C3#49611yescharacteristic
282C22×C3#49711yescharacteristic
283C2×D3—82no—
284C2×D3—82no—
285C2×D3—82no—
286C2×D3—161no—
287C2×D3—161no—
288C2×D3—161no—
289C2×D3—161no—
290C3×C4—82no—
291C3×C4—82no—
292C3×C4#44912yes—
293C3×C4#44912yes—
294C3×C4#45212yes—
295C3×C4#46312yes—
296C3×C4#46312yes—
297C3×C4#46412yes—
298Dic3—161no—
299C23—122no—
300C23—241no—
301C23—241no—
302C23—241no—
303C23—241no—
304C23—241no—
305C23—241no—
306C23#94011yescharacteristic
307Q8—44no—
308Q8—44no—
309Q8—82no—
310Q8—82no—
311D4—124no—
312D4—124no—
313D4—124no—
314D4—242no—
315D4—242no—
316D4—242no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—241no—
324C2×C4—241no—
325C2×C4—241no—
326C2×C4#91412yes—
327C2×C4#91612yes—
328C2×C4#91812yes—
329C2×C4#91912yes—
330C2×C4#93711yescharacteristic
331C2×C4#93711yescharacteristic
332C2×C4#93811yescharacteristic
333C2×C4#93911yescharacteristic
334C2×C4#93911yescharacteristic
335C2×C4#94912yes—
336C2×C4#95012yes—
337C2×C4#95311yescharacteristic
338C8—42no—
339C8—122no—
340C8—122no—
341C2×C3#171111yescharacteristic
342C2×C3#171711yescharacteristic
343C2×C3#171911yescharacteristic
344C2×C3#172011yescharacteristic
345C2×C3#201211yescharacteristic
346C2×C3#201211yescharacteristic
347C2×C3#201511yescharacteristic
348D3—162no—
349C22—242no—
350C22—242no—
351C22—242no—
352C22—481no—
353C22—481no—
354C22—481no—
355C22—481no—
356C22#505811yescharacteristic
357C22#505911yescharacteristic
358C22#516811yescharacteristic
359C22#517611yescharacteristic
360C22#519011yescharacteristic
361C22#519111yescharacteristic
362C22#522311yescharacteristic
363C4—82no—
364C4—82no—
365C4—481no—
366C4#505712yes—
367C4#516512yes—
368C4#517212yes—
369C4#518512yes—
370C4#518712yes—
371C4#523812yes—
372C3#1031111yescharacteristic, complemented, Sylow
373C2—482no—
374C2#7121511yescharacteristic
375C2#7121811yescharacteristic
376C2#7122311yescharacteristic
377C2#7122511yescharacteristic
378C2#7172811yescharacteristic
379C2#7205511yescharacteristic
380C2#7207211yescharacteristic
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC23×C3#127
prime radicalC2×#874C22
hypercenterC2×#487D3
hypocenterC3#10311
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. #71785C2semidirect
7. C3⋊#1273C2semidirect
28. C2×#487D3semidirect
372. C3#10311semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4872
C2×#50882
#717872
#717852
#697722
C3⋊#12732
#103113