Finite Group Databasebeta

← C3×#3502⋊C3 #11919752

Subgroups — C3×#3502⋊C3 #11919752

Order 1152.

620 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 162 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#557⋊C3C212yesmaximal, complemented
3C3×#557⋊C3C211yesmaximal, characteristic, complemented
4C3×#519⋊C3C212yesmaximal
5C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
6C3×#3502C311yesmaximal, characteristic, complemented
7#3502⋊C3C313yesmaximal, direct product factor
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C32×C22⋊Q8—41nomaximal
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2212yes—
13C3×C4×SL(2,𝔽3)C2212yes—
14C3×Q82—31no—
15C3×Q82C2×C311yescharacteristic, complemented
16C3×#557—31no—
17C3×#557—32no—
18C3×#557C2×C311yescharacteristic, complemented
19C3×#557C2×C312yescomplemented
20C3×#544—32no—
21C3×#533—31no—
22C3×#533—31no—
23C3×#519C2×C312yes—
24C2×C3×C4×Q8C2×C311yescharacteristic
25Q8×SL(2,𝔽3)C2×C313yescomplemented
26#557⋊C3C2×C316yescomplemented
27#557⋊C3C2×C313yescomplemented
28#519⋊C3C2×C316yes—
29C2×C4×SL(2,𝔽3)C2×C313yes—
30C2×C32×Q8—41no—
31C22×C32×C4—41no—
32C2×C3×SL(2,𝔽3)Q812yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C32×C4⋊C4—41no—
35C32×C4⋊C4—42no—
36C32×C22⋊C4—42no—
37#3502C3211yescharacteristic, complemented, Sylow
38C22×C3×Q8C22×C311yescharacteristic
39C22×SL(2,𝔽3)C22×C313yes—
40C3×C4⋊Q8—32no—
41C3×C4⋊Q8—31no—
42C3×C4⋊Q8—34no—
43C3×C4⋊Q8—31no—
44C3×C4⋊Q8—31no—
45C3×C4⋊Q8—31no—
46C3×C4⋊Q8—32no—
47C3×#125—32no—
48C3×#125—32no—
49C3×#125—32no—
50C3×#125—34no—
51C3×#125—32no—
52C3×C22⋊Q8—32no—
53C3×C22⋊Q8—31no—
54C3×C22⋊Q8—32no—
55C3×C22⋊Q8—43no—
56C3×C22⋊Q8A411yescharacteristic
57C3×C4×Q8—32no—
58C3×C4×Q8—32no—
59C3×C4×Q8—31no—
60C3×C4×Q8—31no—
61C3×C4×Q8—32no—
62C3×C4×Q8—32no—
63C3×C4×Q8—32no—
64C3×C4×Q8—31no—
65C3×C4×Q8—31no—
66C3×C4×Q8C22×C311yescharacteristic
67C3×C4×Q8C22×C312yes—
68C3×C4×Q8C22×C311yescharacteristic
69C3×C4×Q8C22×C312yes—
70C3×#117—32no—
71C3×#117—32no—
72C2×C3×C4⋊C4—31no—
73C2×C3×C42—31no—
74C4×SL(2,𝔽3)C22×C313yes—
75C4×SL(2,𝔽3)C22×C313yes—
76C4×SL(2,𝔽3)C22×C316yes—
77C4×SL(2,𝔽3)C22×C316yes—
78C23×C32—41no—
79C32×Q8—82no—
80C2×C32×C4—41no—
81C2×C32×C4—42no—
82C2×C32×C4—42no—
83C2×C32×C4—41no—
84C2×C32×C4—82no—
85C3×SL(2,𝔽3)C2×Q811yescharacteristic
86Q82—31no—
87Q82C2×C3211yescharacteristic, complemented
88#557—31no—
89#557—32no—
90#557C2×C3212yescomplemented
91#557C2×C3211yescharacteristic, complemented
92#544—32no—
93#533—31no—
94#533—31no—
95#519C2×C3212yes—
96C2×C4×Q8C2×C3211yescharacteristic
