Finite Group Databasebeta

← #11779716

Subgroups — #11779716

Order 1152.

924 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 143 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
1D4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2#10592718C211yesmaximal, characteristic, complemented
3#10592545C211yesmaximal, characteristic, complemented
4#10592393C211yesmaximal, characteristic, complemented
5C32×D4⋊C8C211yesmaximal, characteristic, complemented
6#10589351C211yesmaximal, characteristic, complemented
7#10589325C211yesmaximal, characteristic, complemented
8#68196—34nomaximal
9C2×D4×C3⋊D3—21no—
10C22×C4⋊(C3⋊D3)—21no—
11C32×C4×D4C2211yescharacteristic
12C2×#3754—21no—
13#64139—21no—
14#64104—21no—
15#64091—22no—
16#64083C2211yescharacteristic
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18#64078C2211yescharacteristic, complemented
19C32×C4⋊C8C2211yescharacteristic, complemented
20C32×C4×C8C2211yescharacteristic, complemented
21#63648—21no—
22#63648—22no—
23#63645—21no—
24#63637—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
26D4×D12—34no—
27#5019—34no—
28#4846—34no—
29#4694—34no—
30C3×D4⋊C8D314yescomplemented
31#4485—34no—
32#4459—34no—
33C23×C3⋊D3—22no—
34C2×C32×D4D411yescharacteristic
35C22×C32×C4—21no—
36C2×#638—22no—
37D4×C3⋊D3—42no—
38D4×C3⋊D3—42no—
39C2×C4⋊(C3⋊D3)—22no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)—42no—
43C2×C4⋊(C3⋊D3)—42no—
44C2×C4⋊(C3⋊D3)D411yescharacteristic
45C2×C4⋊(C3⋊D3)D412yes—
46C2×C4×C3⋊D3—21no—
47C2×#634—21no—
48C2×C32×C8—21no—
49C2×C32×C8—21no—
50C32×C4⋊C4D411yescharacteristic
51C32×C22⋊C4—21no—
52C32×C42C2311yescharacteristic
53#3754—42no—
54#3752—22no—
55#3752—21no—
56C4⋊(C3⋊Dic3)—21no—
57C4⋊(C3⋊Dic3)—21no—
58C4⋊(C3⋊Dic3)D411yescharacteristic
59C2×C3⋊(C3⋊C8)—21no—
60#1639—91noSylow
61C2×D3×D4—64no—
62C22×D12—64no—
63C3×C4×D4C2×D314yes—
64C2×#214—64no—
65#930—64no—
66#895—64no—
67#882—68no—
68#874—34no—
69C4×D12—34no—
70#869—34no—
71C3×C4⋊C8C2×D314yescomplemented
72C3×C4×C8C2×D314yescomplemented
73#821—68no—
74#821—64no—
75#818—64no—
76#810—64no—
77C4⋊(C3⋊C8)—34no—
78C23×C32—21no—
79C22×C3⋊D3—21no—
80C22×C3⋊D3—22no—
81C22×C3⋊D3—44no—
82C22×C3⋊D3—44no—
83C22×C3⋊D3—82no—
84C32×D4—21no—
85C32×D4D812yes—
86C2×C32×C4—21no—
87C2×C32×C4—41no—
88C2×C32×C4—41no—
89C2×C32×C4C2×D411yescharacteristic
90C2×C32×C4C2×D411yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92#638—44no—
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—41no—
95C4⋊(C3⋊D3)—21no—
96C4⋊(C3⋊D3)—42no—
97C4⋊(C3⋊D3)—44no—
98C4⋊(C3⋊D3)—41no—
99C4⋊(C3⋊D3)—41no—
