Finite Group Databasebeta

← #10823313

Subgroups — #10823313

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic
3C2×#4964C212yesmaximal
4#71587C211yesmaximal, characteristic
5#71578C212yesmaximal
6#69573C214yesmaximal, complemented
7#69543C214yesmaximal, complemented
8#23562—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×#926C2212yes—
13C22×#208C2212yes—
14#5519—22no—
15#5519—22no—
16#5517—24no—
17#5500C2212yes—
18#5500C2212yes—
19#5483—22no—
20#5470—22no—
21C3×#389—22no—
22C3×#385—22no—
23C3×#384C2212yes—
24C3×#382C2212yes—
25#4964—24no—
26#4964C2218yes—
27C2×#808—22no—
28C2×#807—22no—
29#4530C2212yes—
30#4530C2212yes—
31#4523C2218yes—
32#3458—31no—
33C2×#428—32no—
34#2317D311yescharacteristic
35#2061—34no—
36#2060—34no—
37#1742—31no—
38#1733—32no—
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C3×#117—22no—
42C3×#117—22no—
43C3×#117—22no—
44C3×#117Q814yescomplemented
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4—22no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—24no—
50C2×C3×C4⋊C4C2312yes—
51C2×C3×C4⋊C4C2311yescharacteristic
52C2×C3×C4⋊C4C2311yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4D412yes—
55C2×C3×C42D412yes—
56C2×C3×C42C2312yes—
57C2×#217—22no—
58#926—22no—
59#926—22no—
60#926D414yes—
61#926C2314yes—
62C2×C4⋊Dic3D412yes—
63C2×C4⋊Dic3D412yes—
64#924—24no—
65#924—24no—
66C2×C4×Dic3—22no—
67C2×#208—22no—
68C2×#208—24no—
69C2×#208Q818yescomplemented
70C2×#208C2314yes—
71C22×C3⋊C8—22no—
72#890—24no—
73#890—24no—
74#888—24no—
75#888—24no—
76#878—28no—
77#878—28no—
78#870—24no—
79#869—22no—
80#869—22no—
81C4×Dic6—24no—
82C4×Dic6—24no—
83C3×#95—24no—
84#808—28no—
85#807—28no—
86C22×M16—32no—
87#544—32no—
88#544—32no—
89#544—62no—
90#544—62no—
91#543—62no—
92#533—64no—
93#533—62no—
94C2×#117—32no—
95C2×#117C2×D311yescharacteristic
96C2×#117C2×D311yescharacteristic
97C22×C4⋊C4C2×D311yescharacteristic
98#428—38no—
99#428—64no—
100C2×#107—62no—
101C2×#106—62no—
102#389—22no—
103#385—22no—
104#384C2×D312yes—
105#382C2×D312yes—
106#343—32no—
107#343—32no—
108#339—38no—
109C24×C3C2×D411yescharacteristic
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—42no—
113C22×C3×C4—82no—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×D412yes—
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×Q812yes—
120C22×C3×C4C2×Q814yes—
121C22×C3×C4#4112yes—
122C22×C3×C4C2411yescharacteristic
123C22×Dic3—22no—
124C22×Dic3—22no—
125C2×Dic6—24no—
126C2×Dic6—24no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—84no—
134C3×C22⋊C4—24no—
135C3×C22⋊C4—24no—
136C3×C42—22no—
137C3×C42—22no—
138C3×C42—22no—
139C3×C42C2×Q814yes—
140#217—24no—
141#217—24no—
142C4⋊Dic3—24no—
143C4⋊Dic3—24no—
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3C2×D414yes—
147C4⋊Dic3C2×D414yes—
148Dic3⋊C4—28no—
149Dic3⋊C4—28no—
150C4×Dic3—24no—
151C4×Dic3—24no—
152#208—28no—
153#208—24no—
154#208C2×Q818yes—
155C2×C3⋊C8—28no—
156C2×C3⋊C8—24no—
157C23×C4—21no—
158C23×C4C22×D311yescharacteristic
159C2×M16—38no—
160C2×M16—34no—
161C2×M16—64no—
162C2×M16—62no—
163C22×C8—62no—
164C4⋊Q8—62no—
165C4⋊Q8—62no—
166C4⋊Q8—64no—
167C4⋊Q8—64no—
168#125—64no—
169#125—64no—
170#125—64no—
171C22⋊Q8—68no—
172C22⋊Q8—68no—
173C22⋊Q8—64no—
174C22⋊Q8—64no—
175C4×Q8—64no—
176C4×Q8—64no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—34no—
181#117—34no—
182#117—62no—
183#117—62no—
184#117Dic614yescomplemented
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4C22×D312yes—
193C2×C4⋊C4C22×D311yescharacteristic
194C2×C4⋊C4C22×D311yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4#6712yes—
198C2×C42—62no—
199C2×C42#6712yes—
200C2×C42C22×D312yes—
