Finite Group Databasebeta

← C3×#3522⋊C3 #11919764

Subgroups — C3×#3522⋊C3 #11919764

Order 1152.

672 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 150 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C212yesmaximal, complemented
4C3×#520⋊C3C212yesmaximal, complemented
5C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C3×#3522C311yesmaximal, characteristic, complemented
7#3522⋊C3C313yesmaximal, direct product factor
8C2×C3×#41⋊C3C2212yes—
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C32×C22⋊Q8—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2212yes—
14C3×#559—31no—
15C3×Q82C2×C311yescharacteristic, complemented
16C3×#554—31no—
17C3×#550C2×C311yescharacteristic, complemented
18C3×#548C2×C312yes—
19C3×#547—32no—
20C3×#542—32no—
21C3×#532—31no—
22C3×#529—31no—
23C3×#520C2×C312yescomplemented
24C2×C3×C4×Q8C2×C311yescharacteristic
25Q8×SL(2,𝔽3)C2×C313yescomplemented
26C4⋊(#41⋊C3)C2×C313yescomplemented
27#548⋊C3C2×C316yescomplemented
28#520⋊C3C2×C316yescomplemented
29C2×C4×SL(2,𝔽3)C2×C313yescomplemented
30C2×C32×Q8—41no—
31C22×C32×C4—41no—
32C3×#41⋊C3—22no—
33C2×C3×SL(2,𝔽3)—21no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C32×C4⋊C4—42no—
36C32×C4⋊C4—41no—
37C32×C22⋊C4—42no—
38#3522C3211yescharacteristic, complemented, Sylow
39C2×C3×#41—32no—
40C2×C3×#41C22×C312yes—
41C22×C3×Q8C22×C311yescharacteristic
42C2×#41⋊C3C22×C316yes—
43C22×SL(2,𝔽3)C22×C313yes—
44C3×C4⋊Q8—31no—
45C3×C4⋊Q8—32no—
46C3×#127—31no—
47C3×#126—32no—
48C3×#125—31no—
49C3×#124—32no—
50C3×#124—31no—
51C3×#124—32no—
52C3×#123—34no—
53C3×#123—32no—
54C3×C22⋊Q8—34no—
55C3×C22⋊Q8—32no—
56C3×C22⋊Q8—43no—
57C3×C22⋊Q8A411yescharacteristic
58C3×#121—32no—
59C3×#121—31no—
60C3×#121—32no—
61C3×C4×Q8—31no—
62C3×C4×Q8—31no—
63C3×C4×Q8—31no—
64C3×C4×Q8C22×C311yescharacteristic
65C3×C4×Q8C22×C312yes—
66C3×C4×Q8C22×C311yescharacteristic
67C3×C4×D4—32no—
68C3×C4×D4—31no—
69C3×C4×D4—32no—
70C3×C4×D4—32no—
71C3×C4×D4—32no—
72C3×#117—32no—
73C2×C3×C4⋊C4—31no—
74C2×C3×C4⋊C4—32no—
75C2×C3×C42—31no—
76C4×SL(2,𝔽3)C22×C313yes—
77C4×SL(2,𝔽3)C22×C313yes—
78C4×SL(2,𝔽3)C22×C316yes—
79C23×C32—41no—
80C32×Q8—82no—
81C2×C32×C4—41no—
82C2×C32×C4—42no—
83C2×C32×C4—42no—
84C2×C32×C4—41no—
85C2×C32×C4—82no—
86C3×SL(2,𝔽3)#4111yescharacteristic, complemented
87#559—31no—
88Q82C2×C3211yescharacteristic, complemented
89#554—31no—
90#550C2×C3211yescharacteristic, complemented
91#548C2×C3212yes—
92#547—32no—
93#542—32no—
94#532—31no—
95#529—31no—
96#520C2×C3212yescomplemented
97C2×C4×Q8C2×C3211yescharacteristic
98C3×#41—22no—
99C3×#41—62no—
100C3×#41—124no—
101C2×C3×Q8—21no—
102C2×C3×Q8—32no—
103C2×C3×Q8—31no—
104C2×C3×Q8—43no—
105C2×C3×Q8—61no—
106C2×C3×Q8C2×A411yescharacteristic
