Finite Group Databasebeta

← #11787174

Subgroups — #11787174

Order 1152.

834 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C22×#3792C211yesmaximal, characteristic, complemented
2C2×#64090C211yesmaximal, characteristic
3C2×C22⋊#622C211yesmaximal, characteristic, complemented
4C2×#63778C212yesmaximal, complemented
5#10590207C212yesmaximal, complemented
6C22×#638C411yescharacteristic
7C23×C3⋊Dic3C2211yescharacteristic
8C22×C4×C3⋊D3C2211yescharacteristic
9C2×#3792—22no—
10C2×#3792—24no—
11C22×#622C2212yes—
12C22×#622C2212yes—
13C2×C32×C22⋊C4C411yescharacteristic
14C2×#3752C411yescharacteristic
15C2×#3750C411yescharacteristic
16C2×C4×C3⋊Dic3C2211yescharacteristic
17#64090—22no—
18#64090—22no—
19C22⋊#622—22no—
20#63778—24no—
21C2×#67⋊C4—62no—
22C24×C32C2×C411yescharacteristic
23C23×C3⋊D3C2×C411yescharacteristic
24C22×C32×C4C2×C411yescharacteristic
25C22×C32×C4C2×C411yescharacteristic
26C2×#638—22no—
27C2×#638—22no—
28C2×#638—22no—
29C22×C3⋊Dic3—22no—
30C22×C3⋊Dic3—22no—
31C22×C3⋊Dic3C2×C411yescharacteristic
32C22×C3⋊Dic3C2×C411yescharacteristic
33C22×C3⋊Dic3D412yes—
34C22×C3⋊Dic3D412yes—
35C22×C3⋊Dic3C2311yescharacteristic
36C2×C4×C3⋊D3—21no—
37C2×C4×C3⋊D3—21no—
38C2×C4×C3⋊D3—22no—
39C2×C4×C3⋊D3—22no—
40#3792—28no—
41C2×#622—22no—
42C2×#622—22no—
43C2×#622—22no—
44C2×#622—22no—
45C2×#622—22no—
46C2×#622D414yescomplemented
47C32×C22⋊C4—22no—
48#3752—22no—
49#3750—22no—
50C4×C3⋊Dic3—21no—
51C4×C3⋊Dic3—21no—
52#1870—91nomaximal, Sylow
53C22×#67—62no—
54C23×Dic3—62no—
55C22×C4×D3—62no—
56C2×C3×C22⋊C4—22no—
57C2×(C2×D3)⋊C4—62no—
58C2×Dic3⋊C4—62no—
59C2×C4×Dic3—62no—
60#67⋊C4—68no—
61#67⋊C4—68no—
62C23×C32—21no—
63C23×C32—21no—
64C23×C32—22no—
65C23×C32—22no—
66C23×C32C22⋊C412yes—
67C23×C32C22×C411yescharacteristic
68C22×C3⋊D3—21no—
69C22×C3⋊D3—21no—
70C22×C3⋊D3—22no—
71C22×C3⋊D3—22no—
72C22×C3⋊D3M1612yescomplemented
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4—22no—
76C2×C32×C4—22no—
77#638—28no—
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3—21no—
80C2×C3⋊Dic3—22no—
81C2×C3⋊Dic3—22no—
82C2×C3⋊Dic3—24no—
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3C22⋊C412yes—
85C2×C3⋊Dic3M1612yescomplemented
86C2×C3⋊Dic3C2×D412yes—
87C2×C3⋊Dic3C2×D411yescharacteristic
88C2×C3⋊Dic3C2×D411yescharacteristic
89C2×C3⋊Dic3#4112yes—
90C4×C3⋊D3—41no—
91C4×C3⋊D3—42no—
92C4×C3⋊D3—41no—
93#622—24no—
94#622—42no—
95C22×M16—91no—
96C2×C4×D4—91no—
97C2×C22⋊C8—92no—
98C2×C22⋊C8—91no—
99#336—92no—
100C24×C3—22no—
101C23×D3—62no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C2×#67—68no—
105C2×#67—68no—
106C2×#67—68no—
107C22×Dic3—62no—
108C22×Dic3—62no—
109C22×Dic3—64no—
110C22×Dic3—64no—
111C22×Dic3—62no—
112C22×Dic3—124no—
113C22×Dic3—124no—
114C2×C4×D3—64no—
115C2×C4×D3—64no—
116C2×C4×D3—124no—
117C2×C4×D3—124no—
118C3×C22⋊C4—28no—
119(C2×D3)⋊C4—68no—
120Dic3⋊C4—68no—
121C4×Dic3—64no—
122C4×Dic3—64no—
123C22×C32—22no—
124C22×C32—22no—
125C22×C32—22no—
126C22×C32—42no—
127C22×C32—42no—
128C22×C32C2×C22⋊C411yescharacteristic
129C22×C32C4×D412yes—
130C22×C32C2×M1611yescharacteristic
131C22×C32C2×M1611yescharacteristic
132C22×C32#13112yes—
133C2×C3⋊D3—22no—
134C2×C3⋊D3—24no—
135C2×C3⋊D3—41no—
136C2×C3⋊D3—41no—
137C2×C3⋊D3—42no—
138C32×C4—41no—
139C32×C4—41no—
140C3⋊Dic3—22no—
141C3⋊Dic3—22no—
142C3⋊Dic3#12011yescharacteristic
143C3⋊Dic3#12112yes—
144C3⋊Dic3#12311yescharacteristic
145C22×D4—91no—
146C23×C4—91no—
147C23×C4—91no—
148C2×M16—184no—
149C2×M16—182no—
150C22×C8—92no—
151C22×C8—92no—
