Finite Group Databasebeta

← #11788051

Subgroups — #11788051

Order 1152.

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

normal subgroups only

This group has 658 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
1C23×#622C211yesmaximal, characteristic
2C2×#63982C211yesmaximal, characteristic
3C2×#63740C212yesmaximal, complemented
4#10591720C211yesmaximal, characteristic
5#10590075C212yesmaximal, complemented
6C23×C3⋊Dic3C2211yescharacteristic
7C22×#627C2211yescharacteristic
8C22×#622—22no—
9C22×#622—24no—
10C22×#622C2212yes—
11C2×C3×#217—21no—
12C2×C3×#217—21no—
13#63982—22no—
14#63982C414yescomplemented
15C2×#3724C2212yes—
16C2×#3724C2211yescharacteristic
17C2×Dic3⋊Dic3—21no—
18#63740—24no—
19#63579—21no—
20C2×#878—61no—
21#4942—61no—
22C24×C32D411yescharacteristic
23C22×C3⋊Dic3—22no—
24C22×C3⋊Dic3—22no—
25C22×C3⋊Dic3C2×C412yes—
26C22×C3⋊Dic3D411yescharacteristic
27C22×C3⋊Dic3D412yes—
28C22×C3⋊Dic3C2311yescharacteristic
29C22×C3×Dic3—21no—
30C22×C3×Dic3—21no—
31C22×C3×Dic3—22no—
32C2×#627—22no—
33C2×#627—22no—
34C2×#627—21no—
35C2×#627C2×C412yes—
36C2×#622—28no—
37C2×#622—22no—
38C2×#622—22no—
39C2×#622D414yes—
40C3×#217—24no—
41C3×#217—42no—
42#3724—21no—
43#3724—22no—
44#3724—22no—
45#3724C2×C412yes—
46Dic3⋊Dic3—24no—
47#1892—91nomaximal, Sylow
48C23×Dic3—61no—
49C23×Dic3—61no—
50C22×Dic6—61no—
51C2×C3×C22⋊C4—61no—
52C2×C3×C22⋊C4—61no—
53C2×#217—21no—
54C2×#217—21no—
55C2×C4⋊Dic3—61no—
56C2×Dic3⋊C4—61no—
57C2×Dic3⋊C4—62no—
58C2×Dic3⋊C4—61no—
59#878—64no—
60#878—64no—
61C3⋊#95—62no—
62C23×C32—22no—
63C23×C32—22no—
64C23×C32C22⋊C412yes—
65C23×C32SD1612yes—
66C23×C32Q1612yes—
67C23×C32C2×D411yescharacteristic
68C2×C3⋊Dic3—22no—
69C2×C3⋊Dic3—24no—
70C2×C3⋊Dic3—22no—
71C2×C3⋊Dic3—22no—
72C2×C3⋊Dic3C22⋊C412yes—
73C2×C3⋊Dic3C22×C411yescharacteristic
74C2×C3⋊Dic3C2×D412yes—
75C2×C3⋊Dic3C2×D411yescharacteristic
76C2×C3⋊Dic3#4112yes—
77C2×C3×Dic3—22no—
78C2×C3×Dic3—22no—
79C2×C3×Dic3—42no—
80C2×C3×Dic3—42no—
81C2×C3×Dic3—42no—
82C2×C3×Dic3—42no—
83C2×C3×Dic3—42no—
84#627—24no—
85#627—41no—
86#627—41no—
87#622—24no—
88C23×C8—91no—
89C2×C22⋊Q8—91no—
90C2×#103—92no—
91#380—91no—
92#340—92no—
93C24×C3—21no—
94C24×C3—21no—
95C22×C3×C4—61no—
96C22×C3×C4—61no—
97C22×C3×C4—62no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—22no—
101C22×Dic3—61no—
102C22×Dic3—61no—
103C22×Dic3—61no—
104C22×Dic3—62no—
105C22×Dic3—62no—
106C22×Dic3—61no—
107C22×Dic3—62no—
108C22×Dic3—62no—
109C22×Dic3—122no—
110C22×Dic3—122no—
111C22×Dic3—122no—
112C22×Dic3—122no—
113C2×Dic6—62no—
114C2×Dic6—62no—
115C2×Dic6—122no—
116C2×Dic6—122no—
117C3×C22⋊C4—64no—
118C3×C22⋊C4—122no—
119#217—24no—
120#217—42no—
121C4⋊Dic3—64no—
122Dic3⋊C4—62no—
123Dic3⋊C4—64no—
124Dic3⋊C4—62no—
125Dic3⋊C4—122no—
126Dic3⋊C4—122no—
127C22×C32—22no—
128C22×C32—22no—
129C22×C32—24no—
130C22×C32—24no—
131C22×C32#10314yescomplemented
132C22×C32C2×C22⋊C411yescharacteristic
133C22×C32#12112yes—
134C22×C32C2×SD1611yescharacteristic
135C22×C32C2×Q1611yescharacteristic
136C22×C32#13512yes—
137C3⋊Dic3—22no—
138C3⋊Dic3C4×D412yes—
139C3⋊Dic3#12011yescharacteristic
140C3⋊Dic3#12311yescharacteristic
141C3×Dic3—42no—
142C3×Dic3—42no—
143C3×Dic3—82no—
144C22×Q8—91no—
145C23×C4—91no—
146C22×C8—92no—
147C22×C8—184no—
148C22×C8—182no—
149C22⋊Q8—94no—
150C22⋊Q8—182no—
