Finite Group Databasebeta

← C3×#3518⋊C3 #11919762

Subgroups — C3×#3518⋊C3 #11919762

Order 1152.

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

normal subgroups only

This group has 646 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#548⋊C3C212yesmaximal, complemented
4C3×#519⋊C3C212yesmaximal, complemented
5C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
6C3×#3518C311yesmaximal, characteristic, complemented
7#3518⋊C3C313yesmaximal, direct product factor
8C2×C3×#41⋊C3C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2212yes—
10C32×#124—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2212yes—
14C3×Q82C2×C311yescharacteristic, complemented
15C3×#556—32no—
16C3×#556—31no—
17C3×#555—31no—
18C3×D4×Q8C2×C311yescharacteristic
19C3×#548C2×C312yescomplemented
20C3×#542—32no—
21C3×#533—31no—
22C3×#529—31no—
23C3×#519C2×C312yescomplemented
24C3×C4×#41C2×C311yescharacteristic, complemented
25Q8×SL(2,𝔽3)C2×C313yescomplemented
26D4×SL(2,𝔽3)C2×C313yescomplemented
27#548⋊C3C2×C316yescomplemented
28#519⋊C3C2×C316yescomplemented
29C4×#41⋊C3C2×C313yescomplemented
30C2×C32×Q8—41no—
31C2×C32×D4—41no—
32C3×#41⋊C3—21no—
33C2×C3×SL(2,𝔽3)—22no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C32×C22⋊C4—42no—
36C32×C22⋊C4—42no—
37C32×C42—41no—
38#3518C3211yescharacteristic, complemented, Sylow
39C2×C3×#41C22×C311yescharacteristic
40C22×C3×Q8C22×C312yes—
41C2×#41⋊C3C22×C313yes—
42C22×SL(2,𝔽3)C22×C316yes—
43C3×C4⋊Q8—31no—
44C3×C4⋊Q8—32no—
45C3×C4⋊Q8—31no—
46C3×#126—34no—
47C3×#126—32no—
48C3×#125—31no—
49C3×#124—34no—
50C3×#124—32no—
51C3×#124—43no—
52C3×#124A411yescharacteristic
53C3×#123—32no—
54C3×C22⋊Q8—32no—
55C3×C22⋊Q8—31no—
56C3×C22⋊Q8—32no—
57C3×C22⋊Q8—32no—
58C3×#121—32no—
59C3×#121—31no—
60C3×C4×Q8—32no—
61C3×C4×Q8—31no—
62C3×C4×Q8—32no—
63C3×C4×Q8—32no—
64C3×C4×Q8C22×C311yescharacteristic
65C3×C4×Q8C22×C312yes—
66C3×C4×Q8C22×C311yescharacteristic
67C3×C4×D4—31no—
68C3×C4×D4—31no—
69C3×C4×D4—31no—
70C3×C4×D4—32no—
71C3×#117—32no—
72C3×#117—32no—
73C3×#117—32no—
74C3×#117—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—42no—
80C32×Q8—82no—
81C32×D4—82no—
82C2×C32×C4—41no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4—42no—
86C3×SL(2,𝔽3)#4111yescharacteristic
87Q82C2×C3211yescharacteristic, complemented
88#556—32no—
89#556—31no—
90#555—31no—
91D4×Q8C2×C3211yescharacteristic
92#548C2×C3212yescomplemented
93#542—32no—
94#533—31no—
95#529—31no—
96#519C2×C3212yescomplemented
97C4×#41C2×C3211yescharacteristic, complemented
98C3×#41—21no—
99C3×#41—61no—
100C2×C3×Q8—22no—
101C2×C3×Q8—32no—
102C2×C3×Q8—32no—
103C2×C3×Q8—43no—
104C2×C3×Q8—62no—
105C2×C3×Q8C2×A411yescharacteristic
106C2×C3×Q8C23×C311yescharacteristic
107C2×C3×D4—31no—
108C2×C3×D4—32no—
109C2×C3×D4—43no—
110C2×C3×D4C2×A411yescharacteristic
111C22×C3×C4—32no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115#41⋊C3—23no—
116C2×SL(2,𝔽3)—26no—
117C2×SL(2,𝔽3)C23×C313yes—
118C3×C4⋊C4—31no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—31no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—32no—
128C3×C22⋊C4—32no—
129C3×C22⋊C4—34no—
130C3×C22⋊C4—32no—
131C3×C22⋊C4—32no—
132C3×C22⋊C4—32no—
133C3×C22⋊C4—46no—
134C3×C22⋊C4—46no—
135C3×C22⋊C4C2×A412yes—
136C3×C22⋊C4C2×A412yes—
137C3×C42—34no—
138C3×C42—31no—
139C3×C42—31no—
140C3×C42—31no—
141C3×C42—32no—
142C3×C42—43no—
143C3×C42C2×A411yescharacteristic
144C22×C32—41no—
145C22×C32—82no—
146C22×C32—82no—
147C22×C32—82no—
148C32×C4—42no—
149C32×C4—81no—
150C32×C4—82no—
151C32×C4—81no—
152C2×#41C22×C3211yescharacteristic
153C22×Q8C22×C3212yes—
154C4⋊Q8—31no—
155C4⋊Q8—31no—
156C4⋊Q8—32no—
157#126—34no—
158#126—32no—
159#125—31no—
160#124—34no—
161#124—32no—
162#124C3×A411yescharacteristic
163#123—32no—
164C22⋊Q8—32no—
165C22⋊Q8—31no—
166C22⋊Q8—32no—
167C22⋊Q8—32no—
168#121—32no—
169#121—31no—
170C4×Q8—32no—
171C4×Q8—31no—
172C4×Q8—32no—
173C4×Q8—32no—
174C4×Q8C22×C3211yescharacteristic
175C4×Q8C22×C3211yescharacteristic
