Finite Group Databasebeta

← #10690930

Subgroups — #10690930

Order 768.

756 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 135 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#82080C211yesmaximal, characteristic
2C2×#4932C211yesmaximal, characteristic, complemented
3C3×#1817C211yesmaximal, characteristic, complemented
4C2×#4470C212yesmaximal
5#67675C212yesmaximal, complemented
6#9149—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#929—21no—
10C22×C4×Dic3—21no—
11C2×C3×C22⋊C8C2212yes—
12#5213—21no—
13#5212C2211yescharacteristic
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2212yes—
16#4940—22no—
17#4939C412yescomplemented
18#4932—24no—
19#4928C412yescomplemented
20C3×#336—22no—
21#4470—24no—
22#2854—31no—
23C2×#376—31no—
24#1817D311yescharacteristic, complemented
25C2×#327—32no—
26#1294—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C23×Dic3—22no—
32C22×D12C2×C411yescharacteristic
33C22×C3×C8—22no—
34C2×C3×C42—22no—
35C2×C3×C42—22no—
36C2×C3×C42C2×C412yes—
37C2×C3×C42D412yes—
38C2×#217—21no—
39#930—22no—
40#930—22no—
41#929—44no—
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4—22no—
44C2×(C2×D3)⋊C4—21no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C412yes—
47C2×Dic3⋊C4—22no—
48C2×C4×Dic3—24no—
49C2×C4×Dic3—28no—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—22no—
52C3⋊#95—24no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—21no—
56C3⋊#95D414yes—
57C2×#123—61no—
58C2×#121—31no—
59C22×C42—61no—
60C22×C42C2×D311yescharacteristic
61C2×C22⋊C8C2×D312yes—
62#390—32no—
63#389—32no—
64#388—62no—
65#386—61no—
66#380—31no—
67#376—64no—
68C2×#95—32no—
69C2×#95—61no—
70#336—22no—
71#327—64no—
72C24×C3C2×D411yescharacteristic
73C23×D3—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—24no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—42no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4112yes—
88C2×#67—41no—
89C2×#67—41no—
90C2×#67—42no—
91C2×#67—42no—
92C22×Dic3—28no—
93C22×Dic3—24no—
94C22×Dic3—21no—
95C22×Dic3—22no—
96C22×Dic3—44no—
97C22×Dic3—42no—
98C22×Dic3—42no—
99C2×D12—22no—
100C2×D12—42no—
101C2×C3×C8—44no—
102C2×C3×C8—42no—
103C3×C42—24no—
104C3×C42—24no—
105#217—42no—
106(C2×D3)⋊C4—44no—
107(C2×D3)⋊C4—44no—
108(C2×D3)⋊C4—42no—
109C4⋊Dic3—22no—
110C4⋊Dic3—42no—
111Dic3⋊C4—44no—
112C4×Dic3—216no—
113C22×D4—31no—
114C22×D4—61no—
115C23×C4—62no—
116C23×C4D1211yescharacteristic
117C23×C4D1211yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—22no—
120#123—124no—
121#121—62no—
122#121—62no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—32no—
125C2×C4⋊C4—62no—
126C2×C22⋊C4—61no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—61no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—64no—
133C2×C42—68no—
134C2×C42C4×D312yes—
135C2×C42#6712yes—
136C22⋊C8—22no—
137C22⋊C8—22no—
138#95—34no—
139#95—64no—
140#95—62no—
141#95—62no—
142#95—61no—
143C23×C3—22no—
144C23×C3—22no—
145C23×C3C2×C22⋊C411yescharacteristic
146C23×C3#12011yescharacteristic
147C23×C3#12311yescharacteristic
148C23×C3#13512yes—
149C23×C3AGL(1,ℤ/8)12yes—
150C22×D3—41no—
151C22×D3—42no—
152C22×D3—41no—
153C22×D3—41no—
154C22×D3—42no—
155C2×C3×C4—24no—
156C2×C3×C4—24no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4#10414yes—
