Finite Group Databasebeta

← #10697747

Subgroups — #10697747

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5091C211yesmaximal, characteristic, complemented
2C2×#5088C211yesmaximal, characteristic
3C2×#4848C211yesmaximal, characteristic
4C2×C3×#333C211yesmaximal, characteristic, complemented
5#70472C211yesmaximal, characteristic, complemented
6C2×#4485C211yesmaximal, characteristic
7#67703C211yesmaximal, characteristic, complemented
8#10269—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#204—21no—
11C2×#874C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#821—21no—
16#5091—22no—
17#5091—22no—
18#5088—22no—
19#5088—22no—
20C2×#810—21no—
21C2×#809—22no—
22C2×#807—21no—
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4848—22no—
26#4848—22no—
27C3×#333—22no—
28#4550—22no—
29#4485—22no—
30C2×#486—31no—
31C2×#477—31no—
32C2×#460—31no—
33#1984—31no—
34C2×#333D311yescharacteristic, complemented
35C2×#331—31no—
36#1277—31no—
37C22×C3×Q8—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—21no—
54C22×C3⋊C8—21no—
55C22×C3⋊C8D412yes—
56C2×#204—24no—
57C2×#204—42no—
58C2×#204—42no—
59#874—21no—
60#874—21no—
61#874—21no—
62#874D412yes—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71#821—24no—
72#810—24no—
73#809—28no—
74#807—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C22×SD16—61no—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×#127—31no—
82C2×C4×Q8—31no—
83#486—62no—
84#486—62no—
85#477—62no—
86#477—62no—
87#460—62no—
88#460—62no—
89C2×#107—61no—
90C2×C4⋊C8—31no—
91C2×C4⋊C8C2×D311yescharacteristic
92C2×C4⋊C8C2×D311yescharacteristic
93C2×#103—61no—
94C2×#102—62no—
95C2×#102—61no—
96C2×C4×C8—31no—
97#336—62no—
98#333—22no—
99#331—62no—
100C23×D3—41no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—82no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—21no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—22no—
113C2×D12—41no—
114C2×D12—42no—
115C2×D12—41no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—42no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—82no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6SD1612yes—
129C2×C3×C8—22no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—42no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147C4×Dic3—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154#204—48no—
155C22×Q8—21no—
156C22×Q8—31no—
157C22×D4—61no—
158C22×D4—61no—
159C22×D4—61no—
160C2×SD16—64no—
161C2×SD16—122no—
162C2×SD16—122no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—32no—
166C22×C8—61no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8D1212yes—
171#127—32no—
172#127—61no—
173#127—61no—
174#127—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×Q8—61no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—61no—
183C2×C42—61no—
184C2×C42C22×D311yescharacteristic
185#107—64no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—32no—
190C4⋊C8—61no—
191C4⋊C8#6712yes—
192#103—64no—
193#102—68no—
194#102—64no—
195C4×C8—61no—
196C4×C8—61no—
197C23×C3#12111yescharacteristic
198C22×D3—42no—
199C22×D3—82no—
200C22×D3—81no—
201C22×D3—81no—
202C22×D3—82no—
203C3×Q8—82no—
204C3×Q8—82no—
205C2×C3×C4—22no—
206C2×C3×C4—42no—
207C2×C3×C4—82no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12711yescharacteristic
214C2×C3×C4C2×SD1611yescharacteristic
215C2×C3×C4#13512yes—
216C2×C3×C4#13511yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218C2×C3×C4#13711yescharacteristic
219C2×C3×C4#13712yes—
220C2×Dic3—22no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—42no—
225D12—42no—
226D12—42no—
227D12—44no—
228D12—82no—
229D12—82no—
230D12—81no—
231D12—81no—