201#107—68no—
202#106—68no—
203#95—24no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3#12012yes—
207C23×C3C22⋊Q812yes—
208C23×C3C4⋊Q812yes—
209C23×C3C22×D411yescharacteristic
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—24no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—44no—
218C2×C3×C4—42no—
219C2×C3×C4—84no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4C22⋊Q814yes—
223C2×C3×C4C22⋊Q814yes—
224C2×C3×C4C22⋊Q812yes—
225C2×C3×C4C4⋊Q814yes—
226C2×C3×C4C4⋊Q814yes—
227C2×C3×C4C4⋊Q812yes—
228C2×C3×C4C22×D412yes—
229C2×C3×C4C22×Q811yescharacteristic
230C2×C3×C4C22×Q812yes—
231C2×C3×C4C2×#4111yescharacteristic
232C2×Dic3—24no—
233C2×Dic3—24no—
234C2×Dic3—82no—
235C2×Dic3—82no—
236Dic6—84no—
237Dic6—84no—
238C3⋊C8—28no—
239C24C2×#6711yescharacteristic
240C2×Q8—64no—
241C2×Q8—64no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—82no—
248C22×C4C2×Dic612yes—
249C22×C4#23512yes—
250C22×C4D3×D412yes—
251C22×C4D3×D412yes—
252C22×C4D3×Q814yes—
253C22×C4C2×#6712yes—
254C22×C4C2×#6712yes—
255C22×C4C2×#6711yescharacteristic
256C22×C4C23×D311yescharacteristic
257M16—38no—
258M16—68no—
259M16—64no—
260C2×C8—68no—
261C2×C8—64no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—34no—
267C4⋊C4—34no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—68no—
271C4⋊C4—68no—
272C4⋊C4—64no—
273C4⋊C4—64no—
274C4⋊C4—62no—
275C4⋊C4—62no—
276C4⋊C4—84no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—64no—
280C22⋊C4—64no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—64no—
285C42—64no—
286C42C2×Dic614yes—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—22no—
290C22×C3—21no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3#39514yes—
294C22×C3C2×#12011yescharacteristic
295C22×C3C2×C22⋊Q811yescharacteristic
296C22×C3C2×C4⋊Q811yescharacteristic
297C3×C4—44no—
298C3×C4—42no—
299C3×C4—42no—
300C3×C4—82no—
301C3×C4#39512yes—
302C3×C4#39512yes—
303C3×C4C2×C22⋊Q812yes—
304C3×C4C2×C4⋊Q812yes—
305Dic3—44no—
306Dic3—44no—
307C23—41no—
308C23—41no—
309C23#92412yes—
310C23#93712yes—
311C23#94412yes—
312C23C22×#6711yescharacteristic
313Q8—244no—
314Q8—244no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—64no—
325C2×C4—64no—
326C2×C4—84no—
327C2×C4—242no—
328C2×C4—242no—
329C2×C4#88814yes—
330C2×C4#88814yes—
331C2×C4#89614yes—
332C2×C4#89614yes—
333C2×C4#92412yes—
334C2×C4#93711yescharacteristic
335C2×C4#93711yescharacteristic
336C2×C4#94412yes—
337C2×C4C22×Dic611yescharacteristic
338C2×C4C2×#23511yescharacteristic
339C2×C4C2×D3×D412yes—
340C2×C4C2×D3×Q812yes—
341C8—68no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—24no—
345C2×C3C2×#39511yescharacteristic
346C2×C3#302111yescharacteristic
347C2×C3#302211yescharacteristic
348C22—21no—
349C22—22no—
350C22—21no—
351C22—22no—
352C22—42no—
353C22—42no—
354C22#497914yes—
355C22C2×#92411yescharacteristic
356C22C2×#93711yescharacteristic
357C22C2×#94411yescharacteristic
358C4—44no—
359C4—42no—
360C4—42no—
361C4—82no—
362C4—124no—
363C4—124no—
364C4#497912yes—
365C4#497912yes—
366C4C2×#88812yes—
367C4C2×#89612yes—
368C3#2356211yescharacteristic, complemented, Sylow
369C2—21no—
370C2—24no—
371C2—21no—
372C2C2×#497911yescharacteristic
373C2#8339511yescharacteristic
374C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2317C2
socleC22×C3C2×#120
prime radicalC24×C3C2×D4
hypercenter#2317D3
hypocenterC3#23562
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
69. C2×#208Q8semidirect
184. #117Dic6semidirect
368. C3#23562semidirect

Maximal subgroups

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

subgroupindex
#857252
C3×#23172
C2×#49642
#715872
#715782
#695732
#695432
#235623