Finite Group Databasebeta

← #11792555

Subgroups — #11792555

Order 1152.

1041 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C22×C3×#215C212yesmaximal, complemented
2C2×#63914C211yesmaximal, characteristic
3#10592928C211yesmaximal, characteristic, complemented
4C2×#63563C212yesmaximal, complemented
5#10589156C211yesmaximal, characteristic, complemented
6#72056—32nomaximal
7C22×C32×Q8C2211yescharacteristic
8C22×C3×D12C2212yes—
9C22×#625—22no—
10C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
11C2×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×#215—24no—
13C2×C3×#215—24no—
14C2×C3×#215—24no—
15C22×C3×C3⋊C8C2212yes—
16#63914—24no—
17#63658—22no—
18#63563—24no—
19C22×C3×SD16—32no—
20C22×#215D312yescomplemented
21C2×#940—32no—
22#5229—32no—
23#5229—61no—
24C2×#832—32no—
25C2×#821—32no—
26#4550—32no—
27C23×C3×D3—22no—
28C2×C32×Q8—21no—
29C2×C32×Q8—22no—
30C2×C32×Q8—22no—
31C2×C32×Q8D412yescomplemented
32C22×C32×C4—21no—
33C22×C32×C4C2311yescharacteristic
34C22×C3⋊Dic3—21no—
35C22×C3⋊Dic3D412yes—
36C2×C3×D12—22no—
37C2×C3×D12—22no—
38C2×C3×D12—22no—
39C2×C3×D12—22no—
40C2×C3×D12—22no—
41C2×C3×D12D414yes—
42C2×#625—28no—
43C2×#625—44no—
44C2×#625—44no—
45C4⋊(C3⋊Dic3)—22no—
46C4×C3⋊Dic3—22no—
47C3×#215—216no—
48C2×C3×C3⋊C8—22no—
49C2×C3×C3⋊C8—22no—
50C2×C3×C3⋊C8D414yescomplemented
51#2014—91noSylow
52C22×C3×Q8—21no—
53C22×C3×Q8C2×D312yes—
54C22×C3×D4—32no—
55C22×#67—64no—
56C22×D12C2×D312yes—
57C2×C3×SD16—64no—
58C2×C3×SD16—64no—
59C2×C3×SD16—64no—
60C22×C3×C8—32no—
61C2×#215—24no—
62C2×#215—24no—
63C2×#215—24no—
64#940—64no—
65#940—64no—
66C2×C4⋊Dic3—32no—
67C2×C4⋊Dic3—61no—
68C2×C4×Dic3—32no—
69C2×C4×Dic3—61no—
70C22×C3⋊C8C2×D312yes—
71#832—64no—
72C3⋊#95—64no—
73C3⋊#95—64no—
74#821—64no—
75C23×C32C2×D411yescharacteristic
76C22×C3×D3—24no—
77C22×C3×D3—42no—
78C22×C3×D3—42no—
79C22×C3×D3—44no—
80C22×C3×D3—44no—
81C32×Q8—24no—
82C32×Q8—42no—
83C2×C32×C4—22no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4C2×D412yes—
87C2×C32×C4C2×D412yes—
88C2×C32×C4C2×D411yescharacteristic
89C2×C32×C4#4111yescharacteristic
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3—41no—
93C2×C3⋊Dic3—42no—
94C2×C3⋊Dic3SD1614yescomplemented
95C3×D12—24no—
96C3×D12—24no—
97C3×D12—42no—
98C3×D12—42no—
99#625—416no—
100C3×C3⋊C8—24no—
101C22×SD16—32no—
102C2×#127—91no—
103C2×#102—92no—
104#391—91no—
105#336—91no—
106C24×C3—62no—
107C23×D3—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—22no—
111C2×C3×Q8—22no—
112C2×C3×Q8—22no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—42no—
116C2×C3×Q8—42no—
117C2×C3×Q8#6714yescomplemented
118C2×C3×D4—34no—
119C2×C3×D4—62no—
120C2×C3×D4—62no—
121C2×C3×D4—62no—
122C2×C3×D4—62no—
123C2×C3×D4—62no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4C22×D312yes—
128C2×#67—616no—
129C2×#67—128no—
130C2×#67—128no—
131C22×Dic3—34no—
132C22×Dic3—62no—
133C22×Dic3—62no—
134C22×Dic3—62no—
135C2×D12—22no—
136C2×D12—22no—
137C2×D12—22no—
138C2×D12—22no—
139C2×D12—22no—
140C2×D12#6714yes—
141C3×SD16—616no—
142C2×C3×C8—34no—
143C2×C3×C8—62no—
144C2×C3×C8—62no—
145#215—216no—
146C4⋊Dic3—62no—
147C4⋊Dic3—62no—
148C4⋊Dic3—122no—
149C4×Dic3—62no—
150C4×Dic3—62no—
151C4×Dic3—122no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8#6714yescomplemented
155C22×C32#12011yescharacteristic
156C22×C32#12111yescharacteristic
157C22×C32#12711yescharacteristic
158C22×C32C2×SD1612yes—
159C22×C32AGL(1,ℤ/8)12yes—
160C2×C3×D3—44no—
161C2×C3×D3—48no—
162C2×C3×D3—84no—
163C2×C3×D3—82no—
164C2×C3×D3—82no—
165C32×C4—42no—
166C32×C4#12012yes—
167C32×C4#12112yes—
168C3⋊Dic3—24no—
169C3⋊Dic3—81no—
170C22×Q8D3211yescharacteristic
171C22×D4—32no—
172C22×D4—182no—
173C2×SD16—64no—
174C2×SD16—64no—
175C2×SD16—64no—
176C22×C8—32no—
177#127—184no—
178C2×C4⋊C4—91no—
179C2×C42—91no—