107C2×C3×Q8C23×C311yescharacteristic
108C2×C3×D4—32no—
109C2×C3×D4—32no—
110C2×C3×D4—32no—
111C2×C3×D4—31no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—34no—
115C22×C3×C4—32no—
116C22×C3×C4—43no—
117C22×C3×C4C2×A411yescharacteristic
118#41⋊C3—26no—
119C2×SL(2,𝔽3)—23no—
120C2×SL(2,𝔽3)C23×C313yes—
121C3×C4⋊C4—31no—
122C3×C4⋊C4—32no—
123C3×C4⋊C4—31no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—31no—
127C3×C4⋊C4—32no—
128C3×C4⋊C4—32no—
129C3×C4⋊C4—46no—
130C3×C4⋊C4—43no—
131C3×C4⋊C4C2×A412yes—
132C3×C4⋊C4C2×A411yescharacteristic
133C3×C22⋊C4—34no—
134C3×C22⋊C4—32no—
135C3×C22⋊C4—32no—
136C3×C22⋊C4—32no—
137C3×C22⋊C4—32no—
138C3×C22⋊C4—46no—
139C3×C22⋊C4C2×A412yes—
140C3×C42—31no—
141C3×C42—31no—
142C3×C42—31no—
143C3×C42—31no—
144C3×C42—32no—
145C22×C32—42no—
146C22×C32—41no—
147C22×C32—81no—
148C22×C32—81no—
149C32×C4—42no—
150C32×C4—81no—
151C32×C4—82no—
152C32×C4—82no—
153C2×#41—32no—
154C2×#41C22×C3212yes—
155C22×Q8C22×C3211yescharacteristic
156C4⋊Q8—31no—
157C4⋊Q8—32no—
158#127—31no—
159#126—32no—
160#125—31no—
161#124—32no—
162#124—31no—
163#124—32no—
164#123—34no—
165#123—32no—
166C22⋊Q8—34no—
167C22⋊Q8—32no—
168C22⋊Q8C3×A411yescharacteristic
169#121—32no—
170#121—31no—
171#121—32no—
172C4×Q8—31no—
173C4×Q8—31no—
174C4×Q8—31no—
175C4×Q8C22×C3211yescharacteristic
176C4×Q8C22×C3211yescharacteristic
177C4×Q8C22×C3212yes—
178C4×D4—32no—
179C4×D4—31no—
180C4×D4—32no—
181C4×D4—32no—
182C4×D4—32no—
183#117—32no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—32no—
186C2×C42—31no—
187C23×C3—32no—
188C23×C3—43no—
189C23×C3C22×A411yescharacteristic
190C3×Q8—22no—
191C3×Q8—31no—
192C3×Q8—61no—
193C3×Q8—61no—
194C3×Q8—86no—
195C3×Q8—122no—
196C3×Q8C3×#4111yescharacteristic
197C3×D4—62no—
198C3×D4—62no—
199C3×D4—64no—
200C3×D4—62no—
201C3×D4—122no—
202C2×C3×C4—22no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—32no—
208C2×C3×C4—43no—
209C2×C3×C4—46no—
210C2×C3×C4—43no—
211C2×C3×C4—46no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—62no—
217C2×C3×C4—86no—
218C2×C3×C4—124no—
219C2×C3×C4C22×A411yescharacteristic
220C2×C3×C4C22×A412yes—
221C2×C3×C4C22×A412yes—
222C2×C3×C4C22×A411yescharacteristic
223SL(2,𝔽3)C3×#4113yescomplemented
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—82no—
228#41—22no—
229#41—62no—
230#41—124no—
231C2×Q8—21no—
232C2×Q8—32no—
233C2×Q8—31no—
234C2×Q8—61no—
235C2×Q8C2×C3×A411yescharacteristic
236C2×Q8C23×C3211yescharacteristic
237C2×D4—32no—
238C2×D4—32no—
239C2×D4—32no—
240C2×D4—31no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—34no—
244C22×C4—32no—
245C22×C4C2×C3×A411yescharacteristic
246C4⋊C4—31no—
247C4⋊C4—32no—
248C4⋊C4—31no—
249C4⋊C4—32no—
250C4⋊C4—32no—
251C4⋊C4—31no—
