Finite Group Databasebeta

← #11778688

Subgroups — #11778688

Order 1152.

708 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 119 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#10596105C211yesmaximal, characteristic
2C32×#484C211yesmaximal, characteristic, complemented
3#10592692C211yesmaximal, characteristic
4#10592669C211yesmaximal, characteristic, complemented
5#10589404C211yesmaximal, characteristic, complemented
6#10589334C211yesmaximal, characteristic
7#10589333C211yesmaximal, characteristic, complemented
8#71176—34nomaximal
9C32×C22⋊Q8C2211yescharacteristic
10C2×C32×C4⋊C4C2211yescharacteristic
11#64155—21no—
12C2×#3752—21no—
13C2×#3752C2211yescharacteristic
14#64133C2211yescharacteristic
15C2×#3750—21no—
16#64106—21no—
17#64105—21no—
18#64105—21no—
19C22⋊C4×C3⋊D3—21no—
20#64087—21no—
21C32×#107—21no—
22C32×#103—21no—
23C32×C22⋊C8C2211yescharacteristic
24#63658C2211yescharacteristic
25#63657C2211yescharacteristic
26#63637—21no—
27#63634—21no—
28#5634—34no—
29C3×#484D314yescomplemented
30#4993—34no—
31#4970—34no—
32#4538—34no—
33#4468—34no—
34#4467—34no—
35C23×C3⋊D3D411yescharacteristic
36C2×C32×Q8D411yescharacteristic
37C22×C32×C4—21no—
38C22×C32×C4C2311yescharacteristic
39C22×C3⋊Dic3—21no—
40C22×C3⋊Dic3D411yescharacteristic
41C2×C4×C3⋊D3—21no—
42C2×C4×C3⋊D3—21no—
43C2×#634—21no—
44C2×#634D411yescharacteristic
45C2×C32×C8—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4D411yescharacteristic
50C32×C22⋊C4—21no—
51#3757—21no—
52#3752—21no—
53#3752—21no—
54#3752—21no—
55#3752—21no—
56#3752—42no—
57C4⋊(C3⋊Dic3)—21no—
58C4⋊(C3⋊Dic3)D411yescharacteristic
59#3750—21no—
60#3750—21no—
61#3750—21no—
62#3750—42no—
63C2×C3⋊(C3⋊C8)—21no—
64#1614—91noSylow
65C3×C22⋊Q8C2×D314yes—
66C2×C3×C4⋊C4C2×D314yes—
67#946—64no—
68C2×(C2×D3)⋊C4—34no—
69C2×(C2×D3)⋊C4—64no—
70#924—34no—
71C2×Dic3⋊C4—64no—
72#897—64no—
73#896—64no—
74#896—64no—
75D3×C22⋊C4—64no—
76#878—64no—
77C3×#107—24no—
78C3×#103—24no—
79C3×C22⋊C8C2×D314yes—
80C3⋊#95—34no—
81#830—34no—
82#810—64no—
83#807—64no—
84C23×C32C2×D411yescharacteristic
85C22×C3⋊D3—21no—
86C22×C3⋊D3—21no—
87C22×C3⋊D3—41no—
88C22×C3⋊D3—41no—
89C22×C3⋊D3Q1612yes—
90C32×Q8—41no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C4×C3⋊D3—81no—
107C4×C3⋊D3—81no—
108#634—41no—
109#634—81no—
110C32×C8—41no—
111C3⋊(C3⋊C8)—41no—
112#543—91no—
113#484—91no—
114#484C3⋊D311yescharacteristic, complemented
115#399—91no—
116#328—91no—
117#328—91no—
118#323—91no—
119C23×D3—34no—
120C2×C3×Q8#6714yes—
121C22×C3×C4—24no—
122C22×C3×C4C22×D314yes—
123C22×Dic3—34no—
124C22×Dic3—64no—
125C2×C4×D3—64no—
126C2×C4×D3—64no—
127C2×Dic6—34no—
128C2×Dic6—64no—
129C2×C3×C8—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—24no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4#6714yes—
134C3×C22⋊C4—24no—
135#217—64no—
136(C2×D3)⋊C4—64no—
137(C2×D3)⋊C4—64no—
138(C2×D3)⋊C4—64no—
139(C2×D3)⋊C4—64no—
140(C2×D3)⋊C4—128no—
141C4⋊Dic3—34no—
142C4⋊Dic3—64no—
143Dic3⋊C4—64no—
144Dic3⋊C4—64no—
145Dic3⋊C4—64no—
146Dic3⋊C4—128no—
147C2×C3⋊C8—64no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32#12011yescharacteristic
151C22×C32C2×Q1611yescharacteristic
152C22×C32#13711yescharacteristic
153C2×C3⋊D3—22no—
154C2×C3⋊D3—41no—
155C2×C3⋊D3—41no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3—81no—
158C32×C4—21no—
159C32×C4—21no—
160C32×C4—41no—
161C32×C4—41no—
162C32×C4—81no—
163C3⋊Dic3—42no—
164C3⋊Dic3—41no—
165C3⋊Dic3—41no—
166C3⋊Dic3—41no—
167C22⋊Q8—91no—
168C22⋊Q8—181no—
169C22⋊Q8—181no—