180#102—184no—
181#95—182no—
182C23×C3—21no—
183C23×C3—64no—
184C23×C3—124no—
185C23×C3—124no—
186C23×C3—122no—
187C23×C3—122no—
188C23×C3D3×D412yes—
189C22×D3—24no—
190C22×D3—44no—
191C22×D3—44no—
192C22×D3—42no—
193C22×D3—42no—
194C3×Q8—24no—
195C3×Q8—24no—
196C3×Q8—44no—
197C3×Q8—44no—
198C3×Q8—42no—
199C3×Q8—42no—
200C3×D4—64no—
201C3×D4—64no—
202C3×D4—122no—
203C3×D4—122no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4D3×D412yes—
215C2×C3×C4D3×D412yes—
216C2×C3×C4#23712yes—
217C2×C3×C4C2×#6712yes—
218C2×C3×C4C2×#6712yes—
219C2×C3×C4C2×#6712yes—
220#67—1232no—
221C2×Dic3—38no—
222C2×Dic3—64no—
223C2×Dic3—64no—
224C2×Dic3—64no—
225C2×Dic3—122no—
226C2×Dic3—122no—
227C2×Dic3—122no—
228C2×Dic3—124no—
229C2×Dic3—124no—
230C2×Dic3—122no—
231D12—24no—
232D12—24no—
233D12—42no—
234D12—42no—
235C3×C8—64no—
236C3⋊C8—24no—
237C2×C32#39211yescharacteristic
238C2×C32#45012yes—
239C2×C32#46012yes—
240C2×C32#49211yescharacteristic
241C2×C32#49611yescharacteristic
242C3×D3—84no—
243C24—62no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8#62512yescomplemented
248C2×D4—34no—
249C2×D4—62no—
250C2×D4—62no—
251C2×D4—62no—
252C2×D4—62no—
253C2×D4—62no—
254C2×D4—188no—
255C2×D4—364no—
256C2×D4—364no—
257C22×C4—21no—
258C22×C4—92no—
259C22×C4—181no—
260C22×C4C2×D3211yescharacteristic
261SD16—616no—
262C2×C8—34no—
263C2×C8—62no—
264C2×C8—62no—
265C4⋊C4—182no—
266C42—182no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—128no—
273C22×C3—124no—
274C22×C3—244no—
275C22×C3—242no—
276C22×C3—242no—
277C22×C3D3×SD1614yes—
278C22×C3#91414yes—
279C22×C3#93712yes—
280C22×C3#93912yes—
281C22×C3#94012yes—
282C2×D3—48no—
283C2×D3—44no—
284C2×D3—84no—
285C2×D3—82no—
286C2×D3—82no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—44no—
290C3×C4—44no—
291C3×C4#93712yes—
292C3×C4#93812yes—
293C3×C4#93912yes—
294C3×C4#95312yes—
295Dic3—68no—
296Dic3—124no—
297Dic3—242no—
298Dic3—242no—
299C32#201411yescharacteristic, complemented, Sylow
300C23—64no—
301C23—124no—
302C23—124no—
303C23—122no—
304C23—122no—
305C23#381111yescharacteristic
306Q8—24no—
307Q8—42no—
308D4—64no—
309D4—64no—
310D4—122no—
311D4—122no—
312D4—3616no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—94no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4—362no—
320C2×C4—361no—
321C2×C4#380511yescharacteristic
322C2×C4C2×#62511yescharacteristic
323C2×C4D3×#6712yes—
324C2×C4D3×#6712yes—
325C8—64no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—22no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—244no—
332C2×C3#516314yes—
333C2×C3#516914yes—
334C2×C3#517512yes—
335C2×C3#517612yes—
336C2×C3#522312yes—
337D3—84no—
338C22—128no—
339C22—124no—
340C22—242no—
341C22—242no—
342C22—244no—
343C22#6394012yes—
344C22#6394112yes—
345C22#6397911yescharacteristic
346C22#6398011yescharacteristic
347C22#6398311yescharacteristic
348C4—42no—
349C4—184no—
350C4—721no—
351C4#6397012yes—
352C4#6397612yes—
353C3—21no—
354C3#7205612yescomplemented
355C2—244no—
356C2#1059162011yescharacteristic
357C2#1059162111yescharacteristic
358C2#1059162212yes—
359C2#1059165212yes—
360C2#1059172311yescharacteristic
subgroupG/H
centerC23#3811
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×Q8C22
socleC23×C32C2×D4
prime radicalC2×#63914C2
hypercenterC22×Q8D32
hypocenterC32#2014
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×#215C2semidirect
3. #10592928C2semidirect
4. C2×#63563C2semidirect
5. #10589156C2semidirect
20. C22×#215D3semidirect
31. C2×C32×Q8D4semidirect
50. C2×C3×C3⋊C8D4semidirect
94. C2×C3⋊Dic3SD16semidirect
117. C2×C3×Q8#67semidirect
154. C2×C3⋊C8#67semidirect
247. C2×Q8#625semidirect
299. C32#2014semidirect
354. C3#72056semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2152
C2×#639142
#105929282
C2×#635632
#105891562
#720563