Finite Group Databasebeta

← #85803⋊C3 #11923176

Subgroups — #85803⋊C3 #11923176

Order 1152.

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

normal subgroups only

This group has 543 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
1C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2#5812⋊C3C212yesmaximal
3#5570⋊C3C211yesmaximal, characteristic, complemented
4#5569⋊C3C212yesmaximal, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#85803C311yesmaximal, characteristic, complemented
7#3502⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C3×#924—41nomaximal
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2212yes—
14C2×C3×C4×Q8C2×C311yescharacteristic
15#5812C2×C312yes—
16#5570—31no—
17#5570C2×C311yescharacteristic, complemented
18#5569—32no—
19#5569C2×C312yescomplemented
20Q8×Dic6—31no—
21Q8×Dic6C2×C311yescharacteristic, complemented
22#5500—32no—
23#5470—31no—
24#5470—31no—
25Q8×SL(2,𝔽3)—31no—
26#557⋊C3—32no—
27#557⋊C3—31no—
28#519⋊C3—32no—
29C2×C4×SL(2,𝔽3)—21no—
30C2×C4×SL(2,𝔽3)D311yescharacteristic
31C22×C32×C4—41no—
32C2×C3×Dic6—41no—
33C2×C3×SL(2,𝔽3)Q812yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3×#217—42no—
36C3×C4⋊Dic3—41no—
37C3×Dic3⋊C4—42no—
38#3502—31noSylow
39C22×C3×Q8C22×C311yescharacteristic
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)C2×D311yescharacteristic
42C3×C22⋊Q8—121no—
43C3×C4×Q8—31no—
44C3×C4×Q8—31no—
45C3×C4×Q8C22×C311yescharacteristic
46C3×C4×Q8C22×C311yescharacteristic
47C2×C3×C4⋊C4—31no—
48C2×C3×C42—31no—
49Q8×Dic3—32no—
50Q8×Dic3—32no—
51Q8×Dic3C22×C312yes—
52Q8×Dic3C22×C312yes—
53#926—32no—
54#926—32no—
55#924—32no—
56#924—31no—
57#924—32no—
58#924A411yescharacteristic
59#890—34no—
60#890—32no—
61#890—32no—
62#888—32no—
63#888—34no—
64#888—32no—
65#870—32no—
66#870—32no—
67#869—31no—
68#869—31no—
69#869—31no—
70#869—31no—
71C4×Dic6—32no—
72C4×Dic6—32no—
73C4×Dic6—31no—
74C4×Dic6—31no—
75C4×Dic6—32no—
76C4×SL(2,𝔽3)—21no—
77C4×SL(2,𝔽3)—21no—
78C4×SL(2,𝔽3)—32no—
79C4×SL(2,𝔽3)—32no—
80C4×SL(2,𝔽3)C2×D311yescharacteristic
81C4×SL(2,𝔽3)C2×D311yescharacteristic
82C23×C32—41no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4—82no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—42no—
88C3×Dic6—82no—
89C3×SL(2,𝔽3)C2×Q811yescharacteristic
90Q82—31no—
91Q82—91no—
92#557—32no—
93#557—31no—
94#557—91no—
95#557—92no—
96#544—92no—
97#533—91no—
98#533—91no—
99#519—32no—
100C2×C4×Q8C3×D311yescharacteristic
101C2×C3×Q8—31no—
102C2×C3×Q8—32no—
103C2×C3×Q8—121no—
104C2×C3×Q8C3×Q812yescomplemented
105C2×C3×Q8C23×C311yescharacteristic
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—41no—
109C22×C3×C4—81no—
110C22×C3×C4C2×A411yescharacteristic
111C2×Dic6—31no—
112C2×Dic6—32no—
113C2×Dic6—32no—
114C2×Dic6C2×A411yescharacteristic
115C2×SL(2,𝔽3)—22no—
116C2×SL(2,𝔽3)—21no—
117C2×SL(2,𝔽3)Dic612yescomplemented
118C2×SL(2,𝔽3)C22×D311yescharacteristic
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—122no—
123C3×C4⋊C4—121no—
124C3×C22⋊C4—122no—
125C3×C42—31no—
126C3×C42—31no—
127C3×C42—31no—
128C3×C42—31no—
129#217—32no—
130#217C2×A412yes—
131C4⋊Dic3—32no—
132C4⋊Dic3—31no—
133C4⋊Dic3—32no—
134C4⋊Dic3—32no—
135C4⋊Dic3—32no—
136C4⋊Dic3—32no—
137C4⋊Dic3—32no—
138C4⋊Dic3C2×A411yescharacteristic
139Dic3⋊C4—34no—
140Dic3⋊C4—32no—
141Dic3⋊C4—32no—
142Dic3⋊C4—32no—
143Dic3⋊C4C2×A412yes—
144C4×Dic3—34no—
145C4×Dic3—32no—
146C4×Dic3—32no—
147C22×C32—42no—
148C22×C32—41no—
149C22×C32—81no—
150C22×C32—81no—
151C32×C4—42no—
152C32×C4—81no—
153C3×Dic3—82no—
154C3×Dic3—82no—
155C22×Q8C2×C3×D311yescharacteristic
156C4⋊Q8—91no—
157C4⋊Q8—91no—
158C4⋊Q8—91no—
159C4⋊Q8—91no—
160C4⋊Q8—92no—
161C4⋊Q8—94no—
162C4⋊Q8—92no—
163#125—92no—
164#125—92no—
165#125—92no—
166#125—92no—
167#125—94no—
168C22⋊Q8—31no—
169C22⋊Q8—92no—
