Finite Group Databasebeta

← #11778331

Subgroups — #11778331

Order 1152.

655 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 655 conjugacy classes of subgroups — too many to draw. Only its 80 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#10594930C211yesmaximal, characteristic
2#10592929C211yesmaximal, characteristic, complemented
3C3×#5227C212yesmaximal, complemented
4#10589150C212yesmaximal, complemented
5#10589148C211yesmaximal, characteristic, complemented
6#70975—32nomaximal
7C22×C32×Q8C2211yescharacteristic
8C2×#3805—21no—
9C2×#3750—21no—
10C2×#3750C2211yescharacteristic
11C2×C3×#215—22no—
12C3×#930C2212yes—
13#63980—21no—
14#63899—22no—
15#63885—22no—
16#63658—21no—
17#63658C2211yescharacteristic
18C3×#835—22no—
19C3×#830C2212yes—
20#5597—32no—
21C3×#448—32no—
22#5230—32no—
23#5230—61no—
24#5227D312yescomplemented
25#4537—32no—
26#4470—32no—
27#4469—32no—
28C2×C32×Q8—22no—
29C2×C32×Q8—42no—
30C22×C32×C4—21no—
31C22×C32×C4C2311yescharacteristic
32C22×C3⋊Dic3—21no—
33C22×C3⋊Dic3D411yescharacteristic
34C22×C3⋊Dic3D411yescharacteristic
35C2×C3×#67—22no—
36C2×C3×D12D412yes—
37C2×#627—21no—
38C2×#625—21no—
39C2×#625—21no—
40#3805—42no—
41#3805—42no—
42C2×D3×Dic3—22no—
43#3750—22no—
44#3750—42no—
45C3×#215—44no—
46C3×(C2×D3)⋊C4—22no—
47C3×C4⋊Dic3D412yes—
48C2×C3×C3⋊C8—22no—
49#3724—21no—
50(C2×D3)⋊Dic3—22no—
51#1604—91noSylow
52C22×C3×Q8—21no—
53C22×C3×Q8C2×D312yes—
54C2×#237—62no—
55C2×C3×SD16—62no—
56C3×#121—32no—
57C2×#215—22no—
58#939—62no—
59#939—62no—
60#930C2×D312yes—
61C2×Dic3⋊C4—32no—
62C2×Dic3⋊C4—61no—
63C2×Dic3⋊C4—62no—
64C2×Dic3⋊C4—62no—
65#896—62no—
66#884—62no—
67#878—62no—
68C3×#103—62no—
69C3×C22⋊C8—32no—
70#835—22no—
71C3⋊#95—32no—
72C3⋊#95—61no—
73C3⋊#95—62no—
74C3⋊#95—62no—
75#830C2×D312yes—
76C23×C32C2×D411yescharacteristic
77C22×C3×D3—22no—
78C32×Q8—24no—
79C32×Q8—42no—
80C2×C32×C4—22no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—41no—
84C2×C32×C4C2×D412yes—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—21no—
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—41no—
90C2×C3⋊Dic3—41no—
91C2×C3⋊Dic3—41no—
92C2×C3⋊Dic3SD1612yescomplemented
93C3×#67—42no—
94C3×#67—42no—
95C2×C3×Dic3—22no—
96C3×D12—42no—
97#627—42no—
98#625—42no—
99#625—81no—
100D3×Dic3—44no—
101C3×C3⋊C8—42no—
102#537—91no—
103#448—32no—
104#395—91no—
105#327—92no—
106#324—91no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8—42no—
110C2×C3×Q8—44no—
111C2×C3×Q8—44no—
112C2×C3×D4—32no—
113C2×C3×D4—62no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4C22×D312yes—
118C2×#67—22no—
119C2×#67—62no—
120C2×#67—62no—
121C22×Dic3—32no—
122C22×Dic3—32no—
123C22×Dic3—61no—
124C22×Dic3—61no—
125C22×Dic3—62no—
126C22×Dic3—62no—
127C22×Dic3—62no—
128#237—124no—
129#237—124no—
130C2×D12#6712yes—
131C2×C4×D3—62no—
132C2×Dic6—62no—
133C3×SD16—124no—
134C2×C3×C8—62no—
135C3×C4⋊C4—32no—
136C3×C22⋊C4—62no—
137#217—62no—
138#215—44no—
139(C2×D3)⋊C4—22no—
140(C2×D3)⋊C4—62no—
141C4⋊Dic3#6712yes—
142Dic3⋊C4—62no—
143Dic3⋊C4—62no—
144Dic3⋊C4—62no—
145Dic3⋊C4—122no—
146Dic3⋊C4—124no—
147Dic3⋊C4—124no—
148C2×C3⋊C8—22no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32#12011yescharacteristic
152C22×C32C2×SD1611yescharacteristic
153C22×C32AGL(1,ℤ/8)11yescharacteristic
154C2×C3×D3—42no—
155C2×C3×D3—42no—
156C2×C3×D3—42no—
157C32×C4—22no—
158C32×C4—42no—
159C3⋊Dic3—22no—
160C3⋊Dic3—42no—
161C3⋊Dic3—41no—
162C3×Dic3—42no—
163C2×#41—181no—
164C22×Q8D3211yescharacteristic
165C2×SD16—62no—
166C22⋊Q8—182no—
