Finite Group Databasebeta

← #10697744

Subgroups — #10697744

Order 768.

658 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4851C211yesmaximal, characteristic, complemented
2C2×#4848C211yesmaximal, characteristic
3C2×#4693C211yesmaximal, characteristic
4C2×#4486C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic
6#69788C211yesmaximal, characteristic, complemented
7C3×#1277C211yesmaximal, characteristic, complemented
8#10268—31nomaximal, Sylow
9C22×#204—21no—
10C22×#204—22no—
11C2×#874C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C4×Dic6C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×C3×C4×C8C2211yescharacteristic
17C2×#821—21no—
18C2×#817—21no—
19C2×#816—21no—
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21#4851—22no—
22#4851—22no—
23#4848—22no—
24#4848—22no—
25#4693—22no—
26#4693—22no—
27#4550—22no—
28#4486—22no—
29#4485—22no—
30C2×#492—31no—
31C2×#475—31no—
32C2×#460—31no—
33#1984—31no—
34C2×#332—31no—
35C2×#331—31no—
36#1277D311yescharacteristic, complemented
37C22×D12—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C22×C3×C8D412yes—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3D411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3—21no—
51C22×C3⋊C8—21no—
52C2×#204—28no—
53C2×#204—24no—
54C2×#204—44no—
55C2×#204—42no—
56C2×#204—42no—
57C2×#204—44no—
58#874—21no—
59#874—21no—
60#874—21no—
61#874D412yes—
62#869—21no—
63#869—21no—
64#869—21no—
65#869D412yes—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C4⋊C8D412yes—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76#821—24no—
77#817—24no—
78#816—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C22×SD16—61no—
82C22×SD16—62no—
83C2×C4⋊Q8—31no—
84C2×#127—31no—
85C2×C4×Q8—31no—
86#492—62no—
87#492—62no—
88#475—62no—
89#475—62no—
90#460—62no—
91#460—62no—
92C2×#106—61no—
93C2×C4⋊C8—31no—
94C2×C4⋊C8C2×D311yescharacteristic
95C2×C4⋊C8C2×D311yescharacteristic
96C2×#103—61no—
97C2×#102—61no—
98C2×C4×C8C2×D311yescharacteristic
99#336—62no—
100#332—62no—
101#331—62no—
102C23×D3—41no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—41no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—42no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12—42no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—82no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—22no—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C2×Dic6—41no—
131C2×Dic6SD1612yescomplemented
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—22no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C2×C3×C8—41no—
140C2×C3×C8SD1614yescomplemented
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C4⋊Dic3—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—22no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—42no—
157#204—416no—
158#204—48no—
159C22×Q8—31no—
160C22×Q8—61no—
161C22×D4—61no—
162C22×D4—61no—
163C22×D4—61no—
164C2×SD16—64no—
165C2×SD16—68no—
166C2×SD16—124no—
167C2×SD16—122no—
168C2×SD16—122no—
169C2×SD16—124no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—61no—
173C22×C8D1212yes—
174C4⋊Q8—32no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178#127—32no—
179#127—61no—
180#127—61no—
181#127—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C42—61no—
191C2×C42C22×D311yescharacteristic
192#106—64no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8D1212yes—
198C4⋊C8D1212yes—
199#103—64no—
200#102—64no—
201C4×C8—21no—
202C4×C8—21no—
203C23×C3#12111yescharacteristic
204C22×D3—42no—
205C22×D3—82no—
206C22×D3—81no—
207C22×D3—81no—
208C22×D3—82no—
209C2×C3×C4—22no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12711yescharacteristic
216C2×C3×C4C2×SD1612yes—
217C2×C3×C4C2×SD1611yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4AGL(1,ℤ/8)12yes—
220C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
221C2×C3×C4#13711yescharacteristic
222C2×Dic3—22no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—82no—
229D12—44no—
