Finite Group Databasebeta

← C3×#3523⋊C3 #11919766

Subgroups — C3×#3523⋊C3 #11919766

Order 1152.

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

normal subgroups only

This group has 688 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C3×D4×SL(2,𝔽3)C212yesmaximal, complemented
3C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×#519⋊C3C212yesmaximal, complemented
5C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6C3×#3523C311yesmaximal, characteristic, complemented
7#3523⋊C3C313yesmaximal, direct product factor
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2212yes—
10C32×#121—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2212yes—
14C3×#557—31no—
15C3×#557C2×C311yescharacteristic, complemented
16C3×#556—31no—
17C3×#552—32no—
18C3×D4×Q8C2×C312yescomplemented
19C3×D4×Q8C2×C311yescharacteristic, complemented
20C3×#541—32no—
21C3×#533—31no—
22C3×#529—31no—
23C3×#519C2×C312yescomplemented
24C2×C3×C4×Q8C2×C311yescharacteristic, complemented
25#557⋊C3C2×C313yescomplemented
26D4×SL(2,𝔽3)C2×C313yescomplemented
27D4×SL(2,𝔽3)C2×C316yescomplemented
28#519⋊C3C2×C316yescomplemented
29C2×C4×SL(2,𝔽3)C2×C313yescomplemented
30C2×C32×D4—41no—
31C2×C32×D4—42no—
32C22×C32×C4—41no—
33C2×C3×SL(2,𝔽3)—21no—
34C2×C3×SL(2,𝔽3)—22no—
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36C32×C4⋊C4—41no—
37C32×C22⋊C4—42no—
38#3523C3211yescharacteristic, complemented, Sylow
39C22×C3×Q8C22×C311yescharacteristic
40C22×C3×Q8C22×C312yes—
41C22×SL(2,𝔽3)C22×C313yes—
42C22×SL(2,𝔽3)C22×C316yes—
43C3×C4⋊Q8—31no—
44C3×C4⋊Q8—31no—
45C3×C4⋊Q8—32no—
46C3×#126—34no—
47C3×#126—32no—
48C3×#125—32no—
49C3×#125—32no—
50C3×#125—34no—
51C3×#125—31no—
52C3×#124—31no—
53C3×#123—32no—
54C3×C22⋊Q8—31no—
55C3×C22⋊Q8—32no—
56C3×C22⋊Q8—32no—
57C3×C22⋊Q8—32no—
58C3×#121—43no—
59C3×#121A411yescharacteristic
60C3×C4×Q8—32no—
61C3×C4×Q8—31no—
62C3×C4×Q8—32no—
63C3×C4×Q8—32no—
64C3×C4×Q8—31no—
65C3×C4×Q8—31no—
66C3×C4×Q8C22×C311yescharacteristic
67C3×C4×Q8C22×C311yescharacteristic
68C3×C4×Q8C22×C312yes—
69C3×C4×D4—32no—
70C3×C4×D4—31no—
71C3×#117—32no—
72C3×#117—32no—
73C3×#117—32no—
74C2×C3×C4⋊C4—31no—
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—
80C23×C32—42no—
81C32×D4—82no—
82C32×D4—84no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4—42no—
86C2×C32×C4—82no—
87C3×SL(2,𝔽3)#4111yescharacteristic
88#557—31no—
89#557C2×C3211yescharacteristic, complemented
90#556—31no—
91#552—32no—
92D4×Q8C2×C3212yescomplemented
93D4×Q8C2×C3211yescharacteristic, complemented
94#541—32no—
95#533—31no—
96#529—31no—
97#519C2×C3212yescomplemented
98C2×C4×Q8C2×C3211yescharacteristic, complemented
99C2×C3×Q8—21no—
100C2×C3×Q8—22no—
101C2×C3×Q8—31no—
102C2×C3×Q8—32no—
103C2×C3×Q8—31no—
104C2×C3×Q8—61no—
