Finite Group Databasebeta

← #85889⋊C3 #11923220

Subgroups — #85889⋊C3 #11923220

Order 1152.

551 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 91 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#5569⋊C3C212yesmaximal, complemented
5SL(2,𝔽3)×Dic6C212yesmaximal, complemented
6#85889C311yesmaximal, characteristic, complemented
7#3503⋊C3—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#888—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#5569C2×C312yes—
18Q8×Dic6C2×C312yes—
19#5539—32no—
20D4×Dic6—32no—
21#5519—31no—
22#5517—31no—
23#5500—31no—
24C2×#888—31no—
25Q8×SL(2,𝔽3)—32no—
26#557⋊C3—32no—
27#520⋊C3—21no—
28#520⋊C3—31no—
29#520⋊C3D311yescharacteristic
30C4×#41⋊C3—31no—
31C2×C3×Dic6—42no—
32C3×#41⋊C3Q812yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C32×C4⋊C4—41no—
35C3×C4⋊Dic3—41no—
36C3×Dic3⋊C4—42no—
37C3×C4×Dic3—41no—
38#3503—31noSylow
39C2×C3×#41C22×C311yescharacteristic
40C22×Dic6—32no—
41C2×#41⋊C3—21no—
42C2×#41⋊C3C2×D311yescharacteristic
43C3×C4⋊Q8—121no—
44C3×C4×Q8C22×C312yes—
45C3×C4×D4—32no—
46C3×#117—31no—
47C2×C3×C4⋊C4—31no—
48Q8×Dic3C22×C311yescharacteristic
49Q8×Dic3C22×C312yes—
50Q8×Dic3C22×C311yescharacteristic
51D4×Dic3—31no—
52D4×Dic3—31no—
53D4×Dic3—32no—
54#926—31no—
55#926—31no—
56C2×C4⋊Dic3—31no—
57#924—34no—
58C2×Dic3⋊C4—32no—
59C2×C4×Dic3—31no—
60#890—31no—
61#890—31no—
62#890—32no—
63#888—32no—
64#888—32no—
65#888—32no—
66#888—32no—
67#888—31no—
68#888A411yescharacteristic
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×Dic6—84no—
87C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
88Q82—32no—
89#557—32no—
90#554—92no—
91D4×Q8—92no—
92#544—91no—
93#544—91no—
94#533—91no—
95C2×C4⋊Q8—91no—
96#520—31no—
97#520C3×D311yescharacteristic
98C4×#41—31no—
99C3×#41—32no—
100C3×#41C3×Q812yescomplemented
101C2×C3×Q8—122no—
102C2×C3×Q8C23×C311yescharacteristic
103C2×C3×D4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—32no—
106C22×Dic3—32no—
107C22×Dic3—31no—
108C22×Dic3—31no—
109C2×Dic6—32no—
110C2×Dic6—34no—
111C2×Dic6—124no—
112C2×Dic6C2×A412yes—
113#41⋊C3—22no—
114#41⋊C3Dic612yescomplemented
115C2×SL(2,𝔽3)—21no—
116C2×SL(2,𝔽3)C22×D311yescharacteristic
117C3×C4⋊C4—32no—
118C3×C4⋊C4—31no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—81no—
121C3×C4⋊C4—121no—
122C3×C4⋊C4—122no—
123C3×C4⋊C4C2×A411yescharacteristic
124C3×C22⋊C4—32no—
125C3×C42—32no—
126C3×C42—121no—
127#217—32no—
128#217—32no—
129C4⋊Dic3—32no—
130C4⋊Dic3—31no—
131C4⋊Dic3—32no—
132C4⋊Dic3—31no—
133C4⋊Dic3—32no—
134C4⋊Dic3—31no—
135C4⋊Dic3C2×A411yescharacteristic
136Dic3⋊C4—34no—
137Dic3⋊C4—32no—
138Dic3⋊C4—32no—
139Dic3⋊C4—32no—
140Dic3⋊C4C2×A412yes—
141C4×Dic3—31no—
142C4×Dic3—31no—
143C4×Dic3—31no—
144C4×Dic3—32no—
145C4×Dic3C2×A411yescharacteristic
146C22×C32—41no—
147C32×C4—42no—
148C32×C4—82no—
149C3×Dic3—42no—
150C3×Dic3—42no—
151C3×Dic3—82no—
152C2×#41C2×C3×D311yescharacteristic
153C22×Q8—92no—
154C4⋊Q8—31no—
155C4⋊Q8—91no—
156C4⋊Q8—92no—
157C4⋊Q8—92no—
158C4⋊Q8—92no—
159C4⋊Q8—92no—
160C4⋊Q8—92no—
161#125—92no—
162#125—91no—
163#125—92no—
164#125—91no—
165C22⋊Q8—92no—
166C22⋊Q8—92no—
167C22⋊Q8—94no—
168C22⋊Q8—94no—
169C4×Q8—31no—
170C4×Q8—31no—
171C4×Q8—32no—
172C4×Q8—92no—
173C4×Q8—92no—
174C4×Q8C2×C3×D312yes—
175C4×D4—32no—
176C4×D4—91no—
177C4×D4—91no—
