Finite Group Databasebeta

← #10713500

Subgroups — #10713500

Order 768.

552 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82382C211yesmaximal, characteristic
2C3×C4×#99C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4C2×#4473C214yesmaximal
5#13224—31nomaximal, Sylow
6C2×C3×C4×D4C2211yescharacteristic
7C2×#936C2212yes—
8C2×#869C411yescharacteristic, complemented
9C2×C3×#99C2212yes—
10C3×C4×C22⋊C4—21no—
11C2×#834C2212yes—
12#5212—21no—
13#5212—21no—
14#5212—21no—
15#4973C412yescomplemented
16#4953—21no—
17#4953—21no—
18C4×C4⋊Dic3C411yescharacteristic, complemented
19C3×#342—21no—
20#4526—21no—
21#4473—28no—
22#2547—31no—
23C2×#354—34no—
24#1799—31no—
25C4×#99D311yescharacteristic
26C22×C3×D4C2311yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C22×Dic6C2×C412yes—
30C3×C4×D4—41no—
31C3×C4×D4—41no—
32C2×C3×C4⋊C4C2×C411yescharacteristic
33C2×C3×C22⋊C4—22no—
34C2×C3×C22⋊C4D411yescharacteristic
35C2×C3×C22⋊C4D411yescharacteristic
36C2×C3×C42—41no—
37C2×C3×C42C2×C411yescharacteristic
38C2×#217—22no—
39C2×#217—22no—
40C2×#217—22no—
41#936—24no—
42#936—24no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—41no—
47C2×C4×Dic3C2×C412yes—
48#869—41no—
49#869—41no—
50C3×#99—22no—
51C3×#99D414yes—
52C3×#95—41no—
53#834—24no—
54C3⋊#95—21no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57C3⋊#95—41no—
58C2×C4⋊Q8—31no—
59C2×#124—32no—
60C2×C4×D4C2×D311yescharacteristic
61C2×#99—32no—
62C2×#99C2×D312yes—
63#397—61no—
64#397—61no—
65#391—32no—
66#380—61no—
67#380—61no—
68#380—61no—
69C4×C4⋊C4—31no—
70C4×C22⋊C4—21no—
71#354—68no—
72#342—21no—
73#342—61no—
74C24×C3C2×D411yescharacteristic
75C24×C3C2×D411yescharacteristic
76C2×C3×D4—41no—
77C2×C3×D4—41no—
78C2×C3×D4C2×D412yes—
79C2×C3×D4#4112yes—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4—42no—
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22×C411yescharacteristic
92C22×Dic3—22no—
93C22×Dic3—22no—
94C22×Dic3—42no—
95C2×Dic6—24no—
96C2×Dic6—42no—
97C2×Dic6—42no—
98C3×C4⋊C4—41no—
99C3×C22⋊C4—24no—
100C3×C22⋊C4—41no—
101C3×C22⋊C4—41no—
102C3×C22⋊C4—42no—
103C3×C42—21no—
104C3×C42—21no—
105C3×C42—82no—
106#217—44no—
107#217—42no—
108#217—42no—
109#217—42no—
110#217—42no—
111C4⋊Dic3—42no—
112C4⋊Dic3—42no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115C4⋊Dic3—82no—
116C4×Dic3—24no—
117C22×Q8—32no—
118C22×D4C22×D311yescharacteristic
119C23×C4D1211yescharacteristic
120C23×C4#6711yescharacteristic
121C4⋊Q8—121no—
122C4⋊Q8—121no—
123#124—64no—
124#124—64no—
125C4×D4—41no—
126C4×D4—41no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—121no—
131C2×C4⋊C4C4×D311yescharacteristic
132C2×C22⋊C4—22no—
133C2×C22⋊C4—62no—
134C2×C22⋊C4—62no—
135C2×C22⋊C4—62no—
136C2×C22⋊C4D1211yescharacteristic
137C2×C22⋊C4#6711yescharacteristic
138C2×C42—32no—
139C2×C42—41no—
140C2×C42C4×D311yescharacteristic
141#99—22no—
142#99—64no—
143#99D1214yes—
144#95—41no—
145#95—61no—
146#95—62no—
147#95—62no—
148#95—121no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3—41no—
156C23×C3—41no—
157C23×C3C2×C22⋊C411yescharacteristic
158C23×C3#12112yes—
159C23×C3#12411yescharacteristic
160C23×C3#12711yescharacteristic
161C3×D4—42no—
