Finite Group Databasebeta

← #85890⋊C3 #11923222

Subgroups — #85890⋊C3 #11923222

Order 1152.

507 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C3×#520⋊C3C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic
3Dic3×#41⋊C3C211yesmaximal, characteristic
4#5570⋊C3C212yesmaximal, complemented
5#5569⋊C3C212yesmaximal, complemented
6#85890C311yesmaximal, characteristic, complemented
7#3504⋊C3—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#890—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11Dic3×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×#520C2×C311yescharacteristic
15#5827C2×C311yescharacteristic
16Dic3×#41C2×C311yescharacteristic
17#5570C2×C312yes—
18#5569C2×C312yes—
19#5543—32no—
20#5539—32no—
21#5519—31no—
22#5517—31no—
23#5500—31no—
24C2×#890—31no—
25#557⋊C3—32no—
26#557⋊C3—32no—
27#520⋊C3—21no—
28#520⋊C3—31no—
29#520⋊C3D311yescharacteristic
30C4×#41⋊C3—31no—
31C3×#41⋊C3Q812yescomplemented
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C32×C4⋊C4—41no—
34C3×C4⋊Dic3—42no—
35C3×C4⋊Dic3—41no—
36C3×Dic3⋊C4—42no—
37C3×C4×Dic3—41no—
38#3504—31noSylow
39C2×C3×#41C22×C311yescharacteristic
40C2×#41⋊C3—21no—
41C2×#41⋊C3C2×D311yescharacteristic
42C3×#125—121no—
43C3×C4×Q8C22×C312yes—
44C3×C4×D4—32no—
45C3×#117—31no—
46C2×C3×C4⋊C4—31no—
47Q8×Dic3C22×C311yescharacteristic
48Q8×Dic3C22×C312yes—
49Q8×Dic3C22×C311yescharacteristic
50D4×Dic3—31no—
51D4×Dic3—31no—
52D4×Dic3—32no—
53#926—31no—
54#926—31no—
55C2×C4⋊Dic3—32no—
56C2×C4⋊Dic3—31no—
57#924—34no—
58C2×Dic3⋊C4—32no—
59C2×C4×Dic3—31no—
60#890—32no—
61#890—32no—
62#890—32no—
63#890—32no—
64#890—31no—
65#890A411yescharacteristic
66#888—32no—
67#888—31no—
68#888—31no—
69#878—34no—
70#878—32no—
71#878—32no—
72#870—32no—
73#869—32no—
74C4×Dic6—32no—
75C4×Dic6—32no—
76C4×SL(2,𝔽3)—22no—
77C4×SL(2,𝔽3)—31no—
78C4×SL(2,𝔽3)—31no—
79C4×SL(2,𝔽3)—32no—
80C4×SL(2,𝔽3)C2×D312yes—
81C2×C32×C4—41no—
82C2×C32×C4—42no—
83C2×C3×Dic3—41no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—42no—
86C3×SL(2,𝔽3)C2×Q811yescharacteristic
87#557—32no—
88#557—32no—
89#554—92no—
90#554—92no—
91#544—91no—
92#544—91no—
93#533—91no—
94C2×#125—91no—
95#520—31no—
96#520C3×D311yescharacteristic
97C4×#41—31no—
98C3×#41—32no—
99C3×#41C3×Q812yes—
100C2×C3×Q8C23×C311yescharacteristic
101C2×C3×D4—31no—
102C22×C3×C4—31no—
103C22×C3×C4—32no—
104C22×Dic3—32no—
105C22×Dic3—31no—
106C22×Dic3—31no—
107C2×Dic6—32no—
108C2×Dic6—32no—
109#41⋊C3—22no—
110#41⋊C3Dic612yescomplemented
111C2×SL(2,𝔽3)—21no—
112C2×SL(2,𝔽3)C22×D311yescharacteristic
113C3×C4⋊C4—32no—
114C3×C4⋊C4—31no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—81no—
117C3×C4⋊C4—121no—
118C3×C4⋊C4—122no—
119C3×C4⋊C4—122no—
120C3×C4⋊C4C2×A411yescharacteristic
121C3×C22⋊C4—32no—
122C3×C42—32no—
123C3×C42—121no—
124#217—32no—
125#217—32no—
126C4⋊Dic3—34no—
127C4⋊Dic3—31no—
128C4⋊Dic3—32no—
129C4⋊Dic3—31no—
130C4⋊Dic3—32no—
131C4⋊Dic3—31no—
132C4⋊Dic3C2×A412yes—
133C4⋊Dic3C2×A411yescharacteristic
134Dic3⋊C4—34no—
135Dic3⋊C4—32no—
136Dic3⋊C4—32no—
137Dic3⋊C4—32no—
138Dic3⋊C4C2×A412yes—
139C4×Dic3—31no—
140C4×Dic3—32no—
141C4×Dic3—31no—
142C4×Dic3—31no—
143C4×Dic3C2×A411yescharacteristic
144C22×C32—41no—
145C32×C4—42no—
146C32×C4—82no—
147C3×Dic3—82no—
148C3×Dic3—81no—
149C3×Dic3—81no—
150C2×#41C2×C3×D311yescharacteristic
151C4⋊Q8—92no—
152C4⋊Q8—91no—
153C4⋊Q8—92no—
154C4⋊Q8—91no—
155#125—31no—
156#125—91no—
157#125—92no—
158#125—92no—
159#125—92no—
160#125—92no—
161#125—92no—