252C4⋊C4—32no—
253C4⋊C4—32no—
254C4⋊C4C2×C3×A412yes—
255C4⋊C4C2×C3×A411yescharacteristic
256C22⋊C4—34no—
257C22⋊C4—32no—
258C22⋊C4—32no—
259C22⋊C4—32no—
260C22⋊C4—32no—
261C22⋊C4C2×C3×A412yes—
262C42—31no—
263C42—31no—
264C42—31no—
265C42—31no—
266C42—32no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—46no—
270C22×C3—43no—
271C22×C3—62no—
272C22×C3—62no—
273C22×C3—62no—
274C22×C3—83no—
275C22×C3—83no—
276C22×C3#99412yes—
277C22×C3C23×A411yescharacteristic
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—22no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—46no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—61no—
287C3×C4—62no—
288C3×C4—83no—
289C3×C4—86no—
290C3×C4—86no—
291C3×C42−1+4⋊C312yes—
292C32—41noSylow
293C23—32no—
294C23C22×C3×A411yescharacteristic
295Q8—22no—
296Q8—31no—
297Q8—61no—
298Q8—61no—
299Q8—122no—
300Q8C32×#4111yescharacteristic
301D4—62no—
302D4—62no—
303D4—64no—
304D4—62no—
305D4—122no—
306C2×C4—22no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—32no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—62no—
317C2×C4—124no—
318C2×C4C22×C3×A411yescharacteristic
319C2×C4C22×C3×A412yes—
320C2×C4C22×C3×A412yes—
321C2×C4C22×C3×A411yescharacteristic
322C2×C3—22no—
323C2×C3—43no—
324C2×C3—43no—
325C2×C3—43no—
326C2×C3—86no—
327C2×C3—122no—
328C2×C3A4×#4111yescharacteristic
329C2×C3C2×#99411yescharacteristic
330C2×C3C2×2−1+4⋊C311yescharacteristic
331C22—21no—
332C22—21no—
333C22—62no—
334C22—62no—
335C22—62no—
336C22C3×#99412yes—
337C22C23×C3×A411yescharacteristic
338C4—22no—
339C4—21no—
340C4—22no—
341C4—31no—
342C4—31no—
343C4—61no—
344C4—61no—
345C4—61no—
346C4—62no—
347C4C3×2−1+4⋊C312yes—
348C3—43no—
349C3#3522⋊C311yescharacteristic, direct product factor
350C2—22no—
351C2—122no—
352C2C3×A4×#4111yescharacteristic
353C2C2×C3×#99411yescharacteristic
354C2C2×C3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3522C3
hypercenterC3×C22⋊Q8A4
hypocenterQ8C32×#41
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×C4⋊(#41⋊C3)C2semidirect
3. C3×#548⋊C3C2semidirect
4. C3×#520⋊C3C2semidirect
5. C2×C3×C4×SL(2,𝔽3)C2semidirect
6. C3×#3522C3semidirect
7. #3522⋊C3C3direct (N × H)
15. C3×Q82C2×C3semidirect
17. C3×#550C2×C3semidirect
23. C3×#520C2×C3semidirect
25. Q8×SL(2,𝔽3)C2×C3semidirect
26. C4⋊(#41⋊C3)C2×C3semidirect
27. #548⋊C3C2×C3semidirect
28. #520⋊C3C2×C3semidirect
29. C2×C4×SL(2,𝔽3)C2×C3semidirect
38. #3522C32semidirect
86. C3×SL(2,𝔽3)#41semidirect
88. Q82C2×C32semidirect
90. #550C2×C32semidirect
96. #520C2×C32semidirect
223. SL(2,𝔽3)C3×#41semidirect
349. C3#3522⋊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×C4⋊(#41⋊C3)2
C3×#548⋊C32
C3×#520⋊C32
C2×C3×C4×SL(2,𝔽3)2
C3×#35223
#3522⋊C33
C32×C22⋊Q84