Finite Group Databasebeta

← #11794030

Subgroups — #11794030

Order 1152.

789 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 789 conjugacy classes of subgroups — too many to draw. Only its 158 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×#3713C211yesmaximal, characteristic
2C2×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
3C3×#4939C212yesmaximal, complemented
4C2×#63562C212yesmaximal, complemented
5#10589163C211yesmaximal, characteristic, complemented
6#69650—32nomaximal
7C22×C3×D12C2212yes—
8C2×C32×C42C2211yescharacteristic
9C22×C3⋊(C3⋊C8)C2211yescharacteristic
10C22×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3×(C2×D3)⋊C4—22no—
12C2×C3×(C2×D3)⋊C4—22no—
13C2×C3×C4⋊Dic3C2212yes—
14C2×#3713—24no—
15C2×#3713—22no—
16C4⋊(C3⋊(C3⋊C8))—22no—
17#63562—24no—
18C22×#213—32no—
19C3×#390—32no—
20C2×#832—32no—
21C2×#809—32no—
22#4939D312yescomplemented
23C2×C4⋊(C3⋊C8)—32no—
24C2×C4⋊(C3⋊C8)—61no—
25#4529—32no—
26C23×C3×D3—22no—
27C22×C32×C4D411yescharacteristic
28C22×C32×C4D411yescharacteristic
29C22×C32×C4C2311yescharacteristic
30C22×C3×Dic3—22no—
31C2×C3×D12—22no—
32C2×C3×D12—22no—
33C2×C3×D12—22no—
34C2×C3×D12—22no—
35C2×C3×D12—22no—
36C2×C3×D12D414yes—
37C32×C42—21no—
38C32×C42—21no—
39C2×C3⋊(C3⋊C8)—22no—
40C2×C3⋊(C3⋊C8)—21no—
41C2×C3⋊(C3⋊C8)—22no—
42C2×C3⋊(C3⋊C8)D412yes—
43C3×(C2×D3)⋊C4—44no—
44C3×(C2×D3)⋊C4—44no—
45C3×C4⋊Dic3—22no—
46C3×C4⋊Dic3—22no—
47#3713—28no—
48#2054—91noSylow
49C22×C3×D4—32no—
50C22×D12C2×D312yes—
51C2×C3×C4⋊C4—32no—
52C2×C3×C22⋊C4—62no—
53C2×C3×C22⋊C4—62no—
54C2×C3×C42—21no—
55C2×C3×C42C2×D312yes—
56C2×#213—64no—
57C2×#213—64no—
58C2×#213—64no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×C4⋊Dic3C2×D312yes—
62C22×C3⋊C8—32no—
63C22×C3⋊C8—32no—
64C22×C3⋊C8—61no—
65C22×C3⋊C8—61no—
66#832—64no—
67#809—64no—
68C4⋊(C3⋊C8)—64no—
69C4⋊(C3⋊C8)—64no—
70C23×C32C2×D411yescharacteristic
71C22×C3×D3—24no—
72C22×C3×D3—42no—
73C22×C3×D3—42no—
74C22×C3×D3—42no—
75C22×C3×D3—42no—
76C22×C3×D3—42no—
77C22×C3×D3—42no—
78C2×C32×C4—21no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—22no—
82C2×C32×C4D812yescomplemented
83C2×C32×C4C2×D412yes—
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4#4112yes—
86C2×C32×C4#4111yescharacteristic
87C2×C3×Dic3—42no—
88C2×C3×Dic3—42no—
89C2×C3×Dic3—42no—
90C3×D12—24no—
91C3×D12—24no—
92C3×D12—42no—
93C3×D12—42no—
94C3⋊(C3⋊C8)—22no—
95C3⋊(C3⋊C8)—41no—
96C22×D8—91no—
97C2×C4⋊C8—91no—
98C2×#102—92no—
99#390—32no—
100#340—91no—
101C24×C3—62no—
102C23×D3—22no—
103C2×C3×D4—34no—
104C2×C3×D4—62no—
105C2×C3×D4—62no—
106C2×C3×D4—62no—
107C2×C3×D4—62no—
108C2×C3×D4—62no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—62no—
113C22×C3×C4#6712yes—
114C22×C3×C4#6712yes—
115C22×C3×C4C22×D312yes—
116C22×Dic3—22no—
117C2×D12—22no—
118C2×D12—22no—
119C2×D12—22no—
120C2×D12—22no—
121C2×D12—22no—
122C2×D12D1214yes—
123C3×C4⋊C4—62no—
124C3×C4⋊C4—62no—
125C3×C22⋊C4—124no—
126C3×C22⋊C4—124no—
127C3×C42—22no—
128C3×C42—22no—
129C3×C42—22no—
130C3×C42—22no—
131#213—616no—
132(C2×D3)⋊C4—44no—
133(C2×D3)⋊C4—44no—
134C4⋊Dic3—22no—
135C4⋊Dic3—22no—
136C2×C3⋊C8—34no—
137C2×C3⋊C8—62no—
138C2×C3⋊C8—62no—
139C2×C3⋊C8—62no—
140C2×C3⋊C8—62no—
141C2×C3⋊C8—62no—
142C2×C3⋊C8—62no—
143C2×C3⋊C8—62no—
144C2×C3⋊C8—122no—
145C2×C3⋊C8—122no—
146C22×C32#12112yes—
147C22×C32#12111yescharacteristic
148C22×C32C2×D811yescharacteristic
149C22×C32#13511yescharacteristic
150C22×C32AGL(1,ℤ/8)12yes—
151C2×C3×D3—44no—
152C2×C3×D3—44no—
153C2×C3×D3—44no—
154C2×C3×D3—82no—
155C2×C3×D3—82no—
156C2×C3×D3—82no—
157C2×C3×D3—82no—
158C32×C4—22no—
159C32×C4—41no—
160C32×C4#12011yescharacteristic
161C32×C4#12312yes—
162C32×C4#12411yescharacteristic
163C3×Dic3—82no—
164C22×D4—32no—
165C2×D8—182no—
166C2×D8—184no—
167C22×C8—91no—
168C22×C8—91no—
169C2×C4⋊C4—32no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—62no—
172C2×C42D3211yescharacteristic