151C2×C4⋊C4—91no—
152C2×C4⋊C4—92no—
153C2×C4⋊C4—181no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156#103—184no—
157#95—181no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—42no—
167C23×C3—42no—
168C23×C3—42no—
169C23×C3—42no—
170C2×C3×C4—62no—
171C2×C3×C4—62no—
172C2×C3×C4—122no—
173C2×C3×C4—122no—
174C2×C3×C4—122no—
175C2×C3×C4—122no—
176C2×C3×C4—122no—
177C2×Dic3—22no—
178C2×Dic3—22no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187C2×Dic3—62no—
188C2×Dic3—62no—
189C2×Dic3—61no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—122no—
195C2×Dic3—122no—
196C2×Dic3—124no—
197C2×Dic3—122no—
198C2×Dic3—124no—
199C2×Dic3—122no—
200C2×Dic3—122no—
201C2×Dic3—122no—
202Dic6—124no—
203Dic6—122no—
204Dic6—122no—
205C2×C32—24no—
206C2×C32#38611yescharacteristic
207C2×C32C2×#10311yescharacteristic
208C2×C32#41611yescharacteristic
209C2×C32#46512yes—
210C2×C32#46712yes—
211C24GO+(4,𝔽2)11yescharacteristic
212C2×Q8—92no—
213C2×Q8—181no—
214C2×Q8—182no—
215C2×Q8—182no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—62no—
219C22×C4—91no—
220C22×C4—91no—
221C22×C4—92no—
222C22×C4—92no—
223C22×C4—182no—
224C22×C4—182no—
225C2×C8—94no—
226C2×C8—182no—
227C2×C8—182no—
228C2×C8—188no—
229C4⋊C4—92no—
230C4⋊C4—181no—
231C4⋊C4—184no—
232C4⋊C4—182no—
233C4⋊C4—182no—
234C22⋊C4—64no—
235C22⋊C4—122no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—24no—
247C22×C3—24no—
248C22×C3—42no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3—44no—
253C22×C3—44no—
254C22×C3—44no—
255C22×C3—44no—
256C3×C4—122no—
257C3×C4—122no—
258C3×C4—242no—
259Dic3—42no—
260Dic3—42no—
261Dic3—62no—
262Dic3—61no—
263Dic3—61no—
264Dic3—62no—
265Dic3—61no—
266Dic3—61no—
267Dic3—82no—
268Dic3—122no—
269Dic3—122no—
270C32#189211yescharacteristic, complemented, Sylow
271C23—22no—
272C23—22no—
273C23D32⋊C412yes—
274C23#377512yes—
275C23#377612yes—
276C23C2×GO+(4,𝔽2)11yescharacteristic
277Q8—184no—
278Q8—361no—
279Q8—361no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—92no—
285C2×C4—92no—
286C2×C4—92no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—182no—
293C2×C4—184no—
294C2×C4—182no—
295C2×C4—182no—
296C8—184no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—22no—
306C2×C3—22no—
307C2×C3—44no—
308C2×C3—44no—
309C22—22no—
310C22—22no—
311C22—24no—
312C22—24no—
313C22#6374014yescomplemented
314C22C2×D32⋊C411yescharacteristic
315C22#6423712yes—
316C22C2×#377511yescharacteristic
317C22C2×#377611yescharacteristic
318C22#6424212yes—
319C4—91no—
320C4—92no—
321C4—91no—
322C4—122no—
323C4—122no—
324C4—182no—
325C4—242no—
326C3—21no—
327C3—21no—
328C2—24no—
329C2#1059349811yescharacteristic
330C2C2×#6374011yescharacteristic
331C2#1059351511yescharacteristic
332C2#1059352612yes—
333C2#1059352712yes—
subgroupG/H
center, FrattiniC23C2×GO+(4,𝔽2)
commutatorC2×C3⋊Dic3C22×C4
Fitting, prime radicalC24×C32D4
socleC23×C32C2×D4
cosocleC22×C3⋊Dic3C23
hypercenterC24GO+(4,𝔽2)
hypocenterC32#1892
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
3. C2×#63740C2semidirect
5. #10590075C2semidirect
14. #63982C4semidirect
131. C22×C32#103semidirect
270. C32#1892semidirect
313. C22#63740semidirect

Maximal subgroups

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

subgroupindex
C23×#6222
C2×#639822
C2×#637402
#105917202
#105900752
#18929