170C22⋊Q8—181no—
171C22⋊Q8—181no—
172C22⋊Q8—181no—
173C22⋊Q8C2×C3⋊D311yescharacteristic
174C2×C4⋊C4—181no—
175C2×C4⋊C4C2×C3⋊D311yescharacteristic
176C2×C22⋊C4—91no—
177C2×C22⋊C4—181no—
178C2×C22⋊C4—181no—
179#107—21no—
180#107—181no—
181#103—21no—
182#103—181no—
183C22⋊C8—91no—
184C22⋊C8C2×C3⋊D311yescharacteristic
185#95—91no—
186C23×C3D3×D414yes—
187C22×D3—38no—
188C22×D3—64no—
189C22×D3—64no—
190C22×D3—124no—
191C22×D3—124no—
192C3×Q8—44no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4—44no—
196C2×C3×C4—44no—
197C2×C3×C4—44no—
198C2×C3×C4—44no—
199C2×C3×C4D3×D414yes—
200C2×C3×C4C2×#6714yes—
201C2×Dic3—68no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204C2×Dic3—64no—
205C2×Dic3—124no—
206C2×Dic3—124no—
207C2×Dic3—124no—
208C4×D3—244no—
209C4×D3—244no—
210Dic6—124no—
211Dic6—244no—
212C3×C8—44no—
213C3⋊C8—124no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32#45111yescharacteristic
217C2×C32#45311yescharacteristic
218C2×C32#45811yescharacteristic
219C3⋊D3—42no—
220C24—91no—
221C2×Q8—91no—
222C2×Q8—181no—
223C2×Q8#63811yescharacteristic
224C22×C4—21no—
225C22×C4—91no—
226C22×C4—181no—
227C22×C4—181no—
228C22×C4—181no—
229C22×C4C22×C3⋊D311yescharacteristic
230C2×C8—21no—
231C2×C8—181no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—91no—
236C4⋊C4—181no—
237C4⋊C4—181no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C4⋊C4—362no—
241C4⋊C4#63811yescharacteristic
242C22⋊C4—21no—
243C22⋊C4—181no—
244C22⋊C4—181no—
245C22⋊C4—181no—
246C22⋊C4—181no—
247C22⋊C4—181no—
248C22⋊C4—362no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3D3×Q1614yes—
252C22×C3#91814yes—
253C22×C3#93714yes—
254C2×D3—68no—
255C2×D3—124no—
256C2×D3—124no—
257C2×D3—128no—
258C2×D3—244no—
259C3×C4—24no—
260C3×C4—24no—
261C3×C4—44no—
262C3×C4—44no—
263C3×C4—84no—
264Dic3—128no—
265Dic3—124no—
266Dic3—124no—
267Dic3—124no—
268C32#161411yescharacteristic, complemented, Sylow
269C23—92no—
270C23—181no—
271C23—181no—
272C23—361no—
273C23—361no—
274C23D4×C3⋊D311yescharacteristic
275Q8—41no—
276Q8—361no—
277Q8—721no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—182no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—361no—
289C2×C4—361no—
290C2×C4—361no—
291C2×C4—721no—
292C2×C4—721no—
293C2×C4D4×C3⋊D311yescharacteristic
294C2×C4C2×#63811yescharacteristic
295C8—41no—
296C8—361no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3#505314yes—
300C2×C3#518614yes—
301C2×C3#519814yes—
302D3—128no—
303C22—21no—
304C22—21no—
305C22—182no—
306C22—362no—
307C22—361no—
308C22—361no—
309C22—721no—
310C22Q16×C3⋊D311yescharacteristic
311C22#6412711yescharacteristic
312C22#6414611yescharacteristic
313C4—21no—
314C4—21no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—362no—
319C4—361no—
320C4—361no—
321C4—361no—
322C3#7117614yescomplemented
323C2—21no—
324C2—21no—
325C2—362no—
326C2#1059275211yescharacteristic
327C2#1059288511yescharacteristic
328C2#1059289711yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#484C2
socleC22×C32#120
prime radicalC23×C3⋊D3D4
hypercenter#484C3⋊D3
hypocenterC32#1614
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C32×#484C2semidirect
4. #10592669C2semidirect
5. #10589404C2semidirect
7. #10589333C2semidirect
29. C3×#484D3semidirect
114. #484C3⋊D3semidirect
268. C32#1614semidirect
322. C3#71176semidirect

Maximal subgroups

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

subgroupindex
#105961052
C32×#4842
#105926922
#105926692
#105894042
#105893342
#105893332
#711763