97C2×C3×Q8—32no—
98C2×C3×Q8—31no—
99C2×C3×Q8—32no—
100C2×C3×Q8—31no—
101C2×C3×Q8—32no—
102C2×C3×Q8—43no—
103C2×C3×Q8C3×Q812yescomplemented
104C2×C3×Q8C2×A411yescharacteristic
105C2×C3×Q8C23×C311yescharacteristic
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—43no—
109C22×C3×C4C2×A411yescharacteristic
110C2×SL(2,𝔽3)C3×Q816yescomplemented
111C2×SL(2,𝔽3)C23×C313yes—
112C3×C4⋊C4—31no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—32no—
115C3×C4⋊C4—32no—
116C3×C4⋊C4—32no—
117C3×C4⋊C4—32no—
118C3×C4⋊C4—34no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—46no—
127C3×C4⋊C4—43no—
128C3×C4⋊C4C2×A412yes—
129C3×C4⋊C4C2×A411yescharacteristic
130C3×C22⋊C4—32no—
131C3×C22⋊C4—46no—
132C3×C22⋊C4C2×A412yes—
133C3×C42—34no—
134C3×C42—31no—
135C3×C42—31no—
136C3×C42—31no—
137C3×C42—31no—
138C3×C42—32no—
139C3×C42—32no—
140C22×C32—42no—
141C22×C32—41no—
142C22×C32—81no—
143C22×C32—81no—
144C32×C4—42no—
145C32×C4—82no—
146C32×C4—81no—
147C32×C4—82no—
148C22×Q8C22×C3211yescharacteristic
149C4⋊Q8—31no—
150C4⋊Q8—32no—
151C4⋊Q8—34no—
152C4⋊Q8—31no—
153C4⋊Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8—32no—
156#125—32no—
157#125—32no—
158#125—32no—
159#125—34no—
160#125—32no—
161C22⋊Q8—32no—
162C22⋊Q8—31no—
163C22⋊Q8—32no—
164C22⋊Q8C3×A411yescharacteristic
165C4×Q8—32no—
166C4×Q8—32no—
167C4×Q8—31no—
168C4×Q8—32no—
169C4×Q8—31no—
170C4×Q8—32no—
171C4×Q8—32no—
172C4×Q8—31no—
173C4×Q8—31no—
174C4×Q8C22×C3211yescharacteristic
175C4×Q8C22×C3211yescharacteristic
176C4×Q8C22×C3212yes—
177C4×Q8C22×C3212yes—
178#117—32no—
179#117—32no—
180C2×C4⋊C4—31no—
181C2×C42—31no—
182C23×C3—43no—
183C23×C3C22×A411yescharacteristic
184C3×Q8—22no—
185C3×Q8—32no—
186C3×Q8—31no—
187C3×Q8—32no—
188C3×Q8—62no—
189C3×Q8—86no—
190C3×Q8—122no—
191C3×Q8—122no—
192C3×Q8C2×C3×Q811yescharacteristic
193C2×C3×C4—22no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—43no—
203C2×C3×C4—46no—
204C2×C3×C4—46no—
205C2×C3×C4—43no—
206C2×C3×C4—86no—
207C2×C3×C4—121no—
208C2×C3×C4C22×A411yescharacteristic
209C2×C3×C4C22×A412yes—
210C2×C3×C4C22×A412yes—
211C2×C3×C4C22×A411yescharacteristic
212SL(2,𝔽3)C2×C3×Q813yes—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32—82no—
217C2×Q8—32no—
218C2×Q8—31no—
219C2×Q8—32no—
220C2×Q8—31no—
221C2×Q8—32no—
222C2×Q8C32×Q812yescomplemented
223C2×Q8C2×C3×A411yescharacteristic
224C2×Q8C23×C3211yescharacteristic
225C22×C4—31no—
226C22×C4—31no—
227C22×C4C2×C3×A411yescharacteristic
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—32no—
231C4⋊C4—32no—
232C4⋊C4—32no—
