Finite Group Databasebeta

← #82570⋊C3 #11921281

Subgroups — #82570⋊C3 #11921281

Order 1152.

403 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 57 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×#1010⋊C3C211yesmaximal, characteristic
2C3×#520⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#82570C311yesmaximal, characteristic, complemented
5#2898⋊C3—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#1010⋊C3—21no—
8#1010⋊C3C412yescomplemented
9C2×D3×SL(2,𝔽3)—21no—
10C3×C4×SL(2,𝔽3)—21no—
11C3⋊(#131⋊C3)—21no—
12C3×#809—41nomaximal
13C2×#1010C2×C311yescharacteristic
14C2×2−1+4⋊C3—31no—
15C3×#520C2×C311yescharacteristic, complemented
16C2×#948C2×C311yescharacteristic, complemented
17C2×#131⋊C3—31no—
18#520⋊C3—21no—
19#520⋊C3D311yescharacteristic, complemented
20#5004—31no—
21#5002—31no—
22C2×#810—31no—
23#4966—31no—
24C2×C3×D12—41no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28D3×SL(2,𝔽3)—22no—
29C32×C4⋊C4—41no—
30C2×C3×C3⋊C8—41no—
31#2898—31noSylow
32C2×C3×#41C22×C311yescharacteristic
33#1010—21no—
34#1010—32no—
35#1010—61no—
36#1010C3×C412yescomplemented
37C2×D3×Q8—21no—
38C2×#237—61no—
39C2×#235—31no—
40C22×Dic6—31no—
412−1+4⋊C3—32no—
422−1+4⋊C3—61no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3C2×D311yescharacteristic
45C22×SL(2,𝔽3)—61no—
46C3×C4×Q8—21no—
47C3×C4×D4—61no—
48C3×#117—31no—
49C2×C3×C4⋊C4—31no—
50#948—21no—
51#948—61no—
52C2×#208—31no—
53C22×C3⋊C8—31no—
54#131⋊C3—61no—
55C4×SL(2,𝔽3)—22no—
56C4×SL(2,𝔽3)—21no—
57C3×#102—121no—
58#810—32no—
59#810—31no—
60#810—31no—
61#809—31no—
62#809A411yescharacteristic
63C22×C3×D3—81no—
64C2×C32×C4—41no—
65C2×C32×C4—81no—
66C3×D12—42no—
67C3×D12—81no—
68C3×SL(2,𝔽3)C22⋊C411yescharacteristic
69C3×C3⋊C8—81no—
70C2×2−1+4—31no—
71C2×#131—31no—
72#520C3×D311yescharacteristic, complemented
73#419—91no—
74#417—91no—
75#416—91no—
76C2×#103—91no—
77C3×#41—31no—
78C3×#41—31no—
79C3×#41C2×C3×C411yescharacteristic
80C3×#41C3×D411yescharacteristic
81C2×C3×Q8C3×D411yescharacteristic
82C2×C3×D4—31no—
83C2×C3×D4—121no—
84C22×C3×C4—31no—
85C22×C3×C4—61no—
86C2×#67—61no—
87C22×Dic3—61no—
88D3×Q8—22no—
89D3×Q8—62no—
90#237—64no—
91#237—62no—
92#237—62no—
93#235—32no—
94#235—32no—
95#235—61no—
96#235—61no—
97C2×D12C2×A411yescharacteristic
98C2×C4×D3—61no—
99C2×Dic6—32no—
100C2×Dic6—31no—
101C2×Dic6—32no—
102C2×Dic6—61no—
103C2×Dic6—61no—
104C2×Dic6—61no—
105#41⋊C3—21no—
106#41⋊C3—21no—
107#41⋊C3C4×D311yescharacteristic
108#41⋊C3D1211yescharacteristic
109C2×SL(2,𝔽3)—21no—
110C2×SL(2,𝔽3)—62no—
111C2×SL(2,𝔽3)#6711yescharacteristic
112C2×C3×C8—121no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4—81no—
117C3×C4⋊C4C2×A411yescharacteristic
118C3×C22⋊C4—61no—
119C3×C42—61no—
120#208—61no—
121#208—61no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—61no—
124C2×C3⋊C8—61no—
125C2×C3⋊C8C2×A411yescharacteristic
126C22×C32—41no—
127C2×C3×D3—82no—
128C2×C3×D3—161no—
129C2×C3×D3—161no—
130C32×C4—41no—
131C32×C4—41no—
132C32×C4—161no—
1332−1+4—32no—
1342−1+4—61no—
1352−1+4—92no—
1362−1+4—181no—
137C2×#41—91no—
138C2×#41—181no—
139C2×#41C2×C3×D311yescharacteristic
140C22×Q8—61no—
141C22×Q8—91no—
142#131—61no—
143#131—181no—
144C2×M16—91no—
145C22×C8—91no—
146C4×Q8—21no—
147C4×D4—61no—
148#117—31no—
149C2×C4⋊C4—31no—
150#103—91no—
151#103—92no—
152#103—91no—
153#102—31no—
154#102—91no—
155C23×C3—31no—
156C23×C3—241no—
157C22×D3—21no—
158C3×Q8—31no—
159C3×Q8C3×C22⋊C411yescharacteristic
160C3×D4—31no—
161C3×D4—31no—
162C3×D4—32no—
