Finite Group Databasebeta

← #10713501

Subgroups — #10713501

Order 768.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C3×#2547C211yesmaximal, characteristic
2C2×#4541C214yesmaximal
3#71619C211yesmaximal, characteristic
4C4×#834C211yesmaximal, characteristic
5#13224—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic, complemented
7C2×C3×#124C2212yes—
8C2×C3×C4×D4C2211yescharacteristic
9C3×#397—21no—
10C3×#397—21no—
11C3×#391C412yescomplemented
12C3×#380—21no—
13C3×#380—21no—
14C3×C4×C4⋊C4C411yescharacteristic, complemented
15C2×#834C2212yes—
16C2×#834C2212yes—
17#5212—21no—
18C4×#217—21no—
19#4541—28no—
20#4526—21no—
21#4526—21no—
22#2547D311yescharacteristic
23C2×#354—34no—
24#1799—31no—
25C4×#99—31no—
26C22×C3×Q8C2×C412yes—
27C22×C3×D4C2311yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C3×C4⋊Q8—41no—
31C3×C4⋊Q8—41no—
32C3×#124—24no—
33C3×#124—24no—
34C3×C4×D4—41no—
35C3×C4×D4—41no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4D411yescharacteristic
43C2×C3×C22⋊C4D411yescharacteristic
44C2×C3×C42C2×C412yes—
45C2×C3×C42C2×C411yescharacteristic
46C2×#217—22no—
47C2×#217—22no—
48C2×C4⋊Dic3—41no—
49C2×C4×Dic3—41no—
50C3×#95—21no—
51C3×#95—22no—
52C3×#95—22no—
53#834—22no—
54#834—24no—
55#834D414yes—
56C3⋊#95—41no—
57C3⋊#95—41no—
58C2×C4⋊Q8Dic311yescharacteristic, complemented
59C2×#124C2×D312yes—
60C2×C4×D4C2×D311yescharacteristic
61C2×#99—32no—
62C2×#99—32no—
63#397—21no—
64#397—21no—
65#391Dic312yescomplemented
66#380—21no—
67#380—21no—
68#380—61no—
69C4×C4⋊C4Dic311yescharacteristic, complemented
70C4×C22⋊C4—61no—
71#354—68no—
72#342—61no—
73#342—61no—
74C24×C3C2×D411yescharacteristic
75C24×C3C2×D411yescharacteristic
76C2×C3×Q8—24no—
77C2×C3×Q8—42no—
78C2×C3×Q8—42no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4C2×D412yes—
82C2×C3×D4#4112yes—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22×C411yescharacteristic
96C22×Dic3—42no—
97C22×Dic3—42no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—42no—
100C3×C4⋊C4—42no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—44no—
106C3×C22⋊C4—42no—
107C3×C22⋊C4—42no—
108C3×C42—21no—
109C3×C42—21no—
110C3×C42—24no—
111#217—24no—
112#217—42no—
113#217—42no—
114#217—42no—
115C4⋊Dic3—82no—
116C4×Dic3—82no—
117C22×Q8C2×Dic312yes—
118C22×D4C22×D311yescharacteristic
119C23×C4#6711yescharacteristic
120C23×C4#6711yescharacteristic
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123#124—24no—
124#124—24no—
125C4×D4—41no—
126C4×D4—41no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—121no—
131C2×C4⋊C4C2×Dic311yescharacteristic
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—62no—
135C2×C22⋊C4—62no—
136C2×C22⋊C4#6711yescharacteristic
137C2×C22⋊C4#6711yescharacteristic
138C2×C42—121no—
139C2×C42C2×Dic312yes—
140C2×C42C2×Dic311yescharacteristic
141#99—34no—
142#99—62no—
143#99—64no—
144#95—21no—
145#95—22no—
146#95—22no—
147#95—121no—
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×Q8—84no—
162C3×D4—42no—
163C3×D4—42no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—81no—
182C2×C3×C4—81no—
183C2×C3×C4#9911yescharacteristic
184C2×C3×C4#9911yescharacteristic
185C2×C3×C4C4×D412yes—
186C2×Dic3—44no—
187C2×Dic3—82no—
188C2×Dic3—82no—
189C2×Dic3—82no—
190C2×Dic3—82no—
191C2×Dic3—82no—
192C24C2×#6711yescharacteristic
193C24C2×#6711yescharacteristic
194C2×Q8—24no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4D3×D412yes—
200C2×D4#23712yes—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—122no—
210C22×C4—122no—
211C22×C4#21711yescharacteristic
212C22×C4#21711yescharacteristic
213C22×C4#21711yescharacteristic
214C22×C4#21711yescharacteristic
215C22×C4C22×Dic311yescharacteristic
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—242no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—64no—
228C22⋊C4—122no—
229C22⋊C4—122no—
230C22⋊C4—122no—
231C42—21no—
232C42—21no—
233C42—24no—
234C42—242no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—22no—
238C22×C3—21no—
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—82no—
258C3×C4—82no—
259C3×C4#35412yescomplemented
260Dic3—84no—
261Dic3—162no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23#93611yescharacteristic
271C23#93812yes—
272C23#94011yescharacteristic
273C23C2×#21711yescharacteristic
274Q8—84no—
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—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—124no—
297C2×C4—242no—
298C2×C4—242no—
299C2×C4—242no—
300C2×C4—242no—
301C2×C4—242no—
302C2×C4#83411yescharacteristic
303C2×C4#83411yescharacteristic
304C2×C4D4×Dic312yes—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—41no—
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—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22C2×#83411yescharacteristic
328C22#522111yescharacteristic
329C22#524011yescharacteristic
330C4—22no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—82no—
336C4—82no—
337C4—244no—
338C4—482no—
339C4#454112yescomplemented
340C3#1322411yescharacteristic, complemented, Sylow
341C2—21no—
342C2—21no—
343C2—42no—
344C2—41no—
345C2—41no—
346C2#7168611yescharacteristic
347C2C2×#454111yescharacteristic
348C2#7187111yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC22×D4C22×D3
FittingC3×#2547C2
socleC22×C3C2×#99
prime radical, cosocleC22×C3×D4C23
hypercenter#2547D3
hypocenterC3#13224
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
6. C2×C3×C4⋊Q8C4semidirect
11. C3×#391C4semidirect
14. C3×C4×C4⋊C4C4semidirect
58. C2×C4⋊Q8Dic3semidirect
65. #391Dic3semidirect
69. C4×C4⋊C4Dic3semidirect
259. C3×C4#354semidirect
339. C4#4541semidirect
340. C3#13224semidirect

Maximal subgroups

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

subgroupindex
C3×#25472
C2×#45412
#716192
C4×#8342
#132243