Finite Group Databasebeta

← #11787176

Subgroups — #11787176

Order 1152.

834 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 92 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×#3788C211yesmaximal, characteristic, complemented
2C2×#64102C211yesmaximal, characteristic
3C2×#63778C211yesmaximal, characteristic, complemented
4C2×#63778C212yesmaximal, complemented
5#10590207C212yesmaximal, complemented
6C22×C4⋊(C3⋊D3)C411yescharacteristic
7C22×C4×C3⋊D3C2211yescharacteristic
8C22×C4×C3⋊D3C2211yescharacteristic
9C22×#622C2212yes—
10C22×#622C2212yes—
11C2×#3788—22no—
12C2×#3788—24no—
13C2×C32×C4⋊C4C411yescharacteristic
14C2×#3752C411yescharacteristic
15C2×#3752C411yescharacteristic
16C2×C4×C3⋊Dic3C2211yescharacteristic
17#64102—22no—
18#64102—22no—
19#63778—22no—
20#63778—24no—
21C2×#893—62no—
22C23×C3⋊D3C2×C411yescharacteristic
23C23×C3⋊D3C2×C411yescharacteristic
24C22×C32×C4C2×C411yescharacteristic
25C22×C32×C4C2×C411yescharacteristic
26C22×C32×C4C2×C411yescharacteristic
27C22×C3⋊Dic3C2×C411yescharacteristic
28C22×C3⋊Dic3C2311yescharacteristic
29C2×C4⋊(C3⋊D3)—22no—
30C2×C4⋊(C3⋊D3)—22no—
31C2×C4⋊(C3⋊D3)—22no—
32C2×C4×C3⋊D3—22no—
33C2×C4×C3⋊D3—22no—
34C2×C4×C3⋊D3—21no—
35C2×C4×C3⋊D3—21no—
36C2×C4×C3⋊D3—22no—
37C2×C4×C3⋊D3—22no—
38C2×C4×C3⋊D3D412yes—
39C2×C4×C3⋊D3D412yes—
40C2×#622—22no—
41C2×#622—22no—
42C2×#622—22no—
43C2×#622—22no—
44C2×#622—22no—
45C2×#622D414yescomplemented
46#3788—28no—
47C32×C4⋊C4—22no—
48#3752—22no—
49#3752—22no—
50C4×C3⋊Dic3—21no—
51C4×C3⋊Dic3—21no—
52#1870—91nomaximal, Sylow
53C22×D12—62no—
54C22×C4×D3—62no—
55C22×C4×D3—62no—
56C2×C3×C4⋊C4—22no—
57C2×(C2×D3)⋊C4—62no—
58C2×(C2×D3)⋊C4—62no—
59C2×C4×Dic3—62no—
60#893—68no—
61#893—68no—
62C23×C32C22×C411yescharacteristic
63C22×C3⋊D3—21no—
64C22×C3⋊D3—21no—
65C22×C3⋊D3—21no—
66C22×C3⋊D3—21no—
67C22×C3⋊D3—22no—
68C22×C3⋊D3—22no—
69C22×C3⋊D3—22no—
70C22×C3⋊D3—22no—
71C22×C3⋊D3C22⋊C412yes—
72C22×C3⋊D3M1612yescomplemented
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C32×C4—22no—
77C2×C32×C4—22no—
78C2×C32×C4C22⋊C412yes—
79C2×C32×C4M1612yescomplemented
80C2×C3⋊Dic3—21no—
81C2×C3⋊Dic3—22no—
82C2×C3⋊Dic3C2×D412yes—
83C2×C3⋊Dic3C2×D411yescharacteristic
84C2×C3⋊Dic3C2×D411yescharacteristic
85C2×C3⋊Dic3#4112yes—
86C4⋊(C3⋊D3)—28no—
87C4×C3⋊D3—22no—
88C4×C3⋊D3—24no—
89C4×C3⋊D3—22no—
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—
100C23×D3—62no—
101C23×D3—62no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×Dic3—62no—
106C22×Dic3—62no—
107C2×D12—68no—
108C2×D12—68no—
109C2×D12—68no—
110C2×C4×D3—64no—
111C2×C4×D3—64no—
112C2×C4×D3—64no—
113C2×C4×D3—64no—
114C2×C4×D3—124no—
115C2×C4×D3—124no—
116C2×C4×D3—124no—
117C2×C4×D3—124no—
118C3×C4⋊C4—28no—
119(C2×D3)⋊C4—68no—
120(C2×D3)⋊C4—68no—
121C4×Dic3—64no—
122C4×Dic3—64no—
123C22×C32C2×C22⋊C411yescharacteristic
124C22×C32C4×D412yes—
125C22×C32C2×M1611yescharacteristic
126C22×C32C2×M1611yescharacteristic
127C22×C32#13112yes—
128C2×C3⋊D3—22no—
129C2×C3⋊D3—22no—
130C2×C3⋊D3—22no—
131C2×C3⋊D3—22no—
132C2×C3⋊D3—24no—
133C2×C3⋊D3—41no—
134C2×C3⋊D3—41no—
135C2×C3⋊D3—42no—
136C2×C3⋊D3—42no—
137C2×C3⋊D3—42no—
138C32×C4—22no—
139C32×C4—22no—
140C32×C4—41no—
141C3⋊Dic3—41no—
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#15711yescharacteristic
155C2×C22⋊C4—91no—
156C2×C22⋊C4—91no—
157C2×C42—91no—