232D12—81no—
233D12—81no—
234Dic6—22no—
235Dic6—22no—
236Dic6—41no—
237Dic6—41no—
238C3×C8—42no—
239C3×C8—81no—
240C3⋊C8—42no—
241C3⋊C8—42no—
242C24—121no—
243C2×Q8—22no—
244C2×Q8—32no—
245C2×Q8—41no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—82no—
252C2×D4—62no—
253C2×D4—62no—
254C2×D4—62no—
255C2×D4—121no—
256C2×D4—121no—
257C2×D4—121no—
258C2×D4—122no—
259C2×D4—121no—
260C2×D4—122no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—242no—
265C22×C4—41no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4#23511yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271SD16—128no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—62no—
278C2×C8—62no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8—121no—
282C2×C8—121no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C42—121no—
292C42—121no—
293C42C2×D1211yescharacteristic
294C42D3×D411yescharacteristic
295C42D3×D411yescharacteristic
296C42C2×#6711yescharacteristic
297C22×C3#39211yescharacteristic
298C22×C3#46412yes—
299C22×C3#46511yescharacteristic
300C22×C3#46811yescharacteristic
301C22×C3#47212yes—
302C2×D3—82no—
303C2×D3—82no—
304C2×D3—82no—
305C2×D3—162no—
306C2×D3—162no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—82no—
310C3×C4#44811yescharacteristic
311C3×C4#44911yescharacteristic
312C3×C4#45011yescharacteristic
313C3×C4#45211yescharacteristic
314C3×C4#45212yes—
315C3×C4#49511yescharacteristic
316C3×C4#49711yescharacteristic
317Dic3—42no—
318Dic3—81no—
319C23—122no—
320C23—242no—
321C23—242no—
322C23—241no—
323C23—241no—
324C23#88411yescharacteristic
325Q8—62no—
326Q8—62no—
327Q8—82no—
328Q8—82no—
329Q8—121no—
330Q8—121no—
331D4—122no—
332D4—122no—
333D4—124no—
334D4—242no—
335D4—242no—
336D4—241no—
337D4—241no—
338D4—241no—
339D4—241no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—62no—
343C2×C4—82no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4#88211yescharacteristic
349C2×C4#88211yescharacteristic
350C2×C4#88411yescharacteristic
351C2×C4D3×SD1611yescharacteristic
352C2×C4#91411yescharacteristic
353C2×C4#91612yes—
354C2×C4#91811yescharacteristic
355C2×C4#91812yes—
356C2×C4#91911yescharacteristic
357C2×C4#93011yescharacteristic
358C2×C4#93711yescharacteristic
359C2×C4#94011yescharacteristic
360C8—42no—
361C8—81no—
362C8—122no—
363C8—122no—
364C2×C3#167812yes—
365C2×C3#169812yes—
366C2×C3#200011yescharacteristic
367C2×C3#201511yescharacteristic
368C2×C3#201811yescharacteristic
369D3—162no—
370C22—242no—
371C22—242no—
372C22—242no—
373C22—482no—
374C22—482no—
375C22#474912yes—
376C22#479812yes—
377C22#480711yescharacteristic
378C22#481211yescharacteristic
379C22#496011yescharacteristic
380C4—21no—
381C4—21no—
382C4—82no—
383C4—122no—
384C4—241no—
385C4#480611yescharacteristic
386C4#480812yes—
387C4#481011yescharacteristic
388C4#516811yescharacteristic
389C4#516911yescharacteristic
390C4#518711yescharacteristic
391C4#519011yescharacteristic
392C3#1026911yescharacteristic, complemented, Sylow
393C2—482no—
394C2#6851012yes—
395C2#6851412yes—
396C2#7009011yescharacteristic
397C2#7027211yescharacteristic
398C2#7027811yescharacteristic
subgroupG/H
centerC23#884
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#333C2
socleC23×C3#121
prime radicalC2×#874C22
hypercenterC2×#333D3
hypocenterC3#10269
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
1. C2×#5091C2semidirect
4. C2×C3×#333C2semidirect
5. #70472C2semidirect
7. #67703C2semidirect
34. C2×#333D3semidirect
392. C3#10269semidirect

Maximal subgroups

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

subgroupindex
C2×#50912
C2×#50882
C2×#48482
C2×C3×#3332
#704722
C2×#44852
#677032
#102693