Finite Group Databasebeta

← #10711981

Subgroups — #10711981

Order 768.

672 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2366C211yesmaximal, characteristic
2C2×#4545C211yesmaximal, characteristic
3#71763C211yesmaximal, characteristic
4#69597C212yesmaximal, complemented
5#67637C212yesmaximal, complemented
6#12964—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×C3×C4×D4—22no—
9#5798C2211yescharacteristic
10C3×#390—22no—
11C3×#386—21no—
12C3×#386C2212yes—
13C3×C4×C4⋊C4—21no—
14C2×#834C2212yes—
15C2×#833C2211yescharacteristic
16#5215—21no—
17#5210—22no—
18#4545C414yescomplemented
19#4527—22no—
20#4523—21no—
21#2366D311yescharacteristic
22C2×#353—31no—
23#1887—31no—
24#1867—32no—
25#1390—32no—
26C22×C3×D4—22no—
27C22×C3×D4D411yescharacteristic
28C22×C3×D4C2311yescharacteristic
29C23×C3×C4—21no—
30C23×C3×C4D412yes—
31C3×#127—22no—
32C3×#127C2×C412yes—
33C3×#127Q812yes—
34C3×C4×D4—48no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4D412yes—
41C2×C3×C42—22no—
42C2×C3×C42D411yescharacteristic
43C2×#217—22no—
44C2×#217—22no—
45D4×Dic3—41no—
46D4×Dic3—41no—
47#926—21no—
48C2×Dic3⋊C4—42no—
49C2×#208—21no—
50C3×#95—22no—
51#834—22no—
52#834C2×C414yescomplemented
53#833—22no—
54#832—41no—
55#832—41no—
56C3⋊#95—42no—
57C2×#127C2×D311yescharacteristic
58#518—31no—
59C2×C4×D4—22no—
60#418—61no—
61C2×#100—31no—
62C2×#99—32no—
63#390—22no—
64#386—21no—
65#386C2×D312yes—
66#385—62no—
67C4×C4⋊C4—21no—
68#353—34no—
69#342—62no—
70#339—61no—
71C24×C3—21no—
72C24×C3C2×D412yes—
73C2×C3×D4—24no—
74C2×C3×D4—24no—
75C2×C3×D4—21no—
76C2×C3×D4—41no—
77C2×C3×D4—41no—
78C2×C3×D4—44no—
79C2×C3×D4—44no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4C4⋊C412yescomplemented
83C2×C3×D4C22×C411yescharacteristic
84C2×C3×D4C2×Q811yescharacteristic
85C2×C3×D4#4112yes—
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—44no—
90C22×C3×C4—44no—
91C22×C3×C4—42no—
92C22×C3×C4—81no—
93C22×C3×C4C2×D411yescharacteristic
94C22×Dic3—22no—
95C22×Dic3—22no—
96C22×Dic3—42no—
97C3×C4⋊C4—24no—
98C3×C4⋊C4—42no—
99C3×C22⋊C4—44no—
100C3×C22⋊C4—44no—
101C3×C22⋊C4—42no—
102C3×C22⋊C4—81no—
103C3×C42—21no—
104C3×C42—44no—
105C3×C42C4⋊C412yescomplemented
106#217—24no—
107#217—42no—
108#217—42no—
109#217—41no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112Dic3⋊C4—84no—
113C4×Dic3—41no—
114#208—41no—
115#208—41no—
116C2×C3⋊C8—41no—
117C22×D4—22no—
118C22×D4#6711yescharacteristic
119C22×D4C22×D311yescharacteristic
120C23×C4—21no—
121C23×C4#6712yes—
122C2×M16—61no—
123#127—22no—
124#127Dic612yes—
125#127C4×D312yes—
126C4×D4—48no—
127C4×D4—121no—
128C4×D4—121no—
129#117—61no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—122no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—62no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4#6712yes—
139C2×C42—22no—
140C2×C42#6711yescharacteristic
141#102—121no—
142#102—121no—
143#100—62no—
144#99—34no—
145#99—62no—
146#95—22no—
147#95—122no—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—42no—
152C23×C3—42no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—41no—
156C23×C3—41no—
157C23×C3C4×D412yes—
158C23×C3#12011yescharacteristic
159C23×C3C22⋊Q812yes—
160C3×D4—41no—
161C3×D4—41no—
162C3×D4—42no—
163C3×D4—42no—
164C3×D4—48no—
165C3×D4—48no—
166C3×D4—81no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—24no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—44no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—82no—