178C4×D4—92no—
179#117—31no—
180#117—91no—
181#117—91no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—92no—
184C2×C4⋊C4—91no—
185C2×C42—91no—
186C23×C3—31no—
187C3×Q8—31no—
188C3×Q8—244no—
189C3×Q8C2×C3×Q811yescharacteristic
190C3×D4—32no—
191C3×D4—32no—
192C2×C3×C4—31no—
193C2×C3×C4—32no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—81no—
199C2×C3×C4—82no—
200C2×C3×C4—121no—
201C2×C3×C4—121no—
202C2×C3×C4—122no—
203C2×C3×C4—122no—
204C2×C3×C4C22×A412yes—
205C2×C3×C4C22×A411yescharacteristic
206C2×Dic3—31no—
207C2×Dic3—31no—
208C2×Dic3—32no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—122no—
212C2×Dic3C22×A412yes—
213C2×Dic3C22×A411yescharacteristic
214C2×Dic3C22×A411yescharacteristic
215Dic6—24no—
216Dic6—64no—
217Dic6—124no—
218SL(2,𝔽3)—21no—
219SL(2,𝔽3)C2×Dic611yescharacteristic, complemented
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223#41—32no—
224#41C3×Dic612yescomplemented
225C2×Q8—32no—
226C2×Q8—92no—
227C2×Q8—94no—
228C2×Q8—364no—
229C2×Q8C22×C3×D311yescharacteristic
230C2×D4—31no—
231C22×C4—31no—
232C22×C4—32no—
233C22×C4—91no—
234C22×C4—91no—
235C22×C4—92no—
236C4⋊C4—32no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—32no—
240C4⋊C4—92no—
241C4⋊C4—92no—
242C4⋊C4—92no—
243C4⋊C4—92no—
244C4⋊C4—91no—
245C4⋊C4—91no—
246C4⋊C4—94no—
247C4⋊C4—91no—
248C4⋊C4—92no—
249C4⋊C4—92no—
250C4⋊C4D3×A411yescharacteristic
251C22⋊C4—32no—
252C22⋊C4—92no—
253C22⋊C4—92no—
254C42—31no—
255C42—32no—
256C42—91no—
257C42—91no—
258C42—92no—
259C42—91no—
260C22×C3—32no—
261C22×C3—41no—
262C22×C3—61no—
263C22×C3—61no—
264C22×C3—81no—
265C22×C3C23×A411yescharacteristic
266C3×C4—22no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—42no—
270C3×C4—62no—
271C3×C4—82no—
272C3×C4—82no—
273C3×C4—122no—
274C3×C4—122no—
275C3×C4—162no—
276C3×C4—242no—
277C3×C4Q8×A412yes—
278Dic3—22no—
279Dic3—62no—
280Dic3—61no—
281Dic3—61no—
282Dic3#99412yes—
283Dic32−1+4⋊C312yes—
284C32—41noSylow
285C23—31no—
286Q8—31no—
287Q8—64no—
288Q8—184no—
289Q8—364no—
290Q8C2×C3×Dic611yescharacteristic
291D4—32no—
292D4—32no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—92no—
303C2×C4—122no—
304C2×C4—182no—
305C2×C4—182no—
306C2×C4—362no—
307C2×C4C2×D3×A412yes—
308C2×C4C2×D3×A411yescharacteristic
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—62no—
313C2×C3—81no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3C2×Q8×A411yescharacteristic
317C2×C3C2×#99411yescharacteristic
318C2×C3C2×2−1+4⋊C311yescharacteristic
319C22—32no—
320C22—61no—
321C22—61no—
322C22C22×D3×A411yescharacteristic
323C4—22no—
324C4—32no—
325C4—32no—
326C4—31no—
327C4—31no—
328C4—62no—
329C4—62no—
330C4—181no—
331C4—182no—
332C4—181no—
333C4A4×Dic612yes—
334C3—41no—
335C3—81no—
336C3#3503⋊C311yescharacteristic, complemented
337C2—62no—
338C2C2×A4×Dic611yescharacteristic
339C2D3×#99411yescharacteristic
340C2D3×2−1+4⋊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. #5569⋊C3C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #85889C3semidirect
32. C3×#41⋊C3Q8semidirect
87. C3×SL(2,𝔽3)C2×Q8semidirect
100. C3×#41C3×Q8semidirect
114. #41⋊C3Dic6semidirect
219. SL(2,𝔽3)C2×Dic6semidirect
224. #41C3×Dic6semidirect
336. C3#3503⋊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
#5569⋊C32
SL(2,𝔽3)×Dic62
#858893
#3503⋊C33
C3×#8884