100C4⋊(C3⋊D3)—41no—
101C4⋊(C3⋊D3)—41no—
102C4⋊(C3⋊D3)SD1612yescomplemented
103C4×C3⋊D3—41no—
104C4×C3⋊D3—41no—
105#634—41no—
106#634—41no—
107C32×C8—41no—
108C32×C8—41no—
109C3⋊(C3⋊C8)—41no—
110D42—91no—
111#486—91no—
112#474—91no—
113#461—91no—
114#331—91no—
115D4⋊C8—91no—
116D4⋊C8C3⋊D311yescharacteristic, complemented
117C23×D3—68no—
118C2×C3×D4D1214yes—
119C22×C3×C4—24no—
120C2×#67—68no—
121D3×D4—128no—
122D3×D4—128no—
123C2×D12—34no—
124C2×D12—38no—
125C2×D12—68no—
126C2×D12—64no—
127C2×D12—64no—
128C2×D12—128no—
129C2×D12—128no—
130C2×C4×D3—64no—
131C2×Dic6—64no—
132C2×C3×C8—24no—
133C2×C3×C8—24no—
134C3×C4⋊C4D1214yes—
135C3×C22⋊C4—24no—
136C3×C42C22×D314yes—
137#214—128no—
138(C2×D3)⋊C4—68no—
139(C2×D3)⋊C4—64no—
140C4⋊Dic3—34no—
141C4⋊Dic3—64no—
142C4⋊Dic3—64no—
143C2×C3⋊C8—64no—
144C22×C32—22no—
145C22×C32—41no—
146C22×C32—41no—
147C22×C32#12011yescharacteristic
148C2×C3⋊D3—22no—
149C2×C3⋊D3—41no—
150C2×C3⋊D3—41no—
151C2×C3⋊D3—42no—
152C2×C3⋊D3—42no—
153C2×C3⋊D3—42no—
154C2×C3⋊D3—88no—
155C32×C4—21no—
156C32×C4—41no—
157C32×C4C2×D811yescharacteristic
158C32×C4C2×SD1611yescharacteristic
159C32×C4AGL(1,ℤ/8)11yescharacteristic
160C32×C4AGL(1,ℤ/8)11yescharacteristic
161C3⋊Dic3—41no—
162C3⋊Dic3—81no—
163C22×D4—181no—
164C22×D4—181no—
165C2×SD16—181no—
166C4⋊Q8—91no—
167#127—91no—
168#121—181no—
169#121—181no—
170#120—182no—
171C4×D4—91no—
172C4×D4C2×C3⋊D311yescharacteristic
173#107—181no—
174C4⋊C8—91no—
175C4⋊C8C2×C3⋊D311yescharacteristic, complemented
176#103—181no—
177#102—181no—
178#102—182no—
179C4×C8C2×C3⋊D311yescharacteristic, complemented
180C23×C3—24no—
181C22×D3—64no—
182C22×D3—68no—
183C22×D3—1216no—
184C22×D3—1216no—
185C22×D3—248no—
186C3×D4—24no—
187C3×D4D2418yes—
188C2×C3×C4—24no—
189C2×C3×C4—44no—
190C2×C3×C4—44no—
191C2×C3×C4C2×D1214yes—
192C2×C3×C4D3×D414yes—
193C2×C3×C4D3×D414yes—
194#67—1216no—
195C2×Dic3—64no—
196C2×Dic3—124no—
197D12—38no—
198D12—64no—
199D12—128no—
200D12—1216no—
201D12—124no—
202D12—124no—
203D12—124no—
204D12—124no—
205C4×D3—124no—
206C4×D3—124no—
207Dic6—124no—
208Dic6—124no—
209C3×C8—44no—
210C3×C8—44no—
211C3⋊C8—124no—
212C2×C32—42no—
213C2×C32#44711yescharacteristic
214C2×C32#45011yescharacteristic
215C2×C32#45511yescharacteristic
216C3⋊D3—42no—
217C3⋊D3—82no—
218C24—182no—
219C2×Q8—181no—
220C2×D4—91no—
221C2×D4—92no—
222C2×D4—182no—
223C2×D4—182no—
224C2×D4—181no—
225C2×D4—181no—
226C2×D4—362no—
227C2×D4—362no—
228C2×D4—362no—
229C2×D4—362no—
230C2×D4C4⋊(C3⋊D3)11yescharacteristic
231C22×C4—21no—
232C22×C4—181no—
233SD16—362no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—181no—