152C4×D4—182no—
153C4×D4—182no—
154C2×C4⋊C4—91no—
155C2×C22⋊C4—91no—
156C2×C22⋊C4#15711yescharacteristic
157C2×C42—91no—
158C22⋊C8—184no—
159C22⋊C8—182no—
160C23×C3—24no—
161C23×C3—24no—
162C23×C3—22no—
163C23×C3—24no—
164C23×C3—44no—
165C23×C3—44no—
166C22×D3—64no—
167C22×D3—64no—
168C22×D3—64no—
169C22×D3—124no—
170C22×D3—124no—
171C2×C3×C4—24no—
172C2×C3×C4—24no—
173C2×C3×C4—44no—
174C2×C3×C4—44no—
175#67—632no—
176C2×Dic3—64no—
177C2×Dic3—62no—
178C2×Dic3—62no—
179C2×Dic3—64no—
180C2×Dic3—64no—
181C2×Dic3—64no—
182C2×Dic3—64no—
183C2×Dic3—64no—
184C2×Dic3—124no—
185C2×Dic3—128no—
186C2×Dic3—124no—
187C2×Dic3—124no—
188C4×D3—124no—
189C4×D3—128no—
190C4×D3—124no—
191C2×C32—42no—
192C2×C32#38611yescharacteristic
193C2×C32#40711yescharacteristic
194C2×C32#40811yescharacteristic
195C2×C32#43512yes—
196C2×C32#43612yes—
197C3⋊D3—42no—
198C24—91no—
199C24C2×#15711yescharacteristic
200C2×D4—182no—
201C2×D4—182no—
202C2×D4—182no—
203C22×C4—91no—
204C22×C4—91no—
205C22×C4—91no—
206C22×C4—92no—
207C22×C4—92no—
208C22×C4—182no—
209C22×C4—181no—
210C22×C4—181no—
211C22×C4—182no—
212C22×C4—182no—
213C22×C4—182no—
214C22×C4C2×#15711yescharacteristic
215C22×C4C2×#15711yescharacteristic
216M16—188no—
217C2×C8—94no—
218C2×C8—182no—
219C2×C8—182no—
220C2×C8—182no—
221C2×C8—182no—
222C2×C8—182no—
223C4⋊C4—182no—
224C22⋊C4—22no—
225C22⋊C4—182no—
226C42—181no—
227C42—181no—
228C22×C3—28no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—24no—
232C22×C3—24no—
233C22×C3—22no—
234C22×C3—44no—
235C22×C3—44no—
236C22×C3—48no—
237C22×C3—48no—
238C2×D3—68no—
239C2×D3—128no—
240C2×D3—124no—
241C2×D3—124no—
242C2×D3—128no—
243C3×C4—44no—
244C3×C4—44no—
245Dic3—62no—
246Dic3—64no—
247Dic3—62no—
248Dic3—68no—
249Dic3—124no—
250C32#187011yescharacteristic, complemented, Sylow
251C23—21no—
252C23—21no—
253C23—22no—
254C23—22no—
255C23—92no—
256C23—181no—
257C23—182no—
258C23—182no—
259C23—181no—
260C23C22⋊#15712yes—
261C23C22×#15711yescharacteristic
262D4—188no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—92no—
268C2×C4—92no—
269C2×C4—91no—
270C2×C4—91no—
271C2×C4—92no—
272C2×C4—92no—
273C2×C4—182no—
274C2×C4—184no—
275C2×C4—182no—
276C2×C4—181no—
277C2×C4—182no—
278C2×C4—181no—
279C2×C4—361no—
280C2×C4—362no—
281C2×C4—361no—
282C8—184no—
283C8—362no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—48no—
290D3—128no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—182no—
297C22—184no—
298C22—361no—
299C22—361no—
300C22—362no—
301C22#6428311yescharacteristic
302C22#6428311yescharacteristic
303C22#6428712yes—
304C22D4×#15712yes—
305C22C2×C22⋊#15711yescharacteristic
306C4—41no—
307C4—41no—
308C4—91no—
309C4—92no—
310C4—91no—
311C4—182no—
312C4—182no—
313C3—22no—
314C2—42no—
315C2—362no—
316C2#1059374112yes—
317C2#1059374212yes—
318C2#1059380211yescharacteristic
319C2#1059382211yescharacteristic
320C2#1059382611yescharacteristic
subgroupG/H
center, FrattiniC23C22×#157
commutator, socleC23×C32C22×C4
FittingC2×C32×C22⋊C4C4
prime radicalC22×#638C4
cosocleC22×C3⋊Dic3C23
hypercenterC2×C22⋊C4#157
hypocenterC32#1870
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#3792C2semidirect
3. C2×C22⋊#622C2semidirect
4. C2×#63778C2semidirect
5. #10590207C2semidirect
46. C2×#622D4semidirect
72. C22×C3⋊D3M16semidirect
85. C2×C3⋊Dic3M16semidirect
250. C32#1870semidirect

Maximal subgroups

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

subgroupindex
C22×#37922
C2×#640902
C2×C22⋊#6222
C2×#637782
#105902072
#18709