Finite Group Databasebeta

← #11790766

Subgroups — #11790766

Order 1152.

1073 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1073 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C2×#64149C211yesmaximal, characteristic
2C2×C3×#832C212yesmaximal, complemented
3#10591230C211yesmaximal, characteristic, complemented
4C2×#63564C212yesmaximal, complemented
5C2×#63555C211yesmaximal, characteristic, complemented
6#70160—32nomaximal
7C22×C32×D4C2211yescharacteristic
8C22×#638—22no—
9C22×C4⋊(C3⋊D3)C2211yescharacteristic
10#64149—22no—
11#64149C414yescomplemented
12C2×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C4⋊Dic3C2212yes—
14C22×C3×C3⋊C8C2212yes—
15C3×#832—24no—
16#63579—22no—
17#63564—24no—
18#63555—22no—
19C2×#940—32no—
20C2×#940—61no—
21C2×C3×#102—32no—
22C2×#832D312yescomplemented
23C2×#821—32no—
24C2×#815—32no—
25#4977—32no—
26C2×#809—32no—
27C24×C32—21no—
28C23×C3⋊D3—21no—
29C2×C32×D4—22no—
30C2×C32×D4—21no—
31C2×C32×D4—22no—
32C2×C32×D4C2×C412yes—
33C22×C32×C4C2311yescharacteristic
34C2×#638—28no—
35C2×#638—48no—
36C22×C3⋊Dic3D412yes—
37C2×C4⋊(C3⋊D3)—22no—
38C2×C4⋊(C3⋊D3)—22no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4⋊(C3⋊D3)C2×C412yes—
41C22×C3×Dic3—22no—
42C4×C3⋊Dic3C2×C412yes—
43C4×C3⋊Dic3D412yes—
44C3×C4⋊Dic3—22no—
45C3×C4⋊Dic3—22no—
46C2×C3×C3⋊C8—22no—
47C2×C3×C3⋊C8—22no—
48C2×C3×C3⋊C8D414yescomplemented
49#1961—91noSylow
50C22×C3×D4—21no—
51C22×C3×D4C2×D312yes—
52C22×#67—64no—
53C22×#67—64no—
54C22×D12—32no—
55C22×D12—61no—
56C22×C3×C8—32no—
57C2×C3×C4⋊C4—32no—
58#940—38no—
59#940—64no—
60#940—64no—
61#940—64no—
62C2×C4⋊Dic3C2×D312yes—
63C2×C4×Dic3—32no—
64C2×C4×Dic3—61no—
65C22×C3⋊C8C2×D312yes—
66C3×#102—64no—
67#832—24no—
68C3⋊#95—64no—
69#821—64no—
70#815—64no—
71#809—64no—
72C23×C32—22no—
73C23×C32—41no—
74C23×C32—41no—
75C23×C32—44no—
76C23×C32C2×D411yescharacteristic
77C22×C3⋊D3—22no—
78C22×C3⋊D3—41no—
79C22×C3⋊D3—41no—
80C22×C3⋊D3—44no—
81C32×D4—24no—
82C32×D4—42no—
83C2×C32×C4C22×C411yescharacteristic
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4C2×D412yes—
86C2×C32×C4#4112yes—
87#638—416no—
88C2×C3⋊Dic3—24no—
89C2×C3⋊Dic3C22⋊C414yes—
90C4⋊(C3⋊D3)—24no—
91C4⋊(C3⋊D3)—42no—
92C2×C3×Dic3—42no—
93C2×C3×Dic3—44no—
94C3×C3⋊C8—24no—
95C2×#127—91no—
96C2×#102—32no—
97C2×#102—92no—
98C2×#97—91no—
99#382—91no—
100C24×C3—22no—
101C24×C3—22no—
102C23×D3—62no—
103C23×D3—62no—
104C2×C3×D4—22no—
105C2×C3×D4—22no—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C2×C3×D4C4×D314yes—
114C22×C3×C4—21no—
115C22×C3×C4—62no—
116C22×C3×C4C22×D312yes—
117C2×#67—616no—
118C2×#67—616no—
119C2×#67—1216no—
120C2×#67—128no—
121C2×#67—128no—
122C22×Dic3—22no—
123C22×Dic3—34no—
124C22×Dic3—62no—
125C2×D12—34no—
126C2×D12—62no—
127C2×D12—62no—
128C2×D12—62no—
129C2×D12—62no—
130C2×D12—62no—
131C2×D12—62no—
132C2×D12—62no—
133C2×D12—122no—
134C2×D12—122no—
135C2×C3×C8—34no—
136C2×C3×C8—62no—
137C2×C3×C8—62no—
138C3×C4⋊C4—62no—
139C3×C4⋊C4—62no—
140C4⋊Dic3—22no—
141C4⋊Dic3—22no—
142C4×Dic3—34no—
143C4×Dic3—34no—
144C4×Dic3—62no—
145C4×Dic3—62no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8#6714yescomplemented
149C22×C32—42no—
150C22×C32—44no—
151C22×C32—82no—
152C22×C32—82no—
153C22×C32C2×C22⋊C411yescharacteristic
154C22×C32#12411yescharacteristic
155C22×C32#12711yescharacteristic
156C22×C32AGL(1,ℤ/8)14yes—
157C2×C3⋊D3—42no—
158C2×C3⋊D3—44no—
159C2×C3⋊D3—82no—
160C2×C3⋊D3—82no—
161C32×C4C4×D412yes—
162C32×C4#12112yes—
163C3⋊Dic3—24no—
164C3×Dic3—82no—
165C22×D4—91no—
166C22×D4—182no—
167C22×D4D3211yescharacteristic
168C22×C8—32no—
169#127—94no—
170#127—182no—
171C2×C4⋊C4—32no—
172C2×C42—91no—
173#102—64no—
174#102—184no—
175#97—182no—
176#95—182no—
177C23×C3—24no—
178C23×C3—24no—
179C23×C3—21no—
