Finite Group Databasebeta

← #11778274

Subgroups — #11778274

Order 1152.

732 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 117 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#10596116C211yesmaximal, characteristic
2C32×#451C211yesmaximal, characteristic, complemented
3#10592929C211yesmaximal, characteristic, complemented
4#10592926C211yesmaximal, characteristic
5#10589404C211yesmaximal, characteristic, complemented
6#10589336C211yesmaximal, characteristic
7#10589335C211yesmaximal, characteristic, complemented
8#70982—34nomaximal
9C22×C32×Q8C2211yescharacteristic
10C2×C32×Q16—21no—
11C32×C22⋊Q8C2211yescharacteristic
12#64155—21no—
13C2×#3755—21no—
14#64139C2211yescharacteristic
15C2×#3750—21no—
16C2×#3750C2211yescharacteristic
17#64107—21no—
18#64103—21no—
19#64099—21no—
20#64095—21no—
21#64090—21no—
22C32×#103—21no—
23C32×C22⋊C8C2211yescharacteristic
24#63662—21no—
25#63658—21no—
26#63658C2211yescharacteristic
27#63657C2211yescharacteristic
28#5645—34no—
29C3×#451D314yescomplemented
30#5230—34no—
31#5227—34no—
32#4538—34no—
33#4470—34no—
34#4469—34no—
35C2×C32×Q8—21no—
36C2×C32×Q8—21no—
37C2×C32×Q8—42no—
38C2×C32×Q8D411yescharacteristic
39C22×C32×C4—21no—
40C22×C32×C4C2311yescharacteristic
41C2×#638—21no—
42C22×C3⋊Dic3—21no—
43C22×C3⋊Dic3D411yescharacteristic
44C22×C3⋊Dic3D411yescharacteristic
45C2×C4⋊(C3⋊D3)D411yescharacteristic
46C2×C4×C3⋊D3—21no—
47C32×Q16—42no—
48C2×C32×C8—21no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4D411yescharacteristic
51C32×C22⋊C4—21no—
52#3755—42no—
53#3752—21no—
54#3752—21no—
55#3752—21no—
56#3752—21no—
57C4⋊(C3⋊Dic3)—21no—
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)D411yescharacteristic
60#3750—21no—
61#3750—21no—
62#3750—21no—
63#3750—42no—
64C4×C3⋊Dic3—21no—
65C2×C3⋊(C3⋊C8)—21no—
66#1603—91noSylow
67C22×C3×Q8C2×D314yes—
68C2×C3×Q16—24no—
69C3×C22⋊Q8C2×D314yes—
70#946—64no—
71C2×#215—64no—
72#930—34no—
73C2×Dic3⋊C4—34no—
74C2×Dic3⋊C4—64no—
75#898—64no—
76#894—64no—
77#890—64no—
78#886—64no—
79#67⋊C4—64no—
80C3×#103—24no—
81C3×C22⋊C8C2×D314yes—
82#835—64no—
83C3⋊#95—34no—
84C3⋊#95—64no—
85#830—34no—
86C23×C32C2×D411yescharacteristic
87C22×C3⋊D3—21no—
88C32×Q8—22no—
89C32×Q8—22no—
90C32×Q8—41no—
91C32×Q8—41no—
92C32×Q8—41no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—41no—
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101#638—41no—
102#638—41no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—41no—
110C2×C3⋊Dic3—41no—
111C2×C3⋊Dic3—41no—
112C4⋊(C3⋊D3)—41no—
113C4×C3⋊D3—81no—
114C32×C8—41no—
115C3⋊(C3⋊C8)—41no—
116#539—91no—
117#451C3⋊D311yescharacteristic, complemented
118#448—91no—
119#395—91no—
120#328—91no—
121#327—91no—
122#324—91no—
123C2×C3×Q8—24no—
124C2×C3×Q8—24no—
125C2×C3×Q8—48no—
126C2×C3×Q8#6714yes—
127C22×C3×C4—24no—
128C22×C3×C4C22×D314yes—
129C2×#67—64no—
130C22×Dic3—34no—
131C22×Dic3—34no—
132C22×Dic3—64no—
133C2×D12—34no—
134C2×C4×D3—64no—
135C3×Q16—48no—
136C2×C3×C8—24no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4#6714yes—
139C3×C22⋊C4—24no—
140#215—128no—
141(C2×D3)⋊C4—64no—
142(C2×D3)⋊C4—64no—
143(C2×D3)⋊C4—64no—
144(C2×D3)⋊C4—64no—
145C4⋊Dic3—34no—
146C4⋊Dic3—64no—
147C4⋊Dic3—64no—
148Dic3⋊C4—64no—
149Dic3⋊C4—64no—
150Dic3⋊C4—64no—
151Dic3⋊C4—128no—
152C4×Dic3—64no—
153C2×C3⋊C8—64no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32#12011yescharacteristic
157C22×C32#13511yescharacteristic
158C22×C32#13711yescharacteristic
159C2×C3⋊D3—41no—
160C2×C3⋊D3—41no—
161C2×C3⋊D3—41no—
162C32×C4—21no—
163C32×C4—21no—
164C32×C4—41no—
165C32×C4—41no—
166C32×C4—42no—