237C4⋊C4—91no—
238C4⋊C4—181no—
239C4⋊C4—181no—
240C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
241C22⋊C4—21no—
242C22⋊C4—182no—
243C22⋊C4—181no—
244C42C22×C3⋊D311yescharacteristic
245C22×C3—28no—
246C22×C3—44no—
247C22×C3—44no—
248C22×C3#88214yes—
249C2×D3—68no—
250C2×D3—124no—
251C2×D3—124no—
252C2×D3—128no—
253C2×D3—128no—
254C2×D3—128no—
255C2×D3—2432no—
256C3×C4—24no—
257C3×C4—44no—
258C3×C4C2×D2414yes—
259C3×C4#90814yes—
260C3×C4#91114yes—
261C3×C4D3×SD1614yes—
262Dic3—124no—
263Dic3—244no—
264C32#163911yescharacteristic, complemented, Sylow
265C23—21no—
266C23—181no—
267C23—182no—
268C23—364no—
269C23—364no—
270C23—722no—
271Q8—361no—
272Q8—361no—
273D4—21no—
274D4—92no—
275D4—181no—
276D4—362no—
277D4—364no—
278D4—364no—
279D4—361no—
280D4—361no—
281D4—361no—
282D4—361no—
283D4#374512yes—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—181no—
288C2×C4—361no—
289C2×C4—361no—
290C2×C4—361no—
291C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
292C2×C4D4×C3⋊D311yescharacteristic
293C2×C4D4×C3⋊D311yescharacteristic
294C8—41no—
295C8—41no—
296C8—361no—
297C2×C3—48no—
298C2×C3#473214yes—
299C2×C3#477614yes—
300C2×C3#482514yes—
301D3—128no—
302D3—248no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—182no—
307C22—361no—
308C22—361no—
309C22—362no—
310C22—362no—
311C22—362no—
312C22—728no—
313C22#6409111yescharacteristic
314C4—21no—
315C4—41no—
316C4—361no—
317C4—721no—
318C4C2×#374511yescharacteristic
319C4#6411711yescharacteristic
320C4#6412011yescharacteristic
321C4SD16×C3⋊D311yescharacteristic
322C3#6819614yescomplemented
323C2—42no—
324C2—362no—
325C2—722no—
326C2#1059243111yescharacteristic
327C2#1059247511yescharacteristic
328C2#1059252411yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×D4⋊C8C2
socleC22×C32#120
prime radicalD4×C4⋊(C3⋊D3)C2
hypercenterD4⋊C8C3⋊D3
hypocenterC32#1639
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #10592718C2semidirect
3. #10592545C2semidirect
4. #10592393C2semidirect
5. C32×D4⋊C8C2semidirect
6. #10589351C2semidirect
7. #10589325C2semidirect
18. #64078C22semidirect
19. C32×C4⋊C8C22semidirect
20. C32×C4×C8C22semidirect
25. C4⋊(C3⋊(C3⋊C8))C22semidirect
30. C3×D4⋊C8D3semidirect
71. C3×C4⋊C8C2×D3semidirect
72. C3×C4×C8C2×D3semidirect
102. C4⋊(C3⋊D3)SD16semidirect
116. D4⋊C8C3⋊D3semidirect
175. C4⋊C8C2×C3⋊D3semidirect
179. C4×C8C2×C3⋊D3semidirect
264. C32#1639semidirect
322. C3#68196semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊(C3⋊D3)2
#105927182
#105925452
#105923932
C32×D4⋊C82
#105893512
#105893252
#681963