Finite Group Databasebeta

← #10823306

Subgroups — #10823306

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5500C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic
3#82207C212yesmaximal
4#71578C212yesmaximal
5C2×#4530C211yesmaximal, characteristic
6#69573C214yesmaximal, complemented
7#69543C214yesmaximal, complemented
8#23561—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×#926C2212yes—
13C2×#924—22no—
14C22×#208C2212yes—
15#5500—22no—
16#5500—24no—
17#5500C2214yes—
18C2×#890—22no—
19C2×#888—22no—
20C3×#389—22no—
21C3×#385—22no—
22C3×#384C2212yes—
23C3×#382C2212yes—
24#4998—24no—
25#4995—22no—
26#4995—22no—
27#4964C2214yes—
28#4964C2214yes—
29#4530C2214yes—
30#4523C2218yes—
31C2×#544—31no—
32#2911—32no—
33#2317D311yescharacteristic
34#2061—34no—
35#2060—34no—
36C2×#343—31no—
37#1733—32no—
38C23×C3×C4D412yes—
39C23×C3×C4C2311yescharacteristic
40C22×Dic6—22no—
41C3×#117—22no—
42C3×#117—22no—
43C3×#117—22no—
44C3×#117Q814yescomplemented
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—24no—
49C2×C3×C4⋊C4D412yes—
50C2×C3×C4⋊C4C2311yescharacteristic
51C2×C3×C4⋊C4C2311yescharacteristic
52C2×C3×C4⋊C4C2312yes—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C42—21no—
56C2×C3×C42C2312yes—
57C2×#217—22no—
58#926—22no—
59#926—22no—
60#926D414yes—
61#926C2314yes—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3D412yes—
64C2×C4⋊Dic3D412yes—
65#924—28no—
66#924—28no—
67C2×Dic3⋊C4—24no—
68C2×C4×Dic3—22no—
69C2×#208—22no—
70C2×#208—24no—
71C2×#208Q818yescomplemented
72C2×#208C2314yes—
73C22×C3⋊C8—22no—
74#890—28no—
75#890—28no—
76#888—28no—
77#888—28no—
78C3×#95—24no—
79#808—24no—
80#808—24no—
81#807—24no—
82#807—24no—
83C22×M16—32no—
84#544—34no—
85#544—64no—
86#544—62no—
87C2×C4⋊Q8—62no—
88C2×#125—62no—
89C2×C22⋊Q8—62no—
90C2×#117—32no—
91C2×#117C2×D311yescharacteristic
92C2×#117C2×D311yescharacteristic
93C22×C4⋊C4C2×D311yescharacteristic
94#428—34no—
95#428—34no—
96#428—62no—
97#428—62no—
98#427—64no—
99#389—22no—
100#385—22no—
101#384C2×D312yes—
102#382C2×D312yes—
103#343—34no—
104#339—38no—
105C24×C3C2×D411yescharacteristic
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—82no—
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D414yes—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×Q812yes—
117C22×C3×C4C2×Q814yes—
118C22×C3×C4#4112yes—
119C22×C3×C4C2411yescharacteristic
120C22×Dic3—22no—
121C22×Dic3—22no—
122C2×Dic6—28no—
123C2×Dic6—84no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—84no—
130C3×C22⋊C4—24no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42—22no—
134C3×C42—42no—
135C3×C42C2×Q814yescomplemented
136#217—24no—
137#217—24no—
138C4⋊Dic3—28no—
139C4⋊Dic3—22no—
140C4⋊Dic3—22no—
141C4⋊Dic3C2×D414yes—
142C4⋊Dic3C2×D414yes—
143Dic3⋊C4—216no—
144C4×Dic3—24no—
145C4×Dic3—24no—
146#208—28no—
147#208—24no—
148#208C2×Q818yescomplemented
149C2×C3⋊C8—28no—
150C2×C3⋊C8—24no—
151C22×Q8—62no—
152C23×C4#6712yes—
153C23×C4C22×D311yescharacteristic
154C2×M16—38no—
155C2×M16—34no—
156C2×M16—64no—
157C2×M16—62no—
158C22×C8—62no—
159C4⋊Q8—68no—
160C4⋊Q8—68no—
161#125—68no—
162#125—68no—
163C22⋊Q8—68no—
164C22⋊Q8—68no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—34no—
169#117—34no—
170#117—62no—
171#117—62no—
172#117Dic614yescomplemented
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—64no—
181C2×C4⋊C4#6712yes—
182C2×C4⋊C4C22×D311yescharacteristic
183C2×C4⋊C4C22×D311yescharacteristic
184C2×C4⋊C4C22×D312yes—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—62no—
188C2×C42—21no—
189C2×C42—62no—
190C2×C42C22×D312yes—
191#107—64no—
192#107—64no—
193#106—64no—
194#106—64no—