167#121—32no—
168#121—182no—
169#121—181no—
170C2×C4⋊C4—91no—
171C2×C4⋊C4—181no—
172#103—62no—
173C22⋊C8—32no—
174#95—91no—
175#95—181no—
176C23×C3—21no—
177C23×C3—62no—
178C23×C3D3×D412yes—
179C22×D3—22no—
180C3×Q8—24no—
181C3×Q8—24no—
182C3×Q8—44no—
183C3×Q8—42no—
184C3×Q8—42no—
185C3×Q8—44no—
186C3×D4—122no—
187C3×D4—122no—
188C3×D4—122no—
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—22no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—62no—
199C2×C3×C4D3×D412yes—
200C2×C3×C4C2×#6712yes—
201#67—42no—
202#67—42no—
203#67—124no—
204#67—242no—
205C2×Dic3—22no—
206C2×Dic3—34no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—64no—
211C2×Dic3—64no—
212C2×Dic3—62no—
213C2×Dic3—121no—
214C2×Dic3—121no—
215C2×Dic3—121no—
216C2×Dic3—124no—
217C2×Dic3—122no—
218C2×Dic3—122no—
219C2×Dic3—122no—
220C2×Dic3—122no—
221C2×Dic3—122no—
222C2×Dic3—122no—
223D12—42no—
224C4×D3—124no—
225Dic6—124no—
226C3×C8—122no—
227C3⋊C8—42no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32#45011yescharacteristic
231C2×C32#45611yescharacteristic
232C2×C32B(4,𝔽2)11yescharacteristic
233C3×D3—82no—
234#41—362no—
235#41—362no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—181no—
239C2×D4—32no—
240C2×D4—62no—
241C2×D4—181no—
242C2×D4—181no—
243C22×C4—21no—
244C22×C4—91no—
245C22×C4—91no—
246C22×C4—182no—
247C22×C4—181no—
248C22×C4C2×D3211yescharacteristic
249SD16—124no—
250C2×C8—62no—
251C4⋊C4—32no—
252C4⋊C4—181no—
253C4⋊C4—182no—
254C4⋊C4—362no—
255C22⋊C4—62no—
256C22⋊C4—182no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—122no—
265C22×C3—122no—
266C22×C3—122no—
267C22×C3D3×SD1612yes—
268C22×C3#91412yes—
269C22×C3#93712yes—
270C2×D3—42no—
271C2×D3—42no—
272C2×D3—42no—
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—42no—
276C3×C4—44no—
277C3×C4—44no—
278C3×C4—122no—
279Dic3—42no—
280Dic3—64no—
281Dic3—122no—
282Dic3—124no—
283Dic3—124no—
284Dic3—122no—
285Dic3—241no—
286C32#160411yescharacteristic, complemented, Sylow
287C23—62no—
288C23#381111yescharacteristic
289Q8—24no—
290Q8—42no—
291Q8—362no—
292D4—122no—
293D4—122no—
294D4—122no—
295D4—362no—
296D4—721no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—62no—
302C2×C4—92no—
303C2×C4—181no—
304C2×C4—181no—
305C2×C4—182no—
306C2×C4—181no—
307C2×C4—364no—
308C2×C4—361no—
309C2×C4—361no—
310C2×C4—361no—
311C2×C4D3×#6712yes—
312C8—122no—
313C2×C3—22no—
314C2×C3—22no—
315C2×C3—22no—
316C2×C3—22no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—242no—
321C2×C3#503212yes—
322C2×C3#516912yes—
323C2×C3#520512yes—
324D3—82no—
325C22—21no—
326C22—21no—
327C22—122no—
328C22—122no—
329C22—122no—
330C22#6394011yescharacteristic
331C22#6394111yescharacteristic
332C22#6398311yescharacteristic
333C4—22no—
334C4—42no—
335C4—122no—
336C4—182no—
337C4—362no—
338C4—361no—
339C3—21no—
340C3#7097512yescomplemented
341C2—21no—
342C2—21no—
343C2—242no—
344C2#1059165211yescharacteristic
345C2#1059166011yescharacteristic
346C2#1059172511yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC22×C32#120
prime radical#10594930C2
hypercenterC22×Q8D32
hypocenterC32#1604
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #10592929C2semidirect
3. C3×#5227C2semidirect
4. #10589150C2semidirect
5. #10589148C2semidirect
24. #5227D3semidirect
92. C2×C3⋊Dic3SD16semidirect
286. C32#1604semidirect
340. C3#70975semidirect

Maximal subgroups

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

subgroupindex
#105949302
#105929292
C3×#52272
#105891502
#105891482
#709753