Finite Group Databasebeta

← #11778512

Subgroups — #11778512

Order 1152.

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

normal subgroups only

This group has 648 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
2#10592742C211yesmaximal, characteristic, complemented
3#10592435C211yesmaximal, characteristic
4#10592360C211yesmaximal, characteristic, complemented
5C32×#328C211yesmaximal, characteristic, complemented
6#10589335C211yesmaximal, characteristic, complemented
7#10589317C211yesmaximal, characteristic
8#68328—34nomaximal
9C32×C22⋊Q8C2211yescharacteristic
10#64155—21no—
11#64139C2211yescharacteristic
12C2×C4⋊(C3⋊Dic3)—21no—
13C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×#3750C2211yescharacteristic
15#64107—21no—
16#64103—21no—
17#64099—21no—
18#64095—21no—
19#64090—21no—
20C32×C22⋊C8C2211yescharacteristic
21C32×#95C2211yescharacteristic
22#63662—21no—
23#63658—21no—
24#63657C2211yescharacteristic
25#63648—21no—
26#63645—21no—
27#63635—21no—
28#5645—34no—
29#5043—34no—
30#4736—34no—
31#4661—34no—
32C3×#328D314yescomplemented
33#4469—34no—
34#4451—34no—
35C2×C32×Q8D411yescharacteristic
36C22×C32×C4—21no—
37C22×C32×C4C2311yescharacteristic
38C2×#638—21no—
39C22×C3⋊Dic3—21no—
40C22×C3⋊Dic3D411yescharacteristic
41C22×C3⋊Dic3D411yescharacteristic
42C2×C4⋊(C3⋊D3)D411yescharacteristic
43C2×C4×C3⋊D3—21no—
44C2×C32×C8—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4D411yescharacteristic
47C32×C22⋊C4—21no—
48#3752—21no—
49#3752—21no—
50#3752—21no—
51#3752—21no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)—21no—
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)—42no—
57C4⋊(C3⋊Dic3)D411yescharacteristic
58#3750—21no—
59#3750—21no—
60#3750—21no—
61C4×C3⋊Dic3—21no—
62C2×C3⋊(C3⋊C8)—21no—
63#1609—91noSylow
64C3×C22⋊Q8C2×D314yes—
65#946—64no—
66#930—34no—
67C2×C4⋊Dic3—34no—
68C2×C4⋊Dic3—64no—
69C2×Dic3⋊C4—34no—
70#898—64no—
71#894—64no—
72#890—64no—
73#886—64no—
74#67⋊C4—64no—
75C3×C22⋊C8C2×D314yes—
76C3×#95C2×D314yes—
77#835—64no—
78C3⋊#95—64no—
79#830—34no—
80#821—64no—
81#818—64no—
82#808—64no—
83C23×C32C2×D411yescharacteristic
84C22×C3⋊D3—21no—
85C32×Q8—41no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94#638—41no—
95#638—41no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—41no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—41no—
105C4⋊(C3⋊D3)—41no—
106C4×C3⋊D3—81no—
107C32×C8—41no—
108C3⋊(C3⋊C8)—41no—
109#539—91no—
110#483—91no—
111#480—91no—
112#398—91no—
113#328C3⋊D311yescharacteristic, complemented
114#327—91no—
115#324—91no—
116C2×C3×Q8D1214yes—
117C22×C3×C4—24no—
118C22×C3×C4C22×D314yes—
119C2×#67—64no—
120C22×Dic3—34no—
121C22×Dic3—34no—
122C22×Dic3—64no—
123C2×D12—34no—
124C2×C4×D3—64no—
125C2×C3×C8—24no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4D1214yes—
128C3×C22⋊C4—24no—
129(C2×D3)⋊C4—64no—
130(C2×D3)⋊C4—64no—
131(C2×D3)⋊C4—64no—
132(C2×D3)⋊C4—64no—
133C4⋊Dic3—34no—
134C4⋊Dic3—64no—
135C4⋊Dic3—64no—
136C4⋊Dic3—64no—
137C4⋊Dic3—64no—
138C4⋊Dic3—128no—
139Dic3⋊C4—64no—
140Dic3⋊C4—64no—
141Dic3⋊C4—64no—
142C4×Dic3—64no—
143C2×C3⋊C8—64no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32#12011yescharacteristic
147C22×C32#13511yescharacteristic
148C22×C32AGL(1,ℤ/8)11yescharacteristic
149C2×C3⋊D3—41no—
150C2×C3⋊D3—41no—
151C2×C3⋊D3—41no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—42no—
155C32×C4—41no—
156C32×C4—41no—