170C22⋊Q8—92no—
171C22⋊Q8—91no—
172C4×Q8—31no—
173C4×Q8—31no—
174C4×Q8—32no—
175C4×Q8—32no—
176C4×Q8—92no—
177C4×Q8—91no—
178C4×Q8—91no—
179C4×Q8—92no—
180C4×Q8—92no—
181C4×Q8—92no—
182C4×Q8—92no—
183C4×Q8C2×C3×D311yescharacteristic
184C4×Q8C2×C3×D311yescharacteristic
185#117—92no—
186#117—92no—
187C2×C4⋊C4—31no—
188C2×C42—31no—
189C23×C3—41no—
190C23×C3—81no—
191C23×C3C22×A411yescharacteristic
192C3×Q8—32no—
193C3×Q8—31no—
194C3×Q8—32no—
195C3×Q8—242no—
196C3×Q8C2×C3×Q811yescharacteristic
197C2×C3×C4—22no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—82no—
207C2×C3×C4—84no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4—122no—
211C2×C3×C4—122no—
212C2×C3×C4—121no—
213C2×C3×C4C22×A411yescharacteristic
214C2×C3×C4C22×A411yescharacteristic
215C2×Dic3—32no—
216C2×Dic3—32no—
217C2×Dic3C22×A412yes—
218C2×Dic3C22×A412yes—
219Dic6—22no—
220Dic6—62no—
221Dic6—122no—
222Dic6—122no—
223SL(2,𝔽3)—21no—
224SL(2,𝔽3)C2×Dic611yescharacteristic
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—82no—
229C2×Q8—31no—
230C2×Q8—31no—
231C2×Q8—32no—
232C2×Q8—92no—
233C2×Q8—92no—
234C2×Q8—91no—
235C2×Q8C3×Dic612yescomplemented
236C2×Q8C22×C3×D311yescharacteristic
237C22×C4—31no—
238C22×C4—31no—
239C22×C4D3×A411yescharacteristic
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—32no—
243C4⋊C4—32no—
244C4⋊C4—31no—
245C4⋊C4—92no—
246C4⋊C4—92no—
247C4⋊C4—92no—
248C4⋊C4—92no—
249C4⋊C4—92no—
250C4⋊C4—92no—
251C4⋊C4—92no—
252C4⋊C4—92no—
253C4⋊C4—94no—
254C4⋊C4—92no—
255C4⋊C4—91no—
256C22⋊C4—32no—
257C22⋊C4—92no—
258C42—31no—
259C42—31no—
260C42—31no—
261C42—31no—
262C42—92no—
263C42—92no—
264C42—94no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—42no—
268C22×C3—41no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3—82no—
272C22×C3—82no—
273C22×C3—82no—
274C22×C3—81no—
275C22×C3Q8×A412yes—
276C22×C3C23×A411yescharacteristic
277C3×C4—21no—
278C3×C4—31no—
279C3×C4—32no—
280C3×C4—31no—
281C3×C4—42no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—81no—
285C3×C4—82no—
286C3×C4—82no—
287C3×C4—242no—
288C3×C4—242no—
289C3×C42−1+4⋊C312yes—
290Dic3—22no—
291Dic3—22no—
292Dic3—62no—
293Dic3—62no—
294C32—41noSylow
295C23C2×D3×A411yescharacteristic
296Q8—32no—
297Q8—31no—
298Q8—32no—
299Q8—62no—
300Q8—182no—
301Q8—362no—
302Q8—362no—
303Q8C2×C3×Dic611yescharacteristic
304C2×C4—22no—
305C2×C4—32no—
306C2×C4—32no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—32no—
312C2×C4—32no—
313C2×C4—92no—
314C2×C4—92no—
315C2×C4—121no—
316C2×C4C2×D3×A411yescharacteristic
317C2×C4C2×D3×A411yescharacteristic
318C2×C3—22no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—82no—
323C2×C3—84no—
324C2×C3—81no—
325C2×C3—81no—
326C2×C3—81no—
327C2×C3C2×Q8×A411yescharacteristic
328C2×C3C2×2−1+4⋊C311yescharacteristic
329C2×C3#594511yescharacteristic
330C22—21no—
331C22—21no—
332C22A4×Dic612yes—
333C22C22×D3×A411yescharacteristic
334C4—21no—
335C4—31no—
336C4—32no—
337C4—31no—
338C4—62no—
339C4—61no—
340C4—61no—
341C4—62no—
342C4—182no—
343C4—182no—
344C4#1007⋊C312yes—
345C3—41no—
346C3—81no—
347C3#3502⋊C311yescharacteristic, complemented
348C2—22no—
349C2C2×A4×Dic611yescharacteristic
350C2C2×#1007⋊C311yescharacteristic
351C2#5964⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC2×C3×C4×Q8C2×C3
socleC22×C3C23×A4
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #5570⋊C3C2semidirect
4. #5569⋊C3C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #85803C3semidirect
17. #5570C2×C3semidirect
19. #5569C2×C3semidirect
21. Q8×Dic6C2×C3semidirect
33. C2×C3×SL(2,𝔽3)Q8semidirect
104. C2×C3×Q8C3×Q8semidirect
117. C2×SL(2,𝔽3)Dic6semidirect
235. C2×Q8C3×Dic6semidirect
347. C3#3502⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
#5812⋊C32
#5570⋊C32
#5569⋊C32
SL(2,𝔽3)×Dic62
#858033
#3502⋊C33
C3×#9244