158C22⋊C8—184no—
159C22⋊C8—182no—
160C23×C3—22no—
161C22×D3—64no—
162C22×D3—64no—
163C22×D3—64no—
164C22×D3—64no—
165C22×D3—64no—
166C22×D3—64no—
167C22×D3—124no—
168C22×D3—124no—
169C22×D3—124no—
170C22×D3—124no—
171C2×C3×C4—24no—
172C2×C3×C4—24no—
173C2×C3×C4—24no—
174C2×C3×C4—24no—
175C2×C3×C4—24no—
176C2×C3×C4—44no—
177C2×C3×C4—44no—
178C2×Dic3—62no—
179C2×Dic3—62no—
180C2×Dic3—64no—
181C2×Dic3—64no—
182C2×Dic3—64no—
183C2×Dic3—124no—
184D12—632no—
185C4×D3—124no—
186C4×D3—128no—
187C4×D3—124no—
188C4×D3—124no—
189C4×D3—128no—
190C4×D3—124no—
191C2×C32#38611yescharacteristic
192C2×C32#40711yescharacteristic
193C2×C32#40811yescharacteristic
194C2×C32#43512yes—
195C2×C32#43612yes—
196C3⋊D3—42no—
197C3⋊D3—42no—
198C24—91no—
199C24—91no—
200C2×D4—182no—
201C2×D4—182no—
202C2×D4—182no—
203C22×C4—91no—
204C22×C4—92no—
205C22×C4—92no—
206C22×C4—91no—
207C22×C4—182no—
208C22×C4—182no—
209C22×C4—182no—
210C22×C4—181no—
211C22×C4—181no—
212C22×C4—182no—
213C22×C4C2×#15711yescharacteristic
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—22no—
224C22⋊C4—182no—
225C22⋊C4—182no—
226C42—181no—
227C42—181no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—24no—
231C22×C3—24no—
232C22×C3—22no—
233C2×D3—68no—
234C2×D3—68no—
235C2×D3—124no—
236C2×D3—124no—
237C2×D3—128no—
238C2×D3—124no—
239C2×D3—124no—
240C2×D3—128no—
241C2×D3—128no—
242C2×D3—128no—
243C3×C4—28no—
244C3×C4—44no—
245C3×C4—44no—
246Dic3—62no—
247Dic3—64no—
248Dic3—62no—
249Dic3—124no—
250C32#187011yescharacteristic, complemented, Sylow
251C23—92no—
252C23—92no—
253C23—181no—
254C23—181no—
255C23—181no—
256C23—182no—
257C23—182no—
258C23—182no—
259C23—182no—
260C23—181no—
261C23C22×#15711yescharacteristic
262D4—188no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—92no—
269C2×C4—92no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—182no—
273C2×C4—184no—
274C2×C4—182no—
275C2×C4—182no—
276C2×C4—181no—
277C2×C4—361no—
278C2×C4—362no—
279C2×C4—361no—
280C2×C4#378812yescomplemented
281C2×C4C22⋊#15712yes—
282C8—184no—
283C8—362no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—24no—
288C2×C3—24no—
289D3—128no—
290D3—128no—
291C22—182no—
292C22—182no—
293C22—182no—
294C22—182no—
295C22—184no—
296C22—361no—
297C22—361no—
298C22—362no—
299C22—362no—
300C22—362no—
301C22C2×#378811yescharacteristic
302C22#6428311yescharacteristic
303C22D4×#15712yes—
304C22#6428912yes—
305C22C2×C22⋊#15711yescharacteristic
306C4—22no—
307C4—22no—
308C4—41no—
309C4—91no—
310C4—92no—
311C4—91no—
312C4—361no—
313C3—22no—
314C2—362no—
315C2—362no—
316C2#1059375112yes—
317C2#1059375212yes—
318C2#1059380311yescharacteristic
319C2#1059382611yescharacteristic
320C2#1059383411yescharacteristic
subgroupG/H
center, FrattiniC23C22×#157
commutator, socleC23×C32C22×C4
FittingC2×C32×C4⋊C4C4
prime radicalC22×C4⋊(C3⋊D3)C4
cosocleC22×C3⋊Dic3C23
hypercenterC2×C4⋊C4#157
hypocenterC32#1870
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#3788C2semidirect
3. C2×#63778C2semidirect
4. C2×#63778C2semidirect
5. #10590207C2semidirect
45. C2×#622D4semidirect
72. C22×C3⋊D3M16semidirect
79. C2×C32×C4M16semidirect
250. C32#1870semidirect
280. C2×C4#3788semidirect

Maximal subgroups

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

subgroupindex
C22×#37882
C2×#641022
C2×#637782
#105902072
#18709