157C3⋊Dic3—41no—
158C3⋊Dic3—41no—
159C3⋊Dic3—41no—
160C3⋊Dic3—81no—
161#126—181no—
162#125—181no—
163#123—181no—
164#123—181no—
165C22⋊Q8—181no—
166C22⋊Q8C2×C3⋊D311yescharacteristic
167#121—91no—
168C4×D4—181no—
169C2×C4⋊C4—91no—
170C2×C4⋊C4—91no—
171C2×C4⋊C4—181no—
172#107—181no—
173#106—181no—
174#103—181no—
175#102—181no—
176C22⋊C8—91no—
177C22⋊C8C2×C3⋊D311yescharacteristic
178#95—181no—
179#95C2×C3⋊D311yescharacteristic
180C23×C3D3×D414yes—
181C22×D3—64no—
182C3×Q8—44no—
183C2×C3×C4—28no—
184C2×C3×C4—24no—
185C2×C3×C4—24no—
186C2×C3×C4—44no—
187C2×C3×C4—44no—
188C2×C3×C4—44no—
189C2×C3×C4C2×D1214yes—
190C2×C3×C4D3×D414yes—
191#67—124no—
192#67—124no—
193C2×Dic3—64no—
194C2×Dic3—64no—
195C2×Dic3—64no—
196C2×Dic3—64no—
197C2×Dic3—64no—
198C2×Dic3—124no—
199C2×Dic3—124no—
200C2×Dic3—124no—
201C2×Dic3—124no—
202D12—124no—
203C4×D3—244no—
204C3×C8—44no—
205C3⋊C8—124no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32#44911yescharacteristic
209C2×C32#45611yescharacteristic
210C2×C32#45811yescharacteristic
211C3⋊D3—81no—
212C2×Q8C4⋊(C3⋊D3)11yescharacteristic
213C2×D4—91no—
214C2×D4—181no—
215C22×C4—21no—
216C22×C4—91no—
217C22×C4—91no—
218C22×C4—181no—
219C22×C4—181no—
220C22×C4C22×C3⋊D311yescharacteristic
221C2×C8—21no—
222C2×C8—181no—
223C4⋊C4—21no—
224C4⋊C4—91no—
225C4⋊C4—181no—
226C4⋊C4—181no—
227C4⋊C4—181no—
228C4⋊C4—181no—
229C4⋊C4—181no—
230C4⋊C4—181no—
231C4⋊C4—181no—
232C4⋊C4—362no—
233C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—181no—
236C22⋊C4—181no—
237C22⋊C4—181no—
238C22⋊C4—181no—
239C42—181no—
240C22×C3—24no—
241C22×C3—24no—
242C22×C3#88214yes—
243C22×C3#91414yes—
244C22×C3#91914yes—
245C2×D3—124no—
246C2×D3—124no—
247C2×D3—124no—
248C3×C4—24no—
249C3×C4—24no—
250C3×C4—48no—
251C3×C4—44no—
252C3×C4—44no—
253Dic3—124no—
254Dic3—124no—
255Dic3—124no—
256Dic3—244no—
257C32#160911yescharacteristic, complemented, Sylow
258C23—181no—
259C23D4×C3⋊D311yescharacteristic
260Q8—41no—
261D4—361no—
262D4—361no—
263D4—361no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—181no—
273C2×C4—181no—
274C2×C4—181no—
275C2×C4—361no—
276C2×C4—361no—
277C2×C4—361no—
278C2×C4—361no—
279C2×C4—721no—
280C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
281C2×C4D4×C3⋊D311yescharacteristic
282C8—41no—
283C8—361no—
284C2×C3—24no—
285C2×C3—24no—
286C2×C3#474214yes—
287C2×C3#481014yes—
288C2×C3#482614yes—
289D3—244no—
290C22—21no—
291C22—21no—
292C22—361no—
293C22—361no—
294C22—361no—
295C22#6409111yescharacteristic
296C22#6412311yescharacteristic
297C22#6412811yescharacteristic
298C4—21no—
299C4—21no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—361no—
304C4—361no—
305C4—361no—
306C4—721no—
307C3#6832814yescomplemented
308C2—21no—
309C2—21no—
310C2—721no—
311C2#1059244111yescharacteristic
312C2#1059250911yescharacteristic
313C2#1059252511yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#328C2
socleC22×C32#120
prime radical#64139C22
hypercenter#328C3⋊D3
hypocenterC32#1609
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #10592742C2semidirect
4. #10592360C2semidirect
5. C32×#328C2semidirect
6. #10589335C2semidirect
32. C3×#328D3semidirect
113. #328C3⋊D3semidirect
257. C32#1609semidirect
307. C3#68328semidirect

Maximal subgroups

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

subgroupindex
#105961162
#105927422
#105924352
#105923602
C32×#3282
#105893352
#105893172
#683283