Finite Group Databasebeta

← #10823309

Subgroups — #10823309

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
1C3×#3458C211yesmaximal, characteristic
2#82234C211yesmaximal, characteristic
3C2×#4995C212yesmaximal
4#71901C214yesmaximal, complemented
5#71900C214yesmaximal, complemented
6#71587C211yesmaximal, characteristic
7#71574C212yesmaximal
8#23562—31nomaximal, Sylow
9C3×#544—22no—
10C3×#544—22no—
11C3×#544C2212yes—
12C3×#544C2212yes—
13C3×#543—22no—
14C3×#533—24no—
15C3×#533—22no—
16C2×C3×#117C2212yes—
17C2×C3×#117C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×#208C2212yes—
21#4995—24no—
22#4995C2218yes—
23#4977C2212yes—
24#4975C2212yes—
25C2×#808—22no—
26C2×#807—22no—
27#4946—22no—
28#4928—22no—
29#4532C2218yes—
30#4530C2212yes—
31#4530C2212yes—
32#3458D311yescharacteristic
33C2×#428—32no—
34#2317—31no—
35#2061—34no—
36#2060—34no—
37#1742—31no—
38#1733—32no—
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C3×C4⋊Q8—22no—
42C3×C4⋊Q8—22no—
43C3×C4⋊Q8—24no—
44C3×C4⋊Q8—24no—
45C3×#125—24no—
46C3×#125—24no—
47C3×#125—24no—
48C3×C22⋊Q8—28no—
49C3×C22⋊Q8—28no—
50C3×C22⋊Q8—24no—
51C3×C22⋊Q8—24no—
52C3×C4×Q8—24no—
53C3×C4×Q8—24no—
54C3×#117—22no—
55C3×#117—22no—
56C3×#117—22no—
57C3×#117—22no—
58C3×#117D414yes—
59C3×#117C2314yes—
60C2×C3×C4⋊C4D412yes—
61C2×C3×C4⋊C4D412yes—
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4D412yes—
66C2×C3×C42—22no—
67C2×C3×C42D412yes—
68C2×#217—21no—
69#926—22no—
70#926Q814yescomplemented
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4—24no—
77C2×C4×Dic3C2312yes—
78C2×#208—22no—
79C2×#208—24no—
80C2×#208Q818yescomplemented
81C2×#208C2314yes—
82C22×C3⋊C8—22no—
83C3⋊#95—24no—
84#808—28no—
85#807—28no—
86C22×M16—32no—
87#544—22no—
88#544—22no—
89#544C2×D312yes—
90#544C2×D312yes—
91#543—22no—
92#533—24no—
93#533—22no—
94C2×#117—31no—
95C2×#117C2×D312yes—
96C2×#117C2×D311yescharacteristic
97C22×C4⋊C4—31no—
98#428—38no—
99#428—64no—
100C2×#107—62no—
101C2×#106—62no—
102#389—62no—
103#385—62no—
104#384—32no—
105#382—32no—
106#343—32no—
107#343—32no—
108#339—38no—
109C24×C3C2×D411yescharacteristic
110C2×C3×Q8—24no—
111C2×C3×Q8—24no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2×Q814yes—
120C22×C3×C4C2×Q812yes—
121C22×C3×C4#4112yes—
122C22×C3×C4C2411yescharacteristic
123C22×Dic3—22no—
124C22×Dic3—22no—
125C22×Dic3—42no—
126C22×Dic3—82no—
127C3×C4⋊C4—28no—
128C3×C4⋊C4—28no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4C2×D414yes—
136C3×C4⋊C4C2×D414yes—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—24no—
139C3×C22⋊C4—24no—
140C3×C42—24no—
141C3×C42—22no—
142C3×C42—22no—
143C3×C42—24no—
144#217—24no—
145C4⋊Dic3—24no—
146C4⋊Dic3—24no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149Dic3⋊C4—84no—
150C4×Dic3—22no—
151C4×Dic3C2×Q814yes—
152#208—28no—
153#208—24no—
154#208C2×Q818yes—
155C2×C3⋊C8—28no—
156C2×C3⋊C8—24no—
157C23×C4—61no—
158C23×C4C22×D311yescharacteristic
159C2×M16—38no—
160C2×M16—34no—
161C2×M16—64no—
162C2×M16—62no—
163C22×C8—62no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168#125—24no—
169#125—24no—
170#125—24no—
171C22⋊Q8—28no—
172C22⋊Q8—28no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—34no—
182#117—62no—
183#117#6714yes—
184#117C22×D314yes—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—64no—
191C2×C4⋊C4#6712yes—
192C2×C4⋊C4#6712yes—