233C4⋊C4—32no—
234C4⋊C4—34no—
235C4⋊C4—32no—
236C4⋊C4—32no—
237C4⋊C4—32no—
238C4⋊C4—31no—
239C4⋊C4—32no—
240C4⋊C4—32no—
241C4⋊C4—32no—
242C4⋊C4C2×C3×A412yes—
243C4⋊C4C2×C3×A411yescharacteristic
244C22⋊C4—32no—
245C22⋊C4C2×C3×A412yes—
246C42—34no—
247C42—31no—
248C42—31no—
249C42—31no—
250C42—31no—
251C42—32no—
252C42—32no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—46no—
256C22×C3—43no—
257C22×C3—83no—
258C22×C3—83no—
259C22×C3Q8×A412yes—
260C22×C3C23×A411yescharacteristic
261C3×C4—21no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—31no—
265C3×C4—32no—
266C3×C4—31no—
267C3×C4—46no—
268C3×C4—62no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—62no—
272C3×C4—83no—
273C3×C4—86no—
274C3×C4—86no—
275C3×C42−1+4⋊C312yes—
276C32—41noSylow
277C23C22×C3×A411yescharacteristic
278Q8—22no—
279Q8—32no—
280Q8—31no—
281Q8—32no—
282Q8—62no—
283Q8—122no—
284Q8—122no—
285Q8C2×C32×Q811yescharacteristic
286C2×C4—22no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—121no—
296C2×C4C22×C3×A411yescharacteristic
297C2×C4C22×C3×A412yes—
298C2×C4C22×C3×A412yes—
299C2×C4C22×C3×A411yescharacteristic
300C2×C3—22no—
301C2×C3—43no—
302C2×C3—43no—
303C2×C3—43no—
304C2×C3—86no—
305C2×C3C2×Q8×A411yescharacteristic
306C2×C3C2×2−1+4⋊C311yescharacteristic
307C2×C3#594511yescharacteristic
308C22—21no—
309C22—21no—
310C22C3×Q8×A412yes—
311C22C23×C3×A411yescharacteristic
312C4—21no—
313C4—22no—
314C4—22no—
315C4—31no—
316C4—32no—
317C4—31no—
318C4—62no—
319C4—61no—
320C4—61no—
321C4—62no—
322C4C3×2−1+4⋊C312yes—
323C3—43no—
324C3#3502⋊C311yescharacteristic, direct product factor
325C2—22no—
326C2C2×C3×Q8×A411yescharacteristic
327C2C2×C3×2−1+4⋊C311yescharacteristic
328C2C3×#594511yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3502C3
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC3×C22⋊Q8A4
hypocenterQ8C2×C32×Q8
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
2. C3×#557⋊C3C2semidirect
3. C3×#557⋊C3C2semidirect
6. C3×#3502C3semidirect
7. #3502⋊C3C3direct (N × H)
15. C3×Q82C2×C3semidirect
18. C3×#557C2×C3semidirect
19. C3×#557C2×C3semidirect
25. Q8×SL(2,𝔽3)C2×C3semidirect
26. #557⋊C3C2×C3semidirect
27. #557⋊C3C2×C3semidirect
32. C2×C3×SL(2,𝔽3)Q8semidirect
37. #3502C32semidirect
87. Q82C2×C32semidirect
90. #557C2×C32semidirect
91. #557C2×C32semidirect
103. C2×C3×Q8C3×Q8semidirect
110. C2×SL(2,𝔽3)C3×Q8semidirect
222. C2×Q8C32×Q8semidirect
324. C3#3502⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
C3×#557⋊C32
C3×#519⋊C32
C2×C3×C4×SL(2,𝔽3)2
C3×#35023
#3502⋊C33
C32×C22⋊Q84