162C3×D4—42no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—22no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—44no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4—82no—
178C2×C3×C4—82no—
179C2×C3×C4#9911yescharacteristic
180C2×C3×C4#9911yescharacteristic
181C2×C3×C4C4×D412yes—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—82no—
189C2×Dic3—82no—
190C2×Dic3—82no—
191Dic6—84no—
192C24C2×D1211yescharacteristic
193C24C2×#6711yescharacteristic
194C2×Q8—64no—
195C2×Q8—122no—
196C2×Q8—122no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4C2×D1212yes—
200C2×D4#23512yes—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—62no—
209C22×C4—62no—
210C22×C4—122no—
211C22×C4(C2×D3)⋊C411yescharacteristic
212C22×C4(C2×D3)⋊C411yescharacteristic
213C22×C4(C2×D3)⋊C411yescharacteristic
214C22×C4(C2×D3)⋊C411yescharacteristic
215C22×C4C2×C4×D311yescharacteristic
216C4⋊C4—41no—
217C4⋊C4—122no—
218C4⋊C4—122no—
219C4⋊C4—121no—
220C4⋊C4—121no—
221C4⋊C4—242no—
222C22⋊C4—24no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C22⋊C4—122no—
227C22⋊C4—122no—
228C22⋊C4—124no—
229C22⋊C4—122no—
230C22⋊C4—122no—
231C42—21no—
232C42—21no—
233C42—64no—
234C42—82no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—22no—
239C22×C3—21no—
240C22×C3—42no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—81no—
248C22×C3—81no—
249C22×C3#39011yescharacteristic
250C22×C3C2×#9911yescharacteristic
251C22×C3#41011yescharacteristic
252C3×C4—21no—
253C3×C4—22no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—84no—
258C3×C4#35412yescomplemented
259Dic3—82no—
260Dic3—82no—
261Dic3—162no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23#87411yescharacteristic
271C23#87511yescharacteristic
272C23C2×(C2×D3)⋊C411yescharacteristic
273C23#93012yes—
274Q8—244no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—44no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—242no—
300C2×C4—242no—
301C2×C4—242no—
302C2×C4#80611yescharacteristic
303C2×C4#80611yescharacteristic
304C2×C4C4×D1212yes—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—41no—
308C2×C3—42no—
309C2×C3—41no—
310C2×C3#196411yescharacteristic
311C2×C3C2×#35411yescharacteristic
312C2×C3#212911yescharacteristic
313C22—21no—
314C22—21no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22#493911yescharacteristic
328C22C2×#80611yescharacteristic
329C22#500611yescharacteristic
330C4—22no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—84no—
336C4—242no—
337C4—242no—
338C4—482no—
339C4#447312yescomplemented
340C3#1322411yescharacteristic, complemented, Sylow
341C2—21no—
342C2—21no—
343C2—42no—
344C2—41no—
345C2—41no—
346C2#6945611yescharacteristic
347C2C2×#447311yescharacteristic
348C2#7005911yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC22×D4C22×D3
FittingC3×C4×#99C2
socleC22×C3C2×#99
prime radical, cosocleC22×C3×D4C23
hypercenterC4×#99D3
hypocenterC3#13224
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
8. C2×#869C4semidirect
15. #4973C4semidirect
18. C4×C4⋊Dic3C4semidirect
258. C3×C4#354semidirect
339. C4#4473semidirect
340. C3#13224semidirect

Maximal subgroups

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

subgroupindex
#823822
C3×C4×#992
#716192
C2×#44732
#132243