193C2×C4⋊C4C22×D311yescharacteristic
194C2×C4⋊C4C22×D311yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4#6712yes—
198C2×C42—22no—
199C2×C42—32no—
200C2×C42#6712yes—
201#107—68no—
202#106—68no—
203#95—64no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3#12012yes—
207C23×C3C22⋊Q812yes—
208C23×C3C4⋊Q812yes—
209C23×C3C22×D411yescharacteristic
210C3×Q8—84no—
211C3×Q8—84no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—82no—
219C2×C3×C4—82no—
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×Q812yes—
230C2×C3×C4C22×Q811yescharacteristic
231C2×C3×C4C2×#4111yescharacteristic
232C2×Dic3—24no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—44no—
236C2×Dic3—42no—
237C2×Dic3—84no—
238C3⋊C8—28no—
239C24C2×#6711yescharacteristic
240C2×Q8—24no—
241C2×Q8—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—122no—
247C22×C4—242no—
248C22×C4D3×Q812yes—
249C22×C4D3×Q814yes—
250C22×C4#23912yes—
251C22×C4C2×#6712yes—
252C22×C4C2×#6712yes—
253C22×C4C2×#6711yescharacteristic
254C22×C4C2×#6712yes—
255C22×C4C2×#6712yes—
256C22×C4C23×D311yescharacteristic
257M16—38no—
258M16—68no—
259M16—64no—
260C2×C8—68no—
261C2×C8—64no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—64no—
271C4⋊C4—64no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—244no—
275C4⋊C4C2×#6714yes—
276C4⋊C4C2×#6714yes—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—64no—
281C42—24no—
282C42—22no—
283C42—22no—
284C42—24no—
285C42—34no—
286C42—62no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3#39514yes—
294C22×C3C2×#12011yescharacteristic
295C22×C3C2×C22⋊Q811yescharacteristic
296C22×C3C2×C4⋊Q811yescharacteristic
297C3×C4—42no—
298C3×C4—44no—
299C3×C4—42no—
300C3×C4—44no—
301C3×C4#39512yes—
302C3×C4#39512yes—
303C3×C4C2×C22⋊Q812yes—
304C3×C4C2×C4⋊Q812yes—
305Dic3—44no—
306Dic3—82no—
307C23—41no—
308C23—41no—
309C23#94412yes—
310C23#94612yes—
311C23#95312yes—
312C23C22×#6711yescharacteristic
313Q8—84no—
314Q8—84no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—64no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—124no—
327C2×C4—122no—
328C2×C4—244no—
329C2×C4#94412yes—
330C2×C4#94414yes—
331C2×C4#94414yes—
332C2×C4#94612yes—
333C2×C4#94614yes—
334C2×C4#94614yes—
335C2×C4#95311yescharacteristic
336C2×C4#95311yescharacteristic
337C2×C4C2×D3×Q811yescharacteristic
338C2×C4C2×D3×Q812yes—
339C2×C4C2×#23911yescharacteristic
340C2×C4C22×#6712yes—
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#523014yes—
355C22C2×#94411yescharacteristic
356C22C2×#94611yescharacteristic
357C22C2×#95311yescharacteristic
358C4—42no—
359C4—44no—
360C4—42no—
361C4—44no—
362C4—124no—
363C4—242no—
364C4#523012yes—
365C4#523012yes—
366C4C2×#94412yes—
367C4C2×#94612yes—
368C3#2356211yescharacteristic, complemented, Sylow
369C2—21no—
370C2—24no—
371C2—21no—
372C2C2×#523011yescharacteristic
373C2#8412211yescharacteristic
374C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3458C2
socleC22×C3C2×#120
prime radicalC24×C3C2×D4
hypercenter#3458D3
hypocenterC3#23562
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
4. #71901C2semidirect
5. #71900C2semidirect
70. #926Q8semidirect
80. C2×#208Q8semidirect
368. C3#23562semidirect

Maximal subgroups

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

subgroupindex
C3×#34582
#822342
C2×#49952
#719012
#719002
#715872
#715742
#235623