105C2×C3×Q8—62no—
106C2×C3×Q8C23×C311yescharacteristic
107C2×C3×D4—46no—
108C2×C3×D4—43no—
109C2×C3×D4C2×A412yes—
110C2×C3×D4C2×A411yescharacteristic
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C22×C3×C4—32no—
114C22×C3×C4—43no—
115C22×C3×C4C2×A411yescharacteristic
116C2×SL(2,𝔽3)—23no—
117C2×SL(2,𝔽3)—26no—
118C2×SL(2,𝔽3)C23×C313yes—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—32no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—34no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—32no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—32no—
130C3×C4⋊C4—32no—
131C3×C4⋊C4—43no—
132C3×C4⋊C4C2×A411yescharacteristic
133C3×C22⋊C4—32no—
134C3×C22⋊C4—32no—
135C3×C22⋊C4—32no—
136C3×C22⋊C4—32no—
137C3×C22⋊C4—46no—
138C3×C22⋊C4C2×A412yes—
139C3×C42—34no—
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—
149C22×C32—82no—
150C22×C32—82no—
151C22×C32—82no—
152C32×C4—42no—
153C32×C4—81no—
154C32×C4—82no—
155C22×Q8C22×C3211yescharacteristic
156C22×Q8C22×C3212yes—
157C4⋊Q8—31no—
158C4⋊Q8—31no—
159C4⋊Q8—32no—
160#126—34no—
161#126—32no—
162#125—32no—
163#125—32no—
164#125—34no—
165#125—31no—
166#124—31no—
167#123—32no—
168C22⋊Q8—31no—
169C22⋊Q8—32no—
170C22⋊Q8—32no—
171C22⋊Q8—32no—
172#121C3×A411yescharacteristic
173C4×Q8—32no—
174C4×Q8—31no—
175C4×Q8—32no—
176C4×Q8—32no—
177C4×Q8—31no—
178C4×Q8—31no—
179C4×Q8C22×C3211yescharacteristic
180C4×Q8C22×C3211yescharacteristic
181C4×Q8C22×C3212yes—
182C4×D4—32no—
183C4×D4—31no—
184#117—32no—
185#117—32no—
186#117—32no—
187C2×C4⋊C4—31no—
188C2×C42—31no—
189C23×C3—43no—
190C23×C3—46no—
191C23×C3C22×A412yes—
192C23×C3C22×A411yescharacteristic
193C3×Q8—31no—
194C3×Q8—61no—
195C3×Q8—62no—
196C3×Q8—62no—
197C3×Q8—61no—
198C3×Q8—62no—
199C3×Q8C3×#4111yescharacteristic
200C3×D4—24no—
201C3×D4—22no—
202C3×D4—86no—
203C3×D4—812no—
204C2×C3×C4—22no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—32no—
208C2×C3×C4—31no—
209C2×C3×C4—32no—
210C2×C3×C4—31no—
211C2×C3×C4—43no—
212C2×C3×C4—43no—
213C2×C3×C4—46no—
214C2×C3×C4—61no—
215C2×C3×C4—62no—
216C2×C3×C4—61no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—86no—
220C2×C3×C4C22×A411yescharacteristic
221C2×C3×C4C22×A412yes—
222C2×C3×C4C22×A411yescharacteristic
223SL(2,𝔽3)C3×#4113yes—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—82no—
228C2×C32—162no—
229C2×Q8—21no—
230C2×Q8—22no—
231C2×Q8—31no—
232C2×Q8—32no—
233C2×Q8—31no—
234C2×Q8—61no—
235C2×Q8—62no—
236C2×Q8C23×C3211yescharacteristic
237C2×D4C2×C3×A412yes—
238C2×D4C2×C3×A411yescharacteristic
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—32no—
242C22×C4C2×C3×A411yescharacteristic
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—32no—
246C4⋊C4—32no—
247C4⋊C4—32no—
248C4⋊C4—32no—
249C4⋊C4—34no—