167C3⋊Dic3—42no—
168C3⋊Dic3—41no—
169C3⋊Dic3—41no—
170C3⋊Dic3—41no—
171C22×Q8C2×C3⋊D311yescharacteristic
172C2×Q16—21no—
173C2×SD16—181no—
174#126—181no—
175#125—181no—
176#123—181no—
177#123—181no—
178C22⋊Q8—181no—
179C22⋊Q8C2×C3⋊D311yescharacteristic
180#121—91no—
181C4×D4—181no—
182C2×C4⋊C4—91no—
183C2×C4⋊C4—181no—
184#103—21no—
185#103—181no—
186C22⋊C8—91no—
187C22⋊C8C2×C3⋊D311yescharacteristic
188#95—91no—
189#95—181no—
190C23×C3D3×D414yes—
191C22×D3—64no—
192C3×Q8—28no—
193C3×Q8—28no—
194C3×Q8—44no—
195C3×Q8—44no—
196C3×Q8—44no—
197C2×C3×C4—28no—
198C2×C3×C4—24no—
199C2×C3×C4—24no—
200C2×C3×C4—44no—
201C2×C3×C4—44no—
202C2×C3×C4—44no—
203C2×C3×C4D3×D414yes—
204C2×C3×C4C2×#6714yes—
205#67—124no—
206#67—124no—
207C2×Dic3—68no—
208C2×Dic3—64no—
209C2×Dic3—64no—
210C2×Dic3—64no—
211C2×Dic3—64no—
212C2×Dic3—64no—
213C2×Dic3—124no—
214C2×Dic3—124no—
215C2×Dic3—124no—
216D12—124no—
217C4×D3—244no—
218C3×C8—44no—
219C3⋊C8—124no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32#45211yescharacteristic
223C2×C32#45611yescharacteristic
224C2×C32B(4,𝔽2)11yescharacteristic
225C3⋊D3—81no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—42no—
229C2×Q8#63811yescharacteristic
230C2×D4—91no—
231C2×D4—181no—
232C22×C4—21no—
233C22×C4—91no—
234C22×C4—91no—
235C22×C4—181no—
236C22×C4—181no—
237C22×C4C22×C3⋊D311yescharacteristic
238Q16—42no—
239SD16—362no—
240C2×C8—21no—
241C2×C8—181no—
242C4⋊C4—21no—
243C4⋊C4—91no—
244C4⋊C4—181no—
245C4⋊C4—181no—
246C4⋊C4—181no—
247C4⋊C4—181no—
248C4⋊C4—181no—
249C4⋊C4—362no—
250C4⋊C4#63811yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—181no—
253C22⋊C4—181no—
254C22⋊C4—181no—
255C22⋊C4—181no—
256C42—181no—
257C22×C3—24no—
258C22×C3—24no—
259C22×C3#91814yes—
260C22×C3#91914yes—
261C22×C3#93714yes—
262C2×D3—124no—
263C2×D3—124no—
264C2×D3—124no—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4—44no—
268C3×C4—44no—
269C3×C4—48no—
270Dic3—128no—
271Dic3—124no—
272Dic3—124no—
273Dic3—124no—
274C32#160311yescharacteristic, complemented, Sylow
275C23—181no—
276C23D4×C3⋊D311yescharacteristic
277Q8—22no—
278Q8—22no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282D4—361no—
283D4—361no—
284D4—361no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—182no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—361no—
298C2×C4—361no—
299C2×C4—361no—
300C2×C4—721no—
301C2×C4D4×C3⋊D311yescharacteristic
302C2×C4C2×#63811yescharacteristic
303C8—41no—
304C8—361no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3#503214yes—
308C2×C3#518714yes—
309C2×C3#520514yes—
310D3—244no—
311C22—21no—
312C22—21no—
313C22—361no—
314C22—361no—
315C22—361no—
316C22#6412711yescharacteristic
317C22#6412811yescharacteristic
318C22#6414611yescharacteristic
319C4—21no—
320C4—21no—
321C4—41no—
322C4—42no—
323C4—41no—
324C4—362no—
325C4—361no—
326C4—361no—
327C4—361no—
328C3#7098214yescomplemented
329C2—21no—
330C2—21no—
331C2—721no—
332C2#1059273111yescharacteristic
333C2#1059288611yescharacteristic
334C2#1059290411yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#451C2
socleC22×C32#120
prime radical#64139C22
hypercenter#451C3⋊D3
hypocenterC32#1603
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C32×#451C2semidirect
3. #10592929C2semidirect
5. #10589404C2semidirect
7. #10589335C2semidirect
29. C3×#451D3semidirect
117. #451C3⋊D3semidirect
274. C32#1603semidirect
328. C3#70982semidirect

Maximal subgroups

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

subgroupindex
#105961162
C32×#4512
#105929292
#105929262
#105894042
#105893362
#105893352
#709823