Finite Group Databasebeta

← #10692942

Subgroups — #10692942

Order 768.

726 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2436C211yesmaximal, characteristic
2#84062C211yesmaximal, characteristic, complemented
3#72133C211yesmaximal, characteristic, complemented
4#71674C212yesmaximal
5#67616C212yesmaximal, complemented
6#9466—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C2×C3×#124C412yescomplemented
9C2×C3×#121C2211yescharacteristic
10C2×#926—21no—
11C3×#393—22no—
12C3×#391C412yescomplemented
13C3×#386—21no—
14C3×#386—21no—
15C3×C4×C22⋊C4C2211yescharacteristic
16#5229—24no—
17C2×#836—22no—
18#5211—21no—
19#5211C2212yes—
20#5207C2212yes—
21#4537—24no—
22#2436D311yescharacteristic
23#2310—31no—
24#1784—31no—
25#1510—32no—
26#1284—32no—
27C22×C3×Q8—22no—
28C22×C3×Q8—22no—
29C22×C3×Q8—44no—
30C22×C3×Q8C2×C412yes—
31C22×C3×Q8D412yes—
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C3×#124—44no—
38C3×#121—22no—
39C3×#121—22no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C42C2×C412yes—
46C2×#217—22no—
47#926—44no—
48C2×C4⋊Dic3—22no—
49C2×C4×Dic3—22no—
50C2×#208—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3×#95—21no—
54#836—44no—
55C3⋊#95—24no—
56C3⋊#95—22no—
57C3⋊#95D414yes—
58#830—22no—
59#830—22no—
60C23×Q8C2×D311yescharacteristic
61C2×#124Dic312yescomplemented
62C2×#121C2×D311yescharacteristic
63C2×#117—61no—
64C2×#101—62no—
65#407—32no—
66#393—22no—
67#391—64no—
68#391Dic312yescomplemented
69#386—21no—
70#386—21no—
71#381—32no—
72#381—61no—
73C4×C22⋊C4C2×D311yescharacteristic
74#327—64no—
75C24×C3—21no—
76C24×C3C2×D411yescharacteristic
77C2×C3×Q8—24no—
78C2×C3×Q8—24no—
79C2×C3×Q8—24no—
80C2×C3×Q8—24no—
81C2×C3×Q8—42no—
82C2×C3×Q8—42no—
83C2×C3×Q8—44no—
84C2×C3×Q8—44no—
85C2×C3×Q8—48no—
86C2×C3×D4—22no—
87C2×C3×D4—42no—
88C2×C3×D4—44no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—24no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—22no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4C22⋊C412yes—
101C22×C3×C4C22⋊C412yes—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D412yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4112yes—
106C22×Dic3—24no—
107C3×C4⋊C4—22no—
108C3×C22⋊C4—44no—
109C3×C22⋊C4—42no—
110C3×C22⋊C4—44no—
111C3×C22⋊C4—42no—
112C3×C42—42no—
113#217—44no—
114C4⋊Dic3—44no—
115C4×Dic3—44no—
116#208—44no—
117C2×C3⋊C8—42no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—44no—
121C22×Q8C2×Dic312yes—
122C22×Q8#6712yes—
123C22×D4—21no—
124C22×D4C2×Dic311yescharacteristic
125C23×C4#6711yescharacteristic
126C23×C4#6711yescharacteristic
127C23×C4C22×D311yescharacteristic
128C2×M16—62no—
129#124—44no—
130#121—22no—
131#121—22no—
132#117—124no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4C2×Dic311yescharacteristic
135C2×C22⋊C4—21no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—62no—
140C2×C42—62no—
141C2×C42C2×Dic312yes—
142#101—124no—
143C22⋊C8—62no—
144C22⋊C8—62no—
145#95—22no—
146#95—22no—
147#95—21no—
148#95—34no—
149#95—64no—
150#95—62no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—41no—
154C23×C3—42no—
155C23×C3—42no—
156C23×C3—41no—
157C23×C3—42no—
158C23×C3C2×C22⋊C411yescharacteristic
159C23×C3#12011yescharacteristic
160C23×C3#12311yescharacteristic
161C3×Q8—48no—
162C3×Q8—48no—
163C3×D4—82no—
164C3×D4—82no—
165C3×D4—82no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—24no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4#9914yes—
186C2×C3×C4C4×D412yes—
187C2×C3×C4#12112yes—
188C2×Dic3—44no—
189C2×Dic3—44no—
190C2×Dic3—44no—
191C3⋊C8—82no—
192C24—21no—
193C24C2×#6711yescharacteristic
194C2×Q8—24no—
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—48no—
203C2×D4—22no—
204C2×D4—42no—
205C2×D4—44no—
206C2×D4—41no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—64no—
218C22×C4#21712yes—
219C22×C4#21712yes—
220C22×C4D3×D412yes—
221C22×C4#23712yes—
222C22×C4C22×Dic311yescharacteristic
223C22×C4C2×#6711yescharacteristic
224M16—124no—
225C2×C8—122no—
226C4⋊C4—22no—
227C4⋊C4—124no—
228C22⋊C4—44no—
229C22⋊C4—42no—
230C22⋊C4—44no—
231C22⋊C4—42no—
232C22⋊C4—124no—
233C42—42no—
234C42—124no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—44no—
243C22×C3—84no—
244C22×C3—81no—
245C22×C3—81no—
246C22×C3—81no—
247C22×C3#38611yescharacteristic
248C22×C3C2×#9912yes—
249C22×C3#45412yes—
250C3×C4—22no—
251C3×C4—24no—
252C3×C4—22no—
253C3×C4—42no—
254C3×C4—82no—
255C3×C4—81no—
256Dic3—84no—
257C23—22no—
258C23—22no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23#93511yescharacteristic
265C23#93711yescharacteristic
266C23C2×#21711yescharacteristic
267Q8—48no—
268Q8—48no—
269D4—82no—
270D4—82no—
271D4—82no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—124no—
292C2×C4—124no—
293C2×C4—124no—
294C2×C4#83414yes—
295C2×C4D4×Dic312yes—
296C2×C4#93912yes—
297C8—242no—
298C2×C3—22no—
299C2×C3—22no—
300C2×C3—41no—
301C2×C3—161no—
302C2×C3#188811yescharacteristic
303C2×C3#189811yescharacteristic
304C2×C3#191211yescharacteristic
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22—84no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22#506212yes—
318C22C2×#83412yes—
319C22#522011yescharacteristic
320C4—22no—
321C4—24no—
322C4—22no—
323C4—42no—
324C4—82no—
325C4—81no—
326C4—244no—
327C3#946611yescharacteristic, complemented, Sylow
328C2—22no—
329C2—22no—
330C2—41no—
331C2—161no—
332C2#7167011yescharacteristic
333C2#7172411yescharacteristic
334C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2436C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2436D3
hypocenterC3#9466
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
2. #84062C2semidirect
3. #72133C2semidirect
5. #67616C2semidirect
8. C2×C3×#124C4semidirect
12. C3×#391C4semidirect
61. C2×#124Dic3semidirect
68. #391Dic3semidirect
327. C3#9466semidirect

Maximal subgroups

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

subgroupindex
C3×#24362
#840622
#721332
#716742
#676162
#94663