176C4×Q8C22×C3212yes—
177C4×D4—31no—
178C4×D4—31no—
179C4×D4—31no—
180C4×D4—32no—
181#117—32no—
182#117—32no—
183#117—32no—
184#117—31no—
185C2×C42—31no—
186C23×C3—31no—
187C23×C3—46no—
188C23×C3C22×A412yes—
189C3×Q8—22no—
190C3×Q8—31no—
191C3×Q8—62no—
192C3×Q8—62no—
193C3×Q8—86no—
194C3×Q8—122no—
195C3×Q8C3×#4111yescharacteristic
196C3×D4—22no—
197C3×D4—61no—
198C3×D4—61no—
199C3×D4—86no—
200C3×D4—122no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—32no—
204C2×C3×C4—32no—
205C2×C3×C4—31no—
206C2×C3×C4—43no—
207C2×C3×C4—43no—
208C2×C3×C4—46no—
209C2×C3×C4—43no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4—62no—
215C2×C3×C4—121no—
216C2×C3×C4C22×A411yescharacteristic
217C2×C3×C4C22×A411yescharacteristic
218C2×C3×C4C22×A412yes—
219C2×C3×C4C22×A411yescharacteristic
220SL(2,𝔽3)C3×#4113yes—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—162no—
225#41—21no—
226#41—61no—
227C2×Q8—22no—
228C2×Q8—32no—
229C2×Q8—32no—
230C2×Q8—62no—
231C2×Q8C2×C3×A411yescharacteristic
232C2×Q8C23×C3211yescharacteristic
233C2×D4—31no—
234C2×D4—32no—
235C2×D4C2×C3×A411yescharacteristic
236C22×C4—32no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—32no—
243C4⋊C4—32no—
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—32no—
247C4⋊C4—32no—
248C4⋊C4—32no—
249C4⋊C4—32no—
250C22⋊C4—32no—
251C22⋊C4—34no—
252C22⋊C4—32no—
253C22⋊C4—32no—
254C22⋊C4—32no—
255C22⋊C4C2×C3×A412yes—
256C22⋊C4C2×C3×A412yes—
257C42—34no—
258C42—31no—
259C42—31no—
260C42—31no—
261C42—32no—
262C42C2×C3×A411yescharacteristic
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—43no—
267C22×C3—61no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3—86no—
271C22×C3—86no—
272C22×C3—86no—
273C22×C3C23×A411yescharacteristic
274C3×C4—21no—
275C3×C4—22no—
276C3×C4—21no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—46no—
280C3×C4—62no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—62no—
284C3×C4—86no—
285C3×C4—83no—
286C3×C4—83no—
287C3×C42−1+4⋊C312yes—
288C32—41noSylow
289C23—31no—
290C23C22×C3×A412yes—
291Q8—22no—
292Q8—31no—
293Q8—62no—
294Q8—62no—
295Q8—122no—
296Q8C32×#4111yescharacteristic
297D4—22no—
298D4—61no—
299D4—61no—
300D4—122no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—31no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—121no—
312C2×C4C22×C3×A411yescharacteristic
313C2×C4C22×C3×A411yescharacteristic
314C2×C4C22×C3×A412yes—
315C2×C4C22×C3×A411yescharacteristic
316C2×C3—42no—
317C2×C3—43no—
318C2×C3—43no—
319C2×C3—43no—
320C2×C3—121no—
321C2×C3—166no—
322C2×C3A4×#4111yescharacteristic
323C2×C3C2×2−1+4⋊C311yescharacteristic
324C2×C3#594511yescharacteristic
325C22—22no—
326C22—22no—
327C22—22no—
328C22—61no—
329C22—61no—
330C22—61no—
331C22C23×C3×A411yescharacteristic
332C4—21no—
333C4—22no—
334C4—21no—
335C4—31no—
336C4—31no—
337C4—62no—
338C4—61no—
339C4—61no—
340C4—62no—
341C4C3×2−1+4⋊C312yes—
342C3—43no—
343C3#3518⋊C311yescharacteristic, direct product factor
344C2—42no—
345C2—121no—
346C2C3×A4×#4111yescharacteristic
347C2C2×C3×2−1+4⋊C311yescharacteristic
348C2C3×#594511yescharacteristic
subgroupG/H
center, socleC22×C3C23×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC3×#3518C3
hypercenterC3×#124A4
hypocenterQ8C32×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
2. C3×D4×SL(2,𝔽3)C2semidirect
3. C3×#548⋊C3C2semidirect
4. C3×#519⋊C3C2semidirect
5. C3×C4×#41⋊C3C2semidirect
6. C3×#3518C3semidirect
7. #3518⋊C3C3direct (N × H)
14. C3×Q82C2×C3semidirect
19. C3×#548C2×C3semidirect
23. C3×#519C2×C3semidirect
24. C3×C4×#41C2×C3semidirect
25. Q8×SL(2,𝔽3)C2×C3semidirect
26. D4×SL(2,𝔽3)C2×C3semidirect
27. #548⋊C3C2×C3semidirect
28. #519⋊C3C2×C3semidirect
29. C4×#41⋊C3C2×C3semidirect
38. #3518C32semidirect
87. Q82C2×C32semidirect
92. #548C2×C32semidirect
96. #519C2×C32semidirect
97. C4×#41C2×C32semidirect
343. C3#3518⋊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×D4×SL(2,𝔽3)2
C3×#548⋊C32
C3×#519⋊C32
C3×C4×#41⋊C32
C3×#35183
#3518⋊C33
C32×#1244