163C3×D4—122no—
164C3×D4—241no—
165C2×C3×C4—21no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—41no—
172C2×C3×C4—61no—
173C2×C3×C4—81no—
174C2×C3×C4—82no—
175C2×C3×C4—81no—
176C2×C3×C4—241no—
177C2×C3×C4C22×A411yescharacteristic
178#67—62no—
179#67—62no—
180C2×Dic3—62no—
181C2×Dic3—61no—
182C2×Dic3—62no—
183D12—21no—
184D12C4×A412yes—
185C4×D3—62no—
186C4×D3—62no—
187Dic6—32no—
188Dic6—62no—
189Dic6—61no—
190Dic6—62no—
191SL(2,𝔽3)—21no—
192SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
193C3×C8—241no—
194C3⋊C8—21no—
195C3⋊C8—61no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C3×D3—162no—
200#41—31no—
201#41—31no—
202#41—92no—
203#41—92no—
204#41—181no—
205#41—181no—
206#41—184no—
207#41—182no—
208#41—182no—
209#41C3×C4×D311yescharacteristic
210#41C3×D1211yescharacteristic
211C2×Q8—62no—
212C2×Q8—92no—
213C2×Q8—92no—
214C2×Q8—91no—
215C2×Q8—181no—
216C2×Q8—181no—
217C2×Q8—181no—
218C2×Q8—182no—
219C2×Q8C3×#6711yescharacteristic
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—181no—
223C22×C4—31no—
224C22×C4—61no—
225C22×C4—181no—
226C22×C4—181no—
227M16—181no—
228M16—181no—
229C2×C8—31no—
230C2×C8—91no—
231C2×C8—181no—
232C2×C8—181no—
233C4⋊C4—31no—
234C4⋊C4—61no—
235C4⋊C4D3×A411yescharacteristic
236C22⋊C4—61no—
237C42—61no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—41no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—81no—
244C22×C3—242no—
245C22×C3—481no—
246C22×C3—481no—
247C22×C3D4×A411yescharacteristic
248C2×D3—22no—
249C2×D3—41no—
250C2×D3—41no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—81no—
257C3×C4—81no—
258C3×C4—121no—
259C3×C4—161no—
260C3×C4—162no—
261C3×C4C2×C4×A411yescharacteristic
262C3×C4D4×A411yescharacteristic
263Dic3—62no—
264C32—41noSylow
265C23—31no—
266C23—61no—
267Q8—31no—
268Q8—92no—
269Q8—182no—
270Q8—182no—
271Q8—181no—
272Q8C3×(C2×D3)⋊C411yescharacteristic
273D4—31no—
274D4—31no—
275D4—32no—
276D4—32no—
277D4—61no—
278D4—182no—
279D4—182no—
280C2×C4—21no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—61no—
287C2×C4—182no—
288C2×C4—182no—
289C2×C4—182no—
290C2×C4—181no—
291C2×C4—182no—
292C2×C4—241no—
293C2×C4C2×D3×A411yescharacteristic
294C8—61no—
295C8—181no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—61no—
300C2×C3—61no—
301C2×C3—81no—
302C2×C3—81no—
303C2×C3—81no—
304C2×C3—482no—
305C2×C3A4×C22⋊C411yescharacteristic
306C2×C3#577⋊C311yescharacteristic
307C2×C3#578⋊C311yescharacteristic
308D3—42no—
309C22—31no—
310C22—31no—
311C22—61no—
312C22—61no—
313C22—62no—
314C22—121no—
315C22—121no—
316C22A4×#6711yescharacteristic
317C4—31no—
318C4—31no—
319C4—41no—
320C4—121no—
321C4—182no—
322C4C4×D3×A411yescharacteristic
323C4A4×D1211yescharacteristic
324C3—41no—
325C3—81no—
326C3#2898⋊C311yescharacteristic, complemented
327C2—61no—
328C2—61no—
329C2—122no—
330C2A4×(C2×D3)⋊C411yescharacteristic
331C2#5836⋊C311yescharacteristic
332C2#5838⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutatorC3×#41C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC3×#520C2×C3
socleC22×C3D4×A4
prime radicalC2×#1010⋊C3C2
cosocleC2×C3×#41C22×C3
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. C2×C3⋊(#131⋊C3)C2semidirect
4. #82570C3semidirect
8. #1010⋊C3C4semidirect
15. C3×#520C2×C3semidirect
16. C2×#948C2×C3semidirect
19. #520⋊C3D3semidirect
36. #1010C3×C4semidirect
72. #520C3×D3semidirect
326. C3#2898⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1010⋊C32
C3×#520⋊C32
C2×C3⋊(#131⋊C3)2
#825703
#2898⋊C33
C3×#8094