230D12—42no—
231D12—42no—
232D12—82no—
233D12—82no—
234D12—81no—
235D12—81no—
236D12—81no—
237D12—81no—
238Dic6—22no—
239Dic6—22no—
240Dic6—41no—
241Dic6—41no—
242Dic6—44no—
243Dic6—81no—
244Dic6—81no—
245C3×C8—24no—
246C3×C8—42no—
247C3×C8—42no—
248C3⋊C8—81no—
249C24—121no—
250C2×Q8—32no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—62no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—121no—
258C2×Q8—122no—
259C2×Q8—122no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—122no—
267C2×D4—122no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—242no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—121no—
276C22×C4C2×D1211yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4#23911yescharacteristic
279SD16—1216no—
280SD16—128no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—121no—
290C2×C8—122no—
291C2×C8#20414yescomplemented
292C4⋊C4—61no—
293C4⋊C4—62no—
294C4⋊C4—61no—
295C4⋊C4—62no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C42—121no—
301C42—121no—
302C42C2×D1211yescharacteristic
303C42C2×D1211yescharacteristic
304C42D3×D411yescharacteristic
305C42D3×D411yescharacteristic
306C22×C3#39211yescharacteristic
307C22×C3#46012yes—
308C22×C3#46511yescharacteristic
309C22×C3#46811yescharacteristic
310C22×C3#47112yes—
311C2×D3—82no—
312C2×D3—82no—
313C2×D3—82no—
314C2×D3—162no—
315C2×D3—162no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4#44811yescharacteristic
319C3×C4#44911yescharacteristic
320C3×C4#45012yes—
321C3×C4#45011yescharacteristic
322C3×C4#45211yescharacteristic
323C3×C4#49211yescharacteristic
324C3×C4#49611yescharacteristic
325Dic3—42no—
326Dic3—82no—
327Dic3—81no—
328C23—122no—
329C23—241no—
330C23—241no—
331C23—242no—
332C23—242no—
333C23#89511yescharacteristic
334Q8—62no—
335Q8—62no—
336Q8—121no—
337Q8—121no—
338Q8—124no—
339Q8—241no—
340Q8—241no—
341D4—124no—
342D4—122no—
343D4—122no—
344D4—242no—
345D4—242no—
346D4—241no—
347D4—241no—
348D4—241no—
349D4—241no—
350C2×C4—22no—
351C2×C4—62no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—122no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4—242no—
358C2×C4#87411yescharacteristic
359C2×C4#88211yescharacteristic
360C2×C4#88211yescharacteristic
361C2×C4#88211yescharacteristic
362C2×C4#89511yescharacteristic
363C2×C4#89511yescharacteristic
364C2×C4C2×#20412yes—
365C2×C4#90812yes—
366C2×C4#91111yescharacteristic
367C2×C4D3×SD1611yescharacteristic
368C2×C4#91611yescharacteristic
369C2×C4#91811yescharacteristic
370C8—24no—
371C8—42no—
372C8—42no—
373C8—241no—
374C2×C3#166612yes—
375C2×C3#168612yes—
376C2×C3#200011yescharacteristic
377C2×C3#201411yescharacteristic
378C2×C3#201911yescharacteristic
379D3—162no—
380C22—242no—
381C22—242no—
382C22—242no—
383C22—482no—
384C22—482no—
385C22#467111yescharacteristic
386C22#484812yes—
387C22#486411yescharacteristic
388C22#488611yescharacteristic
389C22#489812yes—
390C4—21no—
391C4—21no—
392C4—122no—
393C4—242no—
394C4—241no—
395C4#469311yescharacteristic
396C4#471211yescharacteristic
397C4#473112yes—
398C4#477611yescharacteristic
399C4#478111yescharacteristic
400C4#480611yescharacteristic
401C4#480811yescharacteristic
402C3#1026811yescharacteristic, complemented, Sylow
403C2—482no—
404C2#6854612yes—
405C2#6859212yes—
406C2#6867611yescharacteristic
407C2#6888711yescharacteristic
408C2#6894311yescharacteristic
subgroupG/H
centerC23#895
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1277C2
socleC23×C3#121
prime radicalC2×#874C22
hypercenter#1277D3
hypocenterC3#10268
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
1. C2×#4851C2semidirect
4. C2×#4486C2semidirect
6. #69788C2semidirect
7. C3×#1277C2semidirect
36. #1277D3semidirect
131. C2×Dic6SD16semidirect
140. C2×C3×C8SD16semidirect
291. C2×C8#204semidirect
402. C3#10268semidirect

Maximal subgroups

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

subgroupindex
C2×#48512
C2×#48482
C2×#46932
C2×#44862
C2×#44852
#697882
C3×#12772
#102683