180C23×C3—48no—
181C23×C3—42no—
182C23×C3—42no—
183C23×C3—42no—
184C23×C3—44no—
185C23×C3—44no—
186C23×C3—42no—
187C23×C3D3×D412yes—
188C22×D3—64no—
189C22×D3—64no—
190C22×D3—122no—
191C22×D3—122no—
192C22×D3—128no—
193C22×D3—122no—
194C22×D3—124no—
195C22×D3—124no—
196C22×D3—122no—
197C3×D4—24no—
198C3×D4—24no—
199C3×D4—44no—
200C3×D4—44no—
201C3×D4—42no—
202C3×D4—42no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—124no—
208C2×C3×C4—122no—
209C2×C3×C4C2×C4×D312yes—
210C2×C3×C4#23512yes—
211C2×C3×C4D3×D412yes—
212C2×C3×C4D3×D412yes—
213C2×C3×C4#23712yes—
214C2×C3×C4C2×#6712yes—
215#67—1232no—
216#67—1232no—
217C2×Dic3—38no—
218C2×Dic3—44no—
219C2×Dic3—42no—
220C2×Dic3—64no—
221C2×Dic3—68no—
222C2×Dic3—64no—
223C2×Dic3—64no—
224D12—64no—
225D12—64no—
226D12—124no—
227D12—124no—
228D12—122no—
229D12—122no—
230C3×C8—64no—
231C3⋊C8—24no—
232C2×C32—82no—
233C2×C32#39011yescharacteristic
234C2×C32#41812yes—
235C2×C32#49012yes—
236C2×C32#49612yes—
237C3⋊D3—82no—
238C24—21no—
239C24—181no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—92no—
244C2×D4—188no—
245C2×D4—182no—
246C2×D4—182no—
247C2×D4—181no—
248C2×D4—368no—
249C2×D4#62412yes—
250C22×C4—62no—
251C22×C4—92no—
252C22×C4C2×D3211yescharacteristic
253C2×C8—34no—
254C2×C8—62no—
255C2×C8—62no—
256C4⋊C4—62no—
257C4⋊C4—62no—
258C42—92no—
259C42—92no—
260C22×C3—24no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—48no—
265C22×C3—44no—
266C22×C3—44no—
267C22×C3—48no—
268C22×C3—84no—
269C22×C3—84no—
270C22×C3—82no—
271C22×C3—82no—
272C22×C3—84no—
273C22×C3D3×C22⋊C412yes—
274C22×C3#88512yes—
275C22×C3#91114yes—
276C22×C3#91414yes—
277C22×C3#94012yes—
278C2×D3—124no—
279C2×D3—128no—
280C2×D3—124no—
281C2×D3—128no—
282C2×D3—244no—
283C2×D3—244no—
284C2×D3—242no—
285C2×D3—242no—
286C2×D3—244no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—242no—
290C3×C4#67⋊C412yes—
291C3×C4#88412yes—
292C3×C4C4×#6712yes—
293C3×C4#93912yes—
294Dic3—68no—
295Dic3—68no—
296Dic3—82no—
297C32#196111yescharacteristic, complemented, Sylow
298C23—22no—
299C23—44no—
300C23—41no—
301C23—41no—
302C23—182no—
303C23—364no—
304C23—361no—
305C23—361no—
306C23#381111yescharacteristic
307D4—24no—
308D4—42no—
309D4—184no—
310D4—3616no—
311D4—362no—
312C2×C4—94no—
313C2×C4—124no—
314C2×C4—122no—
315C2×C4—184no—
316C2×C4#380412yes—
317C2×C4C2×#62411yescharacteristic
318C2×C4D3×#6712yes—
319C2×C4#381111yescharacteristic
320C8—64no—
321C2×C3—22no—
322C2×C3—21no—
323C2×C3—22no—
324C2×C3—22no—
325C2×C3—84no—
326C2×C3—84no—
327C2×C3#476414yes—
328C2×C3#477014yes—
329C2×C3#495712yes—
330C2×C3#515412yes—
331C2×C3#517612yes—
332D3—244no—
333D3—244no—
334C22—44no—
335C22—42no—
336C22—82no—
337C22—82no—
338C22—364no—
339C22—362no—
340C22—722no—
341C22—722no—
342C22#6393714yes—
343C22#6395311yescharacteristic
344C22#6397411yescharacteristic
345C22#6397911yescharacteristic
346C4—184no—
347C4—242no—
348C4#6395212yes—
349C4#6397112yes—
350C3—21no—
351C3#7016012yescomplemented
352C2—82no—
353C2—722no—
354C2#1059142612yes—
355C2#1059147512yes—
356C2#1059148012yes—
357C2#1059170411yescharacteristic
subgroupG/H
centerC23#3811
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×D4C22
socleC23×C32C2×D4
prime radicalC2×#64149C2
cosocleC22×C32×C4C23
hypercenterC22×D4D32
hypocenterC32#1961
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#832C2semidirect
3. #10591230C2semidirect
4. C2×#63564C2semidirect
5. C2×#63555C2semidirect
11. #64149C4semidirect
22. C2×#832D3semidirect
48. C2×C3×C3⋊C8D4semidirect
148. C2×C3⋊C8#67semidirect
297. C32#1961semidirect
351. C3#70160semidirect

Maximal subgroups

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

subgroupindex
C2×#641492
C2×C3×#8322
#105912302
C2×#635642
C2×#635552
#701603