180C2×C3×C4—84no—
181C2×C3×C4C2×C4⋊C411yescharacteristic
182C2×C3×C4#12311yescharacteristic
183C2×Dic3—44no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—82no—
187C2×Dic3—82no—
188C2×Dic3—82no—
189C2×Dic3—82no—
190C3⋊C8—81no—
191C24—21no—
192C24C2×#6712yes—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4Dic3⋊C412yescomplemented
203C2×D4C2×Dic611yescharacteristic
204C2×D4C2×C4×D311yescharacteristic
205C2×D4#23512yes—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—62no—
213C22×C4—62no—
214C22×C4—81no—
215C22×C4—122no—
216C22×C4C2×#6711yescharacteristic
217M16—121no—
218M16—121no—
219C2×C8—121no—
220C4⋊C4—24no—
221C4⋊C4—42no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—244no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—42no—
228C22⋊C4—64no—
229C22⋊C4—81no—
230C22⋊C4—122no—
231C22⋊C4—122no—
232C22⋊C4—121no—
233C42—21no—
234C42—44no—
235C42—121no—
236C42Dic3⋊C412yescomplemented
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—42no—
242C22×C3—42no—
243C22×C3—44no—
244C22×C3—42no—
245C22×C3—42no—
246C22×C3—42no—
247C22×C3—82no—
248C22×C3—82no—
249C22×C3—81no—
250C22×C3—81no—
251C22×C3#38511yescharacteristic
252C22×C3B(4,𝔽2)11yescharacteristic
253C22×C3#45811yescharacteristic
254C3×C4—24no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—44no—
258C3×C4—82no—
259Dic3—84no—
260Dic3—81no—
261Dic3—81no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23#92412yes—
272C23C4×#6712yes—
273C23#95311yescharacteristic
274D4—41no—
275D4—41no—
276D4—42no—
277D4—42no—
278D4—48no—
279D4—48no—
280D4—81no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—24no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—82no—
294C2×C4—84no—
295C2×C4—124no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—242no—
299C2×C4—242no—
300C2×C4—242no—
301C2×C4—242no—
302C2×C4C2×Dic3⋊C411yescharacteristic
303C2×C4#92911yescharacteristic
304C8—241no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3—42no—
309C2×C3—82no—
310C2×C3#186711yescharacteristic
311C2×C3#219611yescharacteristic
312C2×C3#219811yescharacteristic
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22—44no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—82no—
324C22—82no—
325C22—81no—
326C22—81no—
327C22#521011yescharacteristic
328C22#524311yescharacteristic
329C22#524411yescharacteristic
330C4—24no—
331C4—21no—
332C4—21no—
333C4—44no—
334C4—82no—
335C4—244no—
336C4—241no—
337C4—241no—
338C3#1296411yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—42no—
342C2—42no—
343C2—82no—
344C2#7165711yescharacteristic
345C2#7190911yescharacteristic
346C2#7194611yescharacteristic
subgroupG/H
centerC22#5244
commutatorC2×C3×D4C22×C4
FrattiniC22×D4C22×D3
FittingC3×#2366C2
socleC22×C3B(4,𝔽2)
prime radicalC2×C3×#127C22
cosocleC22×C3×D4C23
hypercenter#2366D3
hypocenterC3#12964
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2366

Complements

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

NH ≅ G/Ntype
4. #69597C2semidirect
5. #67637C2semidirect
18. #4545C4semidirect
52. #834C2×C4semidirect
82. C2×C3×D4C4⋊C4semidirect
105. C3×C42C4⋊C4semidirect
202. C2×D4Dic3⋊C4semidirect
236. C42Dic3⋊C4semidirect
338. C3#12964semidirect

Maximal subgroups

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

subgroupindex
C3×#23662
C2×#45452
#717632
#695972
#676372
#129643