Finite Group Databasebeta

← C3×#3517⋊C3 #11919760

Subgroups — C3×#3517⋊C3 #11919760

Order 1152.

630 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 146 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×C4⋊(#41⋊C3)C212yesmaximal, complemented
3C3×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
4C3×#520⋊C3C212yesmaximal, complemented
5C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
6C3×#3517C311yesmaximal, characteristic, complemented
7#3517⋊C3C313yesmaximal, direct product factor
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×C3×#41⋊C3C2212yes—
10C32×#125—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2212yes—
14C3×#557C2×C311yescharacteristic
15C3×#553—32no—
16C3×#550—31no—
17C3×#550C2×C311yescharacteristic
18C3×#550C2×C312yes—
19C3×#547—31no—
20C3×#540—32no—
21C3×#532—31no—
22C3×#529—31no—
23C3×#520C2×C312yes—
24C3×C4×#41C2×C311yescharacteristic
25#557⋊C3C2×C313yescomplemented
26C4⋊(#41⋊C3)C2×C313yescomplemented
27C4⋊(#41⋊C3)C2×C316yescomplemented
28#520⋊C3C2×C316yescomplemented
29C4×#41⋊C3C2×C313yescomplemented
30C3×#41⋊C3—21no—
31C3×#41⋊C3—22no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C32×C4⋊C4—42no—
34C32×C4⋊C4—41no—
35C32×C4⋊C4—42no—
36C32×C4⋊C4—41no—
37C32×C42—41no—
38#3517C3211yescharacteristic, complemented, Sylow
39C2×C3×#41—32no—
40C2×C3×#41C22×C311yescharacteristic
41C2×C3×#41C22×C312yes—
42C2×#41⋊C3C22×C313yes—
43C2×#41⋊C3C22×C316yes—
44C3×C4⋊Q8—31no—
45C3×#127—31no—
46C3×#127—32no—
47C3×#126—32no—
48C3×#125—32no—
49C3×#125—43no—
50C3×#125A411yescharacteristic
51C3×#124—31no—
52C3×#123—34no—
53C3×#123—32no—
54C3×C22⋊Q8—31no—
55C3×#121—32no—
56C3×#121—34no—
57C3×#121—31no—
58C3×#121—32no—
59C3×#121—32no—
60C3×#121—32no—
61C3×C4×Q8—31no—
62C3×C4×Q8C22×C311yescharacteristic
63C3×C4×Q8C22×C311yescharacteristic
64C3×C4×Q8C22×C312yes—
65C3×C4×D4—32no—
66C3×C4×D4—31no—
67C3×C4×D4—31no—
68C3×C4×D4—31no—
69C3×C4×D4—32no—
70C3×C4×D4—32no—
71C3×C4×D4—32no—
72C3×#117—32no—
73C3×#117—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—
79C2×C32×C4—41no—
80C2×C32×C4—41no—
81C2×C32×C4—42no—
82C2×C32×C4—41no—
83C2×C32×C4—42no—
84C3×SL(2,𝔽3)#4111yescharacteristic, complemented
85#557C2×C3211yescharacteristic
86#553—32no—
87#550—31no—
88#550C2×C3211yescharacteristic
89#550C2×C3212yes—
90#547—31no—
91#540—32no—
92#532—31no—
93#529—31no—
94#520C2×C3212yes—
95C4×#41C2×C3211yescharacteristic
96C3×#41—21no—
97C3×#41—22no—
98C3×#41—61no—
99C3×#41—62no—
100C3×#41—124no—
101C2×C3×Q8—31no—
102C2×C3×Q8C23×C311yescharacteristic
103C2×C3×D4—31no—
104C2×C3×D4—32no—
105C2×C3×D4—32no—
106C2×C3×D4—34no—
107C2×C3×D4—31no—
108C2×C3×D4—32no—
109C22×C3×C4—34no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—32no—
113C22×C3×C4—31no—
114#41⋊C3—23no—
115#41⋊C3—26no—
116C2×SL(2,𝔽3)C23×C313yes—
117C3×C4⋊C4—31no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—32no—
123C3×C4⋊C4—46no—
124C3×C4⋊C4—43no—
125C3×C4⋊C4—46no—
126C3×C4⋊C4—43no—
127C3×C4⋊C4C2×A412yes—
128C3×C4⋊C4C2×A411yescharacteristic
129C3×C4⋊C4C2×A412yes—
130C3×C4⋊C4C2×A411yescharacteristic
131C3×C22⋊C4—32no—
132C3×C22⋊C4—32no—
133C3×C22⋊C4—32no—
134C3×C22⋊C4—32no—
135C3×C22⋊C4—32no—
136C3×C22⋊C4—32no—
137C3×C42—31no—
138C3×C42—31no—
139C3×C42—31no—
140C3×C42—32no—
141C3×C42—43no—
142C3×C42C2×A411yescharacteristic
143C22×C32—41no—
144C32×C4—42no—
145C32×C4—81no—
146C32×C4—82no—
147C32×C4—81no—
148C32×C4—82no—
149C2×#41—32no—
150C2×#41C22×C3211yescharacteristic
151C2×#41C22×C3212yes—
152C4⋊Q8—31no—
153#127—31no—
154#127—32no—
155#126—32no—
156#125—32no—
157#125C3×A411yescharacteristic
158#124—31no—
159#123—34no—
160#123—32no—
161C22⋊Q8—31no—
162#121—32no—
163#121—34no—
164#121—31no—
165#121—32no—
166#121—32no—
167#121—32no—
168C4×Q8—31no—
169C4×Q8C22×C3211yescharacteristic
170C4×Q8C22×C3211yescharacteristic
171C4×Q8C22×C3212yes—
172C4×D4—32no—
