Finite Group Databasebeta

← C3×#85536 #11922822

Subgroups — C3×#85536 #11922822

Order 1152.

637 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C2×C3×2−1+4⋊C3C211yesmaximal, characteristic
2C2×C3×#984C211yesmaximal, characteristic, complemented
3C3×#5424C211yesmaximal, characteristic, complemented
4C3×#5419C214yesmaximal, complemented
5#85536C311yesmaximal, characteristic, direct product factor
6C3×#3011—31nomaximal
7C3×2−1+4⋊C3—22no—
8C2×C3×#41⋊C3C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2212yes—
10C3×#984—22no—
11C2×C3×#226C2212yes—
12C3×#936—41nomaximal
13C3×#859C2212yes—
14C2×C3×2−1+4D311yescharacteristic
15C2×2−1+4⋊C3—21no—
16C2×2−1+4⋊C3C2×C311yescharacteristic
17C2×#984C2×C311yescharacteristic, complemented
18C2×C3×#137—31no—
19C22×C3×Q16—31no—
20C3×#536—31no—
21#5424C2×C311yescharacteristic, complemented
22#5419C2×C314yescomplemented
23C3×#452—34no—
24C3×#451—34no—
25C3×#419—31no—
26C3×#416—31no—
27C3×#408—31no—
28C2×C32×D4—41no—
29C2×C3×Dic6—41no—
30C3×#41⋊C3D412yes—
31C2×C3×SL(2,𝔽3)—22no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3×#226—22no—
34C3×#217—44no—
35C3×C4×Dic3—41no—
36#3011—31noSylow
37C3×2−1+4—22no—
38C3×2−1+4—62no—
39C2×C3×#41—31no—
40C2×C3×#41—32no—
41C2×C3×#41C2×D311yescharacteristic
42C22×C3×Q8—31no—
43C22×C3×Q8—31no—
44C22×C3×Q8C2×D312yes—
452−1+4⋊C3—24no—
462−1+4⋊C3—22no—
47C2×#41⋊C3—21no—
48C2×#41⋊C3C22×C311yescharacteristic
49C22×SL(2,𝔽3)—22no—
50C22×SL(2,𝔽3)C22×C312yes—
51#984—22no—
52C2×#226C22×C312yes—
53C3×#137—62no—
54C3×#137—62no—
55C2×C3×Q16—34no—
56C2×C3×Q16—32no—
57C2×C3×Q16—64no—
58C2×C3×SD16—32no—
59C2×C3×M16—31no—
60C22×C3×C8—31no—
61C3×C4⋊Q8—61no—
62C3×#124—61no—
63C3×#123—62no—
64C3×C22⋊Q8—34no—
65C3×#117—31no—
66#936—41no—
67#859C22×C312yes—
68C3×#103—34no—
69C3×#103—32no—
70C3×#102—32no—
71C3×C22⋊C8—34no—
72C23×C32—42no—
73C32×D4—82no—
74C2×C32×C4—41no—
75C2×C3×Dic3—42no—
76C2×C3×Dic3—42no—
77C3×Dic6—82no—
78C3×SL(2,𝔽3)C2×D411yescharacteristic
79C2×2−1+4C3×D311yescharacteristic
80C2×#137—31no—
81C22×Q16—31no—
82#536—31no—
83#452—34no—
84#451—34no—
85#419—31no—
86#416—31no—
87#408—31no—
88C3×#41—32no—
89C3×#41—62no—
90C3×#41—62no—
91C3×#41—64no—
92C3×#41—64no—
93C3×#41#6712yes—
94C2×C3×Q8—22no—
95C2×C3×Q8—31no—
96C2×C3×Q8—32no—
97C2×C3×Q8—34no—
98C2×C3×Q8—32no—
99C2×C3×Q8—62no—
100C2×C3×Q8—62no—
101C2×C3×Q8—62no—
102C2×C3×Q8—61no—
103C2×C3×Q8—62no—
104C2×C3×Q8C22×D311yescharacteristic
105C2×C3×D4—31no—
106C2×C3×D4—32no—
107C2×C3×D4—41no—
108C2×C3×D4—81no—
109C2×C3×D4S411yescharacteristic
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C22×C3×C4—61no—
113C22×C3×C4—61no—
114C2×Dic6—41no—
115#41⋊C3—22no—
116#41⋊C3C3×D412yes—
117C2×SL(2,𝔽3)—24no—
118C2×SL(2,𝔽3)—22no—
119C2×SL(2,𝔽3)—21no—
120C2×SL(2,𝔽3)C23×C311yescharacteristic
121#226—22no—
122C3×Q16—68no—
123C3×Q16—62no—
124C3×Q16—62no—
125C3×SD16—64no—
126C3×M16—62no—
127C2×C3×C8—32no—
128C2×C3×C8—32no—
129C2×C3×C8—62no—
130C3×C4⋊C4—32no—
131C3×C4⋊C4—64no—
132C3×C22⋊C4—61no—
133C3×C22⋊C4—64no—
134C3×C42—61no—
135#217—44no—
136C4×Dic3—41no—
137C22×C32—41no—
138C22×C32—82no—
139C22×C32—84no—
140C32×C4—42no—
141C3×Dic3—82no—
142C3×Dic3—82no—
1432−1+4—22no—
1442−1+4—62no—
145C2×#41—31no—
146C2×#41—32no—
147C2×#41C2×C3×D311yescharacteristic
148C22×Q8—31no—
149C22×Q8—31no—
150C22×Q8C2×C3×D312yes—
151#137—62no—
152#137—62no—
