Finite Group Databasebeta

← #10692939

Subgroups — #10692939

Order 768.

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

normal subgroups only

This group has 726 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#84066C211yesmaximal, characteristic
2#84062C211yesmaximal, characteristic, complemented
3C3×#1784C211yesmaximal, characteristic, complemented
4#69978C212yesmaximal
5#67682C212yesmaximal, complemented
6#9466—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C2×#947C412yescomplemented
9C2×#930C2211yescharacteristic
10C2×#926—21no—
11C2×C3×#101—22no—
12C3×#407C2212yes—
13#5231—22no—
14#5229—24no—
15#5229C412yescomplemented
16#5213—21no—
17#5213—21no—
18#5211—21no—
19#5211C2212yes—
20C4×#217C2211yescharacteristic
21#4470—24no—
22#2436—31no—
23#2310—31no—
24#1784D311yescharacteristic, complemented
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—
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C22×#67—21no—
36C22×D12C2×C411yescharacteristic
37C2×C3×M16—22no—
38C2×#217—21no—
39C2×#217—22no—
40#947—44no—
41#930—22no—
42#930—22no—
43C2×(C2×D3)⋊C4—22no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—21no—
46#926—44no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×C4×Dic3—22no—
50C2×C4×Dic3C2×C412yes—
51C3×#101—44no—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—22no—
54C3⋊#95—24no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57C3⋊#95—21no—
58C3⋊#95—22no—
59C3⋊#95D414yes—
60C23×Q8C2×D311yescharacteristic
61C2×#124—32no—
62C2×#121—31no—
63C2×#117—61no—
64C2×#101—22no—
65#407C2×D312yes—
66#393—62no—
67#391—32no—
68#391—64no—
69#386—61no—
70#386—61no—
71#381—32no—
72#381—61no—
73C4×C22⋊C4—31no—
74#327—64no—
75C24×C3C2×D411yescharacteristic
76C23×D3—21no—
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—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—24no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4C22⋊C412yes—
94C22×C3×C4C22⋊C412yes—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D412yes—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4112yes—
99C2×#67—44no—
100C2×#67—41no—
101C2×#67—41no—
102C22×Dic3—21no—
103C22×Dic3—22no—
104C22×Dic3—24no—
105C2×D12—22no—
106C2×D12—42no—
107C3×M16—44no—
108C2×C3×C8—42no—
109#217—44no—
110#217—42no—
111(C2×D3)⋊C4—44no—
112(C2×D3)⋊C4—44no—
113(C2×D3)⋊C4—42no—
114C4⋊Dic3—22no—
115C4⋊Dic3—44no—
116C4×Dic3—44no—
117C4×Dic3—42no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—44no—
121C22×Q8C4×D312yes—
122C22×Q8#6712yes—
123C22×D4—31no—
124C22×D4—61no—
125C23×C4D1211yescharacteristic
126C23×C4D1211yescharacteristic
127C23×C4C22×D311yescharacteristic
128C2×M16—22no—
129#124—124no—
130#121—62no—
131#121—62no—
132#117—124no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—62no—
135C2×C22⋊C4—62no—
136C2×C22⋊C4—61no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4—62no—
139C2×C22⋊C4—61no—
140C2×C42—32no—
141C2×C42—62no—
142#101—44no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145#95—34no—
146#95—64no—
147#95—62no—
148#95—62no—
149#95—62no—
150#95—61no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—41no—
154C23×C3C2×C22⋊C411yescharacteristic
155C23×C3#12011yescharacteristic
156C23×C3#12311yescharacteristic
157C22×D3—42no—
158C22×D3—42no—
159C22×D3—41no—
160C22×D3—42no—
161C3×Q8—48no—
162C3×Q8—48no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—24no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4#9914yes—
178C2×C3×C4C4×D412yes—
179C2×C3×C4#12112yes—
180#67—82no—
181#67—82no—
182C2×Dic3—44no—
183C2×Dic3—42no—
184C2×Dic3—44no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—41no—
188C2×Dic3—42no—
189C2×Dic3—44no—
190D12—82no—
191C3×C8—82no—
192C24—61no—
193C24C2×D1211yescharacteristic
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—62no—
204C2×D4—122no—
205C2×D4—124no—
206C2×D4—121no—
207C2×D4—121no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—64no—
216C22×C4—61no—
217C22×C4—62no—
218C22×C4(C2×D3)⋊C412yes—
219C22×C4(C2×D3)⋊C412yes—
220C22×C4C2×C4×D311yescharacteristic
221C22×C4D3×D412yes—
222C22×C4#23712yes—
223C22×C4C2×#6711yescharacteristic
224M16—44no—
225C2×C8—42no—
226C4⋊C4—62no—
227C4⋊C4—124no—
228C22⋊C4—124no—
229C22⋊C4—124no—
230C22⋊C4—122no—
231C22⋊C4—124no—
232C22⋊C4—122no—
233C42—122no—
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#38611yescharacteristic
244C22×C3C2×#9912yes—
245C22×C3#45412yes—
246C2×D3—84no—
247C2×D3—81no—
248C2×D3—81no—
249C2×D3—81no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—24no—
253C3×C4—42no—
254Dic3—84no—
255Dic3—82no—
256Dic3—81no—
257C23—22no—
258C23—22no—
259C23—41no—
260C23—122no—
261C23—122no—
262C23—122no—
263C23—121no—
264C23#88211yescharacteristic
265C23#88611yescharacteristic
266C23C2×(C2×D3)⋊C411yescharacteristic
267Q8—48no—
268Q8—48no—
269D4—242no—
270D4—242no—
271D4—242no—
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—124no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—124no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—121no—
293C2×C4—124no—
294C2×C4#80614yes—
295C2×C4#67⋊C412yes—
296C2×C4#93912yes—
297C8—82no—
298C2×C3—22no—
299C2×C3—22no—
300C2×C3—41no—
301C2×C3#188811yescharacteristic
302C2×C3#189811yescharacteristic
303C2×C3#191211yescharacteristic
304D3—161no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22—244no—
314C22—241no—
315C22—241no—
316C22—241no—
317C22#475312yes—
318C22C2×#80612yes—
319C22#495811yescharacteristic
320C4—22no—
321C4—24no—
322C4—22no—
323C4—42no—
324C4—244no—
325C4—242no—
326C4—241no—
327C3#946611yescharacteristic, complemented, Sylow
328C2—22no—
329C2—22no—
330C2—41no—
331C2—481no—
332C2#6949611yescharacteristic
333C2#7010411yescharacteristic
334C2#7010511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1784C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1784D3
hypocenterC3#9466
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. #84062C2semidirect
3. C3×#1784C2semidirect
5. #67682C2semidirect
8. C2×#947C4semidirect
15. #5229C4semidirect
24. #1784D3semidirect
327. C3#9466semidirect

Maximal subgroups

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

subgroupindex
#840662
#840622
C3×#17842
#699782
#676822
#94663