173C4×D4—31no—
174C4×D4—31no—
175C4×D4—31no—
176C4×D4—32no—
177C4×D4—32no—
178C4×D4—32no—
179#117—32no—
180#117—31no—
181C2×C4⋊C4—32no—
182C2×C42—31no—
183C23×C3—31no—
184C23×C3—32no—
185C3×Q8—31no—
186C3×Q8—62no—
187C3×Q8C3×#4111yescharacteristic
188C3×D4—61no—
189C3×D4—62no—
190C3×D4—61no—
191C3×D4—62no—
192C3×D4—62no—
193C3×D4—122no—
194C3×D4—122no—
195C3×D4—124no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—32no—
200C2×C3×C4—43no—
201C2×C3×C4—43no—
202C2×C3×C4—46no—
203C2×C3×C4—43no—
204C2×C3×C4—46no—
205C2×C3×C4—61no—
206C2×C3×C4—62no—
207C2×C3×C4—61no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—121no—
211C2×C3×C4—124no—
212C2×C3×C4C22×A411yescharacteristic
213C2×C3×C4C22×A411yescharacteristic
214C2×C3×C4C22×A412yes—
215C2×C3×C4C22×A412yes—
216C2×C3×C4C22×A411yescharacteristic
217SL(2,𝔽3)C3×#4113yescomplemented
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—41no—
221#41—21no—
222#41—22no—
223#41—61no—
224#41—62no—
225#41—124no—
226C2×Q8—31no—
227C2×Q8C23×C3211yescharacteristic
228C2×D4—31no—
229C2×D4—32no—
230C2×D4—32no—
231C2×D4—34no—
232C2×D4—31no—
233C2×D4—32no—
234C22×C4—34no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—32no—
238C22×C4—31no—
239C4⋊C4—31no—
240C4⋊C4—32no—
241C4⋊C4—31no—
242C4⋊C4—32no—
243C4⋊C4—32no—
244C4⋊C4—32no—
245C4⋊C4C2×C3×A412yes—
246C4⋊C4C2×C3×A411yescharacteristic
247C4⋊C4C2×C3×A412yes—
248C4⋊C4C2×C3×A411yescharacteristic
249C22⋊C4—32no—
250C22⋊C4—32no—
251C22⋊C4—32no—
252C22⋊C4—32no—
253C22⋊C4—32no—
254C22⋊C4—32no—
255C42—31no—
256C42—31no—
257C42—31no—
258C42—32no—
259C42C2×C3×A411yescharacteristic
260C22×C3—43no—
261C22×C3—61no—
262C22×C3—62no—
263C22×C3—61no—
264C22×C3—62no—
265C22×C3—61no—
266C22×C3—62no—
267C22×C3C23×A411yescharacteristic
268C3×C4—21no—
269C3×C4—22no—
270C3×C4—21no—
271C3×C4—22no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—46no—
275C3×C4—61no—
276C3×C4—62no—
277C3×C4—61no—
278C3×C4—83no—
279C3×C4—86no—
280C3×C4—83no—
281C3×C4—86no—
282C3×C4#99412yes—
283C32—41noSylow
284C23—31no—
285C23—32no—
286Q8—31no—
287Q8—62no—
288Q8C32×#4111yescharacteristic
289D4—61no—
290D4—62no—
291D4—61no—
292D4—62no—
293D4—62no—
294D4—122no—
295D4—124no—
296D4—122no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—32no—
301C2×C4—61no—
302C2×C4—62no—
303C2×C4—61no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—121no—
307C2×C4—124no—
308C2×C4C22×C3×A411yescharacteristic
309C2×C4C22×C3×A411yescharacteristic
310C2×C4C22×C3×A412yes—
311C2×C4C22×C3×A412yes—
312C2×C4C22×C3×A411yescharacteristic
313C2×C3—43no—
314C2×C3—43no—
315C2×C3—43no—
316C2×C3—121no—
317C2×C3—122no—
318C2×C3A4×#4111yescharacteristic
319C2×C3C2×#99411yescharacteristic
320C2×C3#594511yescharacteristic
321C22—61no—
322C22—62no—
323C22—61no—
324C22—62no—
325C22—61no—
326C22—62no—
327C22C23×C3×A411yescharacteristic
328C4—21no—
329C4—22no—
330C4—21no—
331C4—22no—
332C4—31no—
333C4—31no—
334C4—61no—
335C4—62no—
336C4—61no—
337C4C3×#99412yes—
338C3—43no—
339C3#3517⋊C311yescharacteristic, direct product factor
340C2—121no—
341C2—122no—
342C2C3×A4×#4111yescharacteristic
343C2C2×C3×#99411yescharacteristic
344C2C3×#594511yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3517C3
hypercenterC3×#125A4
hypocenterQ8C32×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
2. C3×C4⋊(#41⋊C3)C2semidirect
3. C3×C4⋊(#41⋊C3)C2semidirect
4. C3×#520⋊C3C2semidirect
5. C3×C4×#41⋊C3C2semidirect
6. C3×#3517C3semidirect
7. #3517⋊C3C3direct (N × H)
25. #557⋊C3C2×C3semidirect
26. C4⋊(#41⋊C3)C2×C3semidirect
27. C4⋊(#41⋊C3)C2×C3semidirect
28. #520⋊C3C2×C3semidirect
29. C4×#41⋊C3C2×C3semidirect
38. #3517C32semidirect
84. C3×SL(2,𝔽3)#41semidirect
217. SL(2,𝔽3)C3×#41semidirect
339. C3#3517⋊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×C4⋊(#41⋊C3)2
C3×#520⋊C32
C3×C4×#41⋊C32
C3×#35173
#3517⋊C33
C32×#1254