173C4⋊C8—182no—
174#102—184no—
175C23×C3—21no—
176C23×C3—64no—
177C23×C3—122no—
178C23×C3—122no—
179C23×C3—122no—
180C23×C3—122no—
181C23×C3—122no—
182C23×C3—122no—
183C23×C3C2×#6712yes—
184C22×D3—24no—
185C22×D3—42no—
186C22×D3—42no—
187C22×D3—42no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—42no—
191C3×D4—64no—
192C3×D4—64no—
193C3×D4—122no—
194C3×D4—122no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—42no—
209C2×C3×C4—122no—
210C2×C3×C4—122no—
211C2×C3×C4—122no—
212C2×C3×C4#21314yescomplemented
213C2×C3×C4C2×D1212yes—
214C2×C3×C4#23512yes—
215C2×C3×C4D3×D412yes—
216C2×C3×C4D3×D412yes—
217C2×C3×C4#23712yes—
218C2×C3×C4#23712yes—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222D12—24no—
223D12—24no—
224D12—42no—
225D12—42no—
226C3⋊C8—64no—
227C3⋊C8—122no—
228C3⋊C8—122no—
229C3⋊C8—122no—
230C2×C32#39411yescharacteristic
231C2×C32#45912yes—
232C2×C32#46312yes—
233C2×C32#46611yescharacteristic
234C2×C32#46911yescharacteristic
235C3×D3—84no—
236C24—62no—
237C2×D4—34no—
238C2×D4—62no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—62no—
243C22×C4—62no—
244C22×C4#62511yescharacteristic
245C22×C4#62511yescharacteristic
246C22×C4C2×D3211yescharacteristic
247D8—188no—
248C2×C8—92no—
249C2×C8—182no—
250C2×C8—182no—
251C2×C8—181no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C22⋊C4—124no—
255C22⋊C4—124no—
256C42—21no—
257C42—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—21no—
263C22×C3—124no—
264C22×C3—124no—
265C22×C3—124no—
266C22×C3—242no—
267C22×C3—242no—
268C22×C3—242no—
269C22×C3—242no—
270C22×C3#93012yes—
271C22×C3C2×#21312yes—
272C22×C3#93212yes—
273C22×C3#93812yes—
274C22×C3#93912yes—
275C22×C3#94912yes—
276C22×C3#95012yes—
277C2×D3—44no—
278C2×D3—44no—
279C2×D3—44no—
280C2×D3—82no—
281C2×D3—82no—
282C2×D3—82no—
283C2×D3—82no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—42no—
290C3×C4—42no—
291C3×C4—242no—
292C3×C4#88212yes—
293C3×C4#88312yes—
294C3×C4#88512yes—
295C3×C4#88612yes—
296Dic3—82no—
297C32#205411yescharacteristic, complemented, Sylow
298C23—64no—
299C23—122no—
300C23—122no—
301C23—122no—
302C23—122no—
303C23—122no—
304C23—122no—
305C23C2×#62511yescharacteristic
306D4—64no—
307D4—64no—
308D4—122no—
309D4—122no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4#371312yescomplemented
318C2×C4#379512yes—
319C2×C4D3×D1212yes—
320C2×C4#380511yescharacteristic
321C2×C4#381111yescharacteristic
322C8—182no—
323C8—361no—
324C2×C3—22no—
325C2×C3—22no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—244no—
330C2×C3#496112yes—
331C2×C3#501512yes—
332C2×C3#501612yes—
333C2×C3#503812yes—
334C2×C3#503912yes—
335C2×C3#505712yes—
336C2×C3#507212yes—
337D3—84no—
338C22—124no—
339C22—124no—
340C22—124no—
341C22—242no—
342C22—242no—
343C22—242no—
344C22—242no—
345C22C2×#371311yescharacteristic
346C22#6382211yescharacteristic
347C22#6382312yes—
348C22#6390812yes—
349C22#6398011yescharacteristic
350C4—22no—
351C4—41no—
352C4—242no—
353C4#6389112yes—
354C4#6391311yescharacteristic
355C4#6392311yescharacteristic
356C3—21no—
357C3#6965012yescomplemented
358C2—244no—
359C2#1059034712yes—
360C2#1059034812yes—
361C2#1059116611yescharacteristic
362C2#1059116711yescharacteristic
363C2#1059139411yescharacteristic
subgroupG/H
centerC23C2×#625
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socleC23×C32C2×D4
prime radicalC22×#3713C2
hypercenterC2×C42D32
hypocenterC32#2054
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C3⋊(C3⋊C8))C2semidirect
3. C3×#4939C2semidirect
4. C2×#63562C2semidirect
5. #10589163C2semidirect
22. #4939D3semidirect
82. C2×C32×C4D8semidirect
212. C2×C3×C4#213semidirect
297. C32#2054semidirect
317. C2×C4#3713semidirect
357. C3#69650semidirect

Maximal subgroups

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

subgroupindex
C22×#37132
C2×C4⋊(C3⋊(C3⋊C8))2
C3×#49392
C2×#635622
#105891632
#696503