153C2×Q16—34no—
154C2×Q16—32no—
155C2×Q16—64no—
156C2×SD16—32no—
157C2×M16—31no—
158C22×C8—31no—
159C4⋊Q8—61no—
160#124—61no—
161#123—62no—
162C22⋊Q8—34no—
163#117—31no—
164#103—34no—
165#103—32no—
166#102—32no—
167C22⋊C8—34no—
168C23×C3—31no—
169C23×C3—42no—
170C23×C3—82no—
171C23×C3C2×S412yes—
172C3×Q8—31no—
173C3×Q8—34no—
174C3×Q8—62no—
175C3×Q8—62no—
176C3×Q8—62no—
177C3×Q8—62no—
178C3×Q8—122no—
179C3×Q8C2×#6711yescharacteristic
180C3×D4—22no—
181C3×D4—32no—
182C3×D4—61no—
183C3×D4—64no—
184C3×D4—82no—
185C3×D4—84no—
186C2×C3×C4—31no—
187C2×C3×C4—32no—
188C2×C3×C4—32no—
189C2×C3×C4—32no—
190C2×C3×C4—41no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—64no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—81no—
197C2×C3×C4—122no—
198C2×C3×C4C2×S411yescharacteristic
199C2×Dic3—42no—
200C2×Dic3—42no—
201Dic6—82no—
202SL(2,𝔽3)—21no—
203SL(2,𝔽3)C2×C3×D411yescharacteristic
204C3×C8—62no—
205C3×C8—62no—
206C2×C32—42no—
207C2×C32—41no—
208C2×C32—162no—
209#41—32no—
210#41—62no—
211#41—62no—
212#41—64no—
213#41—64no—
214#41C3×#6712yes—
215C2×Q8—22no—
216C2×Q8—31no—
217C2×Q8—32no—
218C2×Q8—34no—
219C2×Q8—32no—
220C2×Q8—62no—
221C2×Q8—62no—
222C2×Q8—62no—
223C2×Q8—61no—
224C2×Q8—62no—
225C2×Q8C22×C3×D311yescharacteristic
226C2×D4—31no—
227C2×D4—32no—
228C2×D4C3×S411yescharacteristic
229C22×C4—31no—
230C22×C4—32no—
231C22×C4—61no—
232C22×C4—61no—
233Q16—68no—
234Q16—62no—
235Q16—62no—
236SD16—64no—
237M16—62no—
238C2×C8—32no—
239C2×C8—32no—
240C2×C8—62no—
241C4⋊C4—32no—
242C4⋊C4—64no—
243C22⋊C4—61no—
244C22⋊C4—64no—
245C42—61no—
246C22×C3—24no—
247C22×C3—22no—
248C22×C3—32no—
249C22×C3—41no—
250C22×C3—62no—
251C22×C3—82no—
252C22×C3—84no—
253C22×C3—84no—
254C22×C3—81no—
255C22×C3—164no—
256C22×C3C22×S411yescharacteristic
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—42no—
260C3×C4—62no—
261C3×C4—82no—
262C3×C4—122no—
263C3×C4—122no—
264C3×C4GL(2,ℤ/4)12yes—
265Dic3—82no—
266Dic3—82no—
267C32—41noSylow
268C23—31no—
269C23C2×C3×S412yes—
270Q8—31no—
271Q8—34no—
272Q8—62no—
273Q8—62no—
274Q8—62no—
275Q8—62no—
276Q8—122no—
277Q8C2×C3×#6711yescharacteristic
278D4—22no—
279D4—32no—
280D4—61no—
281D4—64no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—32no—
285C2×C4—32no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—64no—
289C2×C4—62no—
290C2×C4—62no—
291C2×C4—122no—
292C2×C4C2×C3×S411yescharacteristic
293C8—62no—
294C8—62no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3—41no—
298C2×C3—62no—
299C2×C3—82no—
300C2×C3—81no—
301C2×C3—162no—
302C2×C3—164no—
303C2×C3#592712yes—
304C2×C3C2×GL(2,ℤ/4)11yescharacteristic
305C22—24no—
306C22—22no—
307C22—32no—
308C22—62no—
309C22C22×C3×S411yescharacteristic
310C4—31no—
311C4—31no—
312C4—62no—
313C4—122no—
314C4—122no—
315C4C3×GL(2,ℤ/4)12yes—
316C3—41no—
317C3—81no—
318C3#8553611yescharacteristic, direct product factor
319C2—42no—
320C2—62no—
321C2C3×#592712yes—
322C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC2×C3×2−1+4D3
prime radicalC2×C3×2−1+4⋊C3C2
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#984C2semidirect
3. C3×#5424C2semidirect
4. C3×#5419C2semidirect
5. #85536C3direct (N × H)
17. C2×#984C2×C3semidirect
21. #5424C2×C3semidirect
22. #5419C2×C3semidirect
318. C3#85536direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×C3×#9842
C3×#54242
C3×#54192
#855363
C3×#30113
C3×#9364