162C22⋊Q8—92no—
163C22⋊Q8—92no—
164C22⋊Q8—94no—
165C22⋊Q8—94no—
166C4×Q8—31no—
167C4×Q8—31no—
168C4×Q8—32no—
169C4×Q8—92no—
170C4×Q8—92no—
171C4×Q8C2×C3×D312yes—
172C4×D4—32no—
173C4×D4—91no—
174C4×D4—91no—
175C4×D4—92no—
176#117—31no—
177#117—91no—
178#117—91no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—92no—
181C2×C4⋊C4—92no—
182C2×C4⋊C4—91no—
183C2×C42—91no—
184C23×C3—31no—
185C3×Q8—31no—
186C3×Q8C2×C3×Q811yescharacteristic
187C3×D4—32no—
188C3×D4—32no—
189C2×C3×C4—31no—
190C2×C3×C4—32no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4—81no—
196C2×C3×C4—82no—
197C2×C3×C4—121no—
198C2×C3×C4—121no—
199C2×C3×C4—122no—
200C2×C3×C4—122no—
201C2×C3×C4C22×A412yes—
202C2×C3×C4C22×A411yescharacteristic
203C2×Dic3—31no—
204C2×Dic3—31no—
205C2×Dic3—32no—
206C2×Dic3—121no—
207C2×Dic3—121no—
208C2×Dic3—122no—
209C2×Dic3C22×A411yescharacteristic
210C2×Dic3C22×A411yescharacteristic
211C2×Dic3C22×A412yes—
212Dic6—122no—
213Dic6—122no—
214SL(2,𝔽3)—21no—
215SL(2,𝔽3)C2×Dic611yescharacteristic
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219#41—32no—
220#41C3×Dic612yes—
221C2×Q8—92no—
222C2×Q8—92no—
223C2×Q8C22×C3×D311yescharacteristic
224C2×D4—31no—
225C22×C4—31no—
226C22×C4—32no—
227C22×C4—91no—
228C22×C4—91no—
229C22×C4—92no—
230C4⋊C4—32no—
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—32no—
234C4⋊C4—32no—
235C4⋊C4—92no—
236C4⋊C4—92no—
237C4⋊C4—92no—
238C4⋊C4—92no—
239C4⋊C4—91no—
240C4⋊C4—91no—
241C4⋊C4—94no—
242C4⋊C4—94no—
243C4⋊C4—91no—
244C4⋊C4—92no—
245C4⋊C4D3×A411yescharacteristic
246C22⋊C4—32no—
247C22⋊C4—92no—
248C22⋊C4—92no—
249C42—31no—
250C42—32no—
251C42—91no—
252C42—91no—
253C42—92no—
254C42—91no—
255C22×C3—32no—
256C22×C3—41no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—81no—
260C22×C3C23×A411yescharacteristic
261C3×C4—22no—
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—42no—
265C3×C4—62no—
266C3×C4—82no—
267C3×C4—82no—
268C3×C4—162no—
269C3×C4—241no—
270C3×C4—242no—
271C3×C4—241no—
272C3×C4Q8×A412yes—
273Dic3—22no—
274Dic3—21no—
275Dic3—21no—
276Dic3—61no—
277Dic3—61no—
278Dic3—62no—
279C32—41noSylow
280C23—31no—
281Q8—31no—
282Q8—362no—
283Q8—362no—
284Q8C2×C3×Dic611yescharacteristic
285D4—32no—
286D4—32no—
287C2×C4—31no—
288C2×C4—32no—
289C2×C4—32no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—92no—
297C2×C4—122no—
298C2×C4—361no—
299C2×C4—361no—
300C2×C4—362no—
301C2×C4C2×D3×A412yes—
302C2×C4C2×D3×A411yescharacteristic
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—62no—
307C2×C3—81no—
308C2×C3—81no—
309C2×C3—81no—
310C2×C3C2×Q8×A411yescharacteristic
311C2×C3#594511yescharacteristic
312C2×C3#594511yescharacteristic
313C22—32no—
314C22—61no—
315C22—61no—
316C22C22×D3×A411yescharacteristic
317C4—22no—
318C4—31no—
319C4—31no—
320C4—62no—
321C4—61no—
322C4—62no—
323C4—61no—
324C4—181no—
325C4—182no—
326C4—181no—
327C4A4×Dic612yes—
328C3—41no—
329C3—81no—
330C3#3504⋊C311yescharacteristic, complemented
331C2—62no—
332C2C2×A4×Dic611yescharacteristic
333C2#1059662311yescharacteristic
334C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#520C2×C3
socleC22×C3C23×A4
prime radicalC2×C3×#41⋊C3C22
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #5570⋊C3C2semidirect
5. #5569⋊C3C2semidirect
6. #85890C3semidirect
31. C3×#41⋊C3Q8semidirect
110. #41⋊C3Dic6semidirect
330. C3#3504⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#520⋊C32
#5827⋊C32
Dic3×#41⋊C32
#5570⋊C32
#5569⋊C32
#858903
#3504⋊C33
C3×#8904