Finite Group Databasebeta

← #10706331

Subgroups — #10706331

Order 768.

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

normal subgroups only

This group has 440 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
1#82374C211yesmaximal, characteristic
2C3×#1800C211yesmaximal, characteristic
3#71600C211yesmaximal, characteristic
4#67496C214yesmaximal, complemented
5#11903—31nomaximal, Sylow
6C2×C3×C4×D4C2211yescharacteristic
7C22×C4⋊Dic3C411yescharacteristic
8C3×#423—21no—
9C2×C3×#100—21no—
10#5220—21no—
11#5212—21no—
12#5212C411yescharacteristic
13C2×#830—21no—
14#4986C2212yes—
15#4949C412yes—
16#4946—21no—
17#4928—21no—
18C3×#343C2212yes—
19#4528C2212yes—
20#4525—21no—
21#2487—31no—
22#1800D311yescharacteristic
23#1459—31no—
24#1353—34no—
25C22×C3×D4D411yescharacteristic
26C23×C3×C4C2×C411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×Dic3—21no—
29C2×C3×M16—22no—
30C3×C4×D4—41no—
31C3×C4×D4—41no—
32C2×C3×C4⋊C4C2311yescharacteristic
33C2×C3×C22⋊C4C2×C411yescharacteristic
34C2×C3×C22⋊C4D411yescharacteristic
35C2×C3×C42D411yescharacteristic
36C2×#217—21no—
37C2×#217—22no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3—41no—
41C2×C4⋊Dic3—41no—
42C2×C4⋊Dic3C2×C412yes—
43C2×Dic3⋊C4—22no—
44C22×C3⋊C8—41no—
45C3×C4⋊C8—41no—
46C3×C4⋊C8—41no—
47C3×#100—42no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50C3⋊#95C2×C412yes—
51#830—22no—
52#830—41no—
53C2×C4×D4C2×D311yescharacteristic
54C22×C4⋊C4—31no—
55#423—21no—
56C2×#100—21no—
57C2×C22⋊C8—61no—
58#400—32no—
59#399—32no—
60#389—61no—
61#386—61no—
62#385—61no—
63#380—31no—
64#380—61no—
65#343C2×D312yes—
66#324—32no—
67#323—61no—
68C24×C3C22⋊C411yescharacteristic
69C24×C3C22⋊C411yescharacteristic
70C2×C3×D4—41no—
71C2×C3×D4—41no—
72C2×C3×D4SD1612yes—
73C2×C3×D4Q1612yes—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—42no—
79C22×C3×C4—42no—
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—42no—
88C22×Dic3—82no—
89C3×M16—42no—
90C3×M16—42no—
91C2×C3×C8—42no—
92C3×C4⋊C4—41no—
93C3×C22⋊C4—41no—
94C3×C22⋊C4—41no—
95C3×C42—21no—
96C3×C42—21no—
97#217—42no—
98#217—82no—
99C4⋊Dic3—42no—
100C4⋊Dic3—41no—
101C4⋊Dic3—41no—
102C4⋊Dic3—42no—
103Dic3⋊C4—82no—
104C2×C3⋊C8—42no—
105C2×C3⋊C8—82no—
106C22×D4#6711yescharacteristic
107C23×C4—61no—
108C23×C4C4×D311yescharacteristic
109C23×C4D1211yescharacteristic
110C2×M16—22no—
111C22×C8—121no—
112C4×D4—41no—
113C4×D4—41no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—121no—
119C2×C4⋊C4—121no—
120C2×C4⋊C4C22×D311yescharacteristic
121C2×C22⋊C4—61no—
122C2×C22⋊C4—62no—
123C2×C22⋊C4C4×D311yescharacteristic
124C2×C22⋊C4D1211yescharacteristic
125C2×C42#6711yescharacteristic
126C4⋊C8—41no—
127C4⋊C8—41no—
128#100—42no—
129C22⋊C8—62no—
130C22⋊C8—121no—
131#95—32no—
132#95—62no—
133#95—62no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—21no—
137C23×C3—21no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3—41no—
141C23×C3—41no—
142C23×C3#10312yes—
143C23×C3#10312yes—
144C23×C3C2×C22⋊C411yescharacteristic
145C3×D4—42no—
146C3×D4—42no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—81no—
160C2×C3×C4#9911yescharacteristic
161C2×C3×C4#9911yescharacteristic
162C2×C3×C4C2×SD1611yescharacteristic
163C2×C3×C4C2×Q1611yescharacteristic
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C2×Dic3—84no—
171C3×C8—82no—
172C3⋊C8—82no—
173C24(C2×D3)⋊C411yescharacteristic
174C24(C2×D3)⋊C411yescharacteristic
175C2×D4—41no—
176C2×D4—41no—
177C2×D4#21412yes—
178C2×D4#21612yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—62no—
186C22×C4—62no—
187C22×C4—122no—
188C22×C4—242no—
189C22×C4(C2×D3)⋊C411yescharacteristic
190C22×C4(C2×D3)⋊C411yescharacteristic
191C22×C4C2×C4×D311yescharacteristic
192C22×C4C2×D1211yescharacteristic
193C22×C4C2×#6711yescharacteristic
194M16—42no—
195M16—42no—
196C2×C8—42no—
197C2×C8—122no—
198C2×C8—242no—
199C4⋊C4—41no—
200C4⋊C4—122no—
201C4⋊C4—122no—
202C4⋊C4—121no—
203C4⋊C4—121no—
204C4⋊C4—242no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—122no—
208C22⋊C4—242no—
209C42—21no—
210C42—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—81no—
223C22×C3C2×#9911yescharacteristic
224C22×C3C2×#10311yescharacteristic
225C22×C3#42111yescharacteristic
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4—41no—
232C3×C4#35212yes—
233Dic3—82no—
234Dic3—82no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#81012yes—
244C23#81012yes—
245C23C2×(C2×D3)⋊C411yescharacteristic
246D4—42no—
247D4—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—81no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—244no—
268C2×C4#80611yescharacteristic
269C2×C4#80611yescharacteristic
270C2×C4C2×#21411yescharacteristic
271C2×C4C2×#21611yescharacteristic
272C8—82no—
273C8—242no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—42no—
277C2×C3—42no—
278C2×C3#152011yescharacteristic
279C2×C3C2×#35211yescharacteristic
280C2×C3#213411yescharacteristic
281C22—21no—
282C22—21no—
283C22—22no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22C2×#80611yescharacteristic
294C22C2×#81011yescharacteristic
295C22#500511yescharacteristic
296C4—21no—
297C4—21no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—242no—
303C4—242no—
304C4#447612yes—
305C3#1190311yescharacteristic, complemented, Sylow
306C2—21no—
307C2—21no—
308C2—42no—
309C2—42no—
310C2#6951011yescharacteristic
311C2C2×#447611yescharacteristic
312C2#7014911yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1800C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C4⋊C4C23
hypercenter#1800D3
hypocenterC3#11903
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
4. #67496C2semidirect
305. C3#11903semidirect

Maximal subgroups

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

subgroupindex
#823742
C3×#18002
#716002
#674962
#119033