195#95—24no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3#12011yescharacteristic
199C23×C3#12011yescharacteristic
200C23×C3C22⋊Q812yes—
201C23×C3C4⋊Q812yes—
202C23×C3C22×D411yescharacteristic
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—42no—
206C2×C3×C4—44no—
207C2×C3×C4—42no—
208C2×C3×C4—44no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—82no—
214C2×C3×C4#12012yes—
215C2×C3×C4C22⋊Q814yes—
216C2×C3×C4C22⋊Q814yes—
217C2×C3×C4C22⋊Q812yes—
218C2×C3×C4C4⋊Q814yes—
219C2×C3×C4C4⋊Q814yes—
220C2×C3×C4C4⋊Q812yes—
221C2×C3×C4C22×D412yes—
222C2×C3×C4C22×Q811yescharacteristic
223C2×C3×C4C22×Q812yes—
224C2×C3×C4C2×#4111yescharacteristic
225C2×Dic3—24no—
226C2×Dic3—24no—
227C2×Dic3—44no—
228C2×Dic3—44no—
229Dic6—88no—
230C3⋊C8—28no—
231C24C2×#6711yescharacteristic
232C2×Q8—68no—
233C2×Q8—244no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4—82no—
242C22×C4C2×Dic612yes—
243C22×C4#23512yes—
244C22×C4D3×D412yes—
245C22×C4D3×D412yes—
246C22×C4D3×Q814yes—
247C22×C4C2×#6714yes—
248C22×C4C2×#6711yescharacteristic
249C22×C4C23×D311yescharacteristic
250M16—38no—
251M16—68no—
252M16—64no—
253C2×C8—68no—
254C2×C8—64no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—34no—
261C4⋊C4—34no—
262C4⋊C4—68no—
263C4⋊C4—616no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—84no—
267C22⋊C4—24no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—64no—
271C22⋊C4—64no—
272C42—22no—
273C42—42no—
274C42—64no—
275C42—64no—
276C42C2×Dic614yescomplemented
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3#39512yes—
286C22×C3#39512yes—
287C22×C3C2×#12011yescharacteristic
288C22×C3C2×C22⋊Q811yescharacteristic
289C22×C3C2×C4⋊Q811yescharacteristic
290C3×C4—44no—
291C3×C4—44no—
292C3×C4—81no—
293C3×C4—81no—
294C3×C4#39514yes—
295C3×C4C2×C22⋊Q812yes—
296C3×C4C2×C4⋊Q812yes—
297Dic3—44no—
298Dic3—44no—
299C23—41no—
300C23—41no—
301C23#92412yes—
302C23#93711yescharacteristic
303C23#93711yescharacteristic
304C23#94412yes—
305C23C22×#6711yescharacteristic
306Q8—248no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—64no—
318C2×C4—64no—
319C2×C4—82no—
320C2×C4—124no—
321C2×C4—124no—
322C2×C4#88814yes—
323C2×C4#88814yes—
324C2×C4#89614yes—
325C2×C4#89614yes—
326C2×C4#92412yes—
327C2×C4#93712yes—
328C2×C4#94412yes—
329C2×C4C22×Dic611yescharacteristic
330C2×C4C2×#23511yescharacteristic
331C2×C4C2×D3×D412yes—
332C2×C4C2×D3×Q812yes—
333C8—68no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—22no—
337C2×C3—22no—
338C2×C3C2×#39511yescharacteristic
339C2×C3#301911yescharacteristic
340C2×C3#302011yescharacteristic
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—42no—
348C22—42no—
349C22#497912yes—
350C22#497912yes—
351C22C2×#92411yescharacteristic
352C22C2×#93711yescharacteristic
353C22C2×#94411yescharacteristic
354C4—44no—
355C4—44no—
356C4—81no—
357C4—81no—
358C4—124no—
359C4—124no—
360C4#497914yes—
361C4C2×#88812yes—
362C4C2×#89612yes—
363C3#2356111yescharacteristic, complemented, Sylow
364C2—21no—
365C2—22no—
366C2—22no—
367C2—21no—
368C2C2×#497911yescharacteristic
369C2#8339411yescharacteristic
370C2#8341511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2317C2
socleC22×C3C2×#120
prime radicalC24×C3C2×D4
hypercenter#2317D3
hypocenterC3#23561
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
6. #69573C2semidirect
7. #69543C2semidirect
44. C3×#117Q8semidirect
71. C2×#208Q8semidirect
135. C3×C42C2×Q8semidirect
148. #208C2×Q8semidirect
172. #117Dic6semidirect
276. C42C2×Dic6semidirect
363. C3#23561semidirect

Maximal subgroups

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

subgroupindex
C2×#55002
C3×#23172
#822072
#715782
C2×#45302
#695732
#695432
#235613