250C4⋊C4—32no—
251C4⋊C4—32no—
252C4⋊C4—31no—
253C4⋊C4—32no—
254C4⋊C4—32no—
255C4⋊C4C2×C3×A411yescharacteristic
256C22⋊C4—32no—
257C22⋊C4—32no—
258C22⋊C4—32no—
259C22⋊C4—32no—
260C22⋊C4C2×C3×A412yes—
261C42—34no—
262C42—31no—
263C42—31no—
264C42—31no—
265C42—31no—
266C42—32no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—46no—
273C22×C3—43no—
274C22×C3—83no—
275C22×C3—83no—
276C22×C3—86no—
277C22×C3—86no—
278C22×C3—86no—
279C22×C32−1+4⋊C312yes—
280C22×C3C23×A411yescharacteristic
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—46no—
286C3×C4—62no—
287C3×C4—61no—
288C3×C4—61no—
289C3×C4—61no—
290C3×C4—62no—
291C3×C4—83no—
292C3×C4—86no—
293C3×C42−1+4⋊C312yes—
294C32—41noSylow
295C23C22×C3×A412yes—
296C23C22×C3×A411yescharacteristic
297Q8—31no—
298Q8—61no—
299Q8—62no—
300Q8—62no—
301Q8—61no—
302Q8—62no—
303Q8C32×#4111yescharacteristic
304D4—24no—
305D4—22no—
306C2×C4—22no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—32no—
310C2×C4—31no—
311C2×C4—32no—
312C2×C4—31no—
313C2×C4—61no—
314C2×C4—62no—
315C2×C4—61no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4C22×C3×A411yescharacteristic
319C2×C4C22×C3×A412yes—
320C2×C4C22×C3×A411yescharacteristic
321C2×C3—22no—
322C2×C3—42no—
323C2×C3—43no—
324C2×C3—43no—
325C2×C3—43no—
326C2×C3—86no—
327C2×C3—166no—
328C2×C3A4×#4111yescharacteristic
329C2×C3C2×2−1+4⋊C311yescharacteristic
330C2×C3C2×2−1+4⋊C311yescharacteristic
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22C3×2−1+4⋊C312yes—
337C22C23×C3×A411yescharacteristic
338C4—21no—
339C4—22no—
340C4—31no—
341C4—31no—
342C4—62no—
343C4—61no—
344C4—61no—
345C4—61no—
346C4—62no—
347C4C3×2−1+4⋊C312yes—
348C3—43no—
349C3#3523⋊C311yescharacteristic, direct product factor
350C2—22no—
351C2—42no—
352C2C3×A4×#4111yescharacteristic
353C2C2×C3×2−1+4⋊C311yescharacteristic
354C2C2×C3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3523C3
hypercenterC3×#121A4
hypocenterQ8C32×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
2. C3×D4×SL(2,𝔽3)C2semidirect
3. C3×D4×SL(2,𝔽3)C2semidirect
4. C3×#519⋊C3C2semidirect
5. C2×C3×C4×SL(2,𝔽3)C2semidirect
6. C3×#3523C3semidirect
7. #3523⋊C3C3direct (N × H)
15. C3×#557C2×C3semidirect
18. C3×D4×Q8C2×C3semidirect
19. C3×D4×Q8C2×C3semidirect
23. C3×#519C2×C3semidirect
24. C2×C3×C4×Q8C2×C3semidirect
25. #557⋊C3C2×C3semidirect
26. D4×SL(2,𝔽3)C2×C3semidirect
27. D4×SL(2,𝔽3)C2×C3semidirect
28. #519⋊C3C2×C3semidirect
29. C2×C4×SL(2,𝔽3)C2×C3semidirect
38. #3523C32semidirect
89. #557C2×C32semidirect
92. D4×Q8C2×C32semidirect
93. D4×Q8C2×C32semidirect
97. #519C2×C32semidirect
98. C2×C4×Q8C2×C32semidirect
349. C3#3523⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C3×D4×SL(2,𝔽3)2
C3×#519⋊C32
C2×C3×C4×SL(2,𝔽3)2
C3×#35233
#3523⋊C33
C32×#1214