169C2×C3×C4C4×D412yes—
170C2×C3×C4#12112yes—
171#67—82no—
172#67—82no—
173C2×Dic3—28no—
174C2×Dic3—44no—
175C2×Dic3—48no—
176C2×Dic3—48no—
177C2×Dic3—44no—
178C2×Dic3—41no—
179C2×Dic3—42no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184D12—82no—
185C3×C8—82no—
186C24—61no—
187C24C2×D1211yescharacteristic
188C2×D4—62no—
189C2×D4—122no—
190C2×D4—121no—
191C2×D4—121no—
192C2×D4—122no—
193C2×D4—122no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—42no—
202C22×C4—68no—
203C22×C4—64no—
204C22×C4—62no—
205C22×C4—61no—
206C22×C4—124no—
207C22×C4—122no—
208C22×C4—122no—
209C22×C4(C2×D3)⋊C412yes—
210C22×C4(C2×D3)⋊C412yes—
211C22×C4C2×C4×D311yescharacteristic
212C22×C4D3×D412yes—
213C22×C4#23712yes—
214C22×C4C2×#6711yescharacteristic
215C2×C8—44no—
216C2×C8—42no—
217C4⋊C4—62no—
218C4⋊C4—122no—
219C4⋊C4—124no—
220C22⋊C4—124no—
221C22⋊C4—122no—
222C22⋊C4—124no—
223C22⋊C4—122no—
224C42—24no—
225C42—24no—
226C42—616no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—24no—
232C22×C3—24no—
233C22×C3#38611yescharacteristic
234C22×C3#41012yes—
235C22×C3#41611yescharacteristic
236C22×C3#41811yescharacteristic
237C22×C3C2×#10412yes—
238C22×C3#44912yes—
239C22×C3#48212yes—
240C2×D3—81no—
241C2×D3—81no—
242C2×D3—81no—
243C2×D3—82no—
244C2×D3—82no—
245C3×C4—24no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—42no—
249Dic3—48no—
250Dic3—81no—
251Dic3—82no—
252C23—22no—
253C23—22no—
254C23—121no—
255C23—122no—
256C23—121no—
257C23—122no—
258C23—121no—
259C23#88211yescharacteristic
260C23#88611yescharacteristic
261C23#90412yes—
262C23#90812yes—
263C23C2×(C2×D3)⋊C411yescharacteristic
264D4—242no—
265D4—242no—
266D4—242no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—68no—
281C2×C4—128no—
282C2×C4—124no—
283C2×C4—128no—
284C2×C4—124no—
285C2×C4—122no—
286C2×C4—121no—
287C2×C4—122no—
288C2×C4—121no—
289C2×C4—122no—
290C2×C4—121no—
291C2×C4#82514yes—
292C2×C4#67⋊C412yes—
293C2×C4#93912yes—
294C8—82no—
295C2×C3—22no—
296C2×C3—22no—
297C2×C3#150012yes—
298C2×C3#150512yes—
299C2×C3#187511yescharacteristic
300C2×C3#189611yescharacteristic
301C2×C3#189911yescharacteristic
302D3—161no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—241no—
310C22—241no—
311C22—242no—
312C22—241no—
313C22—242no—
314C22#473412yes—
315C22#473712yes—
316C22#495811yescharacteristic
317C22#500612yes—
318C22#511311yescharacteristic
319C22#513111yescharacteristic
320C22C2×#82512yes—
321C4—24no—
322C4—22no—
323C4—22no—
324C4—42no—
325C4—128no—
326C4—241no—
327C4—242no—
328C3#914911yescharacteristic, complemented, Sylow
329C2—22no—
330C2—22no—
331C2—481no—
332C2#6953511yescharacteristic
333C2#6993311yescharacteristic
334C2#6996511yescharacteristic
335C2#7078812yes—
336C2#7083212yes—
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1817C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1817D3
hypocenterC3#9149
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#4932C2semidirect
3. C3×#1817C2semidirect
5. #67675C2semidirect
17. #4939C4semidirect
19. #4928C4semidirect
24. #1817D3semidirect
328. C3#9149semidirect

Maximal subgroups

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

subgroupindex
#820802
C2×#49322
C3×#18172
C2×#44702
#676752
#91493