Finite Group Databasebeta

← C3×#85365 #11922650

Subgroups — C3×#85365 #11922650

Order 1152.

551 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 92 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×#985C211yesmaximal, characteristic
2C3×#520⋊C3C211yesmaximal, characteristic, complemented
3C3×#5426C211yesmaximal, characteristic, complemented
4C3×#5394C212yesmaximal, complemented
5C3×C4×#226C212yesmaximal, complemented
6#85365C311yesmaximal, characteristic, direct product factor
7C3×#2951—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#985C414yescomplemented
10C2×C3×GL(2,𝔽3)C2211yescharacteristic
11C2×C3×#226C2211yescharacteristic
12C3×#892—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yes—
14C3×#859C2212yes—
15C2×#985C2×C311yescharacteristic
16C2×C3×#135—31no—
17C3×#520—31no—
18C3×#520D311yescharacteristic, complemented
19#520⋊C3—21no—
20#520⋊C3C2×C311yescharacteristic, complemented
21#5426C2×C311yescharacteristic, complemented
22#5394C2×C312yescomplemented
23C4×#226C2×C312yescomplemented
24C3×#440—32no—
25C3×#440—32no—
26C3×C4×Q16—32no—
27C3×C4×D8—32no—
28C2×C3×#107—31no—
29C3×#417—31no—
30C3×#417—31no—
31C3×#405—31no—
32C2×C3×C4×D3—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3×GL(2,𝔽3)C2×C412yescomplemented
36C3×#226C2×C412yescomplemented
37C32×C4⋊C4—41no—
38C3×(C2×D3)⋊C4—42no—
39C3×C4⋊Dic3—41no—
40C3×C4×Dic3—42no—
41#2951—31noSylow
42C2×C3×#41—31no—
43C2×C3×#41C2×D311yescharacteristic
44C2×#41⋊C3—21no—
45C2×#41⋊C3C22×C311yescharacteristic
46#985C3×C414yescomplemented
47C2×GL(2,𝔽3)C22×C311yescharacteristic
48C2×#226C22×C311yescharacteristic
49C3×#135—34no—
50C3×#135—34no—
51C2×C3×Q16—31no—
52C2×C3×SD16—31no—
53C2×C3×SD16—31no—
54C2×C3×D8—31no—
55C2×C3×M16—31no—
56C22×C3×C8—31no—
57C3×C4×Q8—32no—
58C3×C4×Q8C2×D312yes—
59C3×C4×D4—32no—
60C3×C4×D4—32no—
61C3×C4×D4—62no—
62C3×#117—31no—
63C3×#117—61no—
64C2×C3×C4⋊C4—31no—
65C2×C3×C4⋊C4—31no—
66C2×C3×C4⋊C4—61no—
67#892—41no—
68C4×SL(2,𝔽3)—22no—
69C4×SL(2,𝔽3)C22×C312yes—
70#859C22×C312yes—
71C3×#107—32no—
72C3×#107—32no—
73C3×#103—32no—
74C3×#103—32no—
75C3×#102—32no—
76C3×#102—32no—
77C3×#97—32no—
78C3×C4×C8—32no—
79C22×C3×D3—41no—
80C2×C32×C4—41no—
81C2×C32×C4—42no—
82C2×C3×Dic3—41no—
83C2×C3×Dic3—42no—
84C3×C4×D3—44no—
85C3×SL(2,𝔽3)C22×C411yescharacteristic
86C2×#135—31no—
87#520—31no—
88#520C3×D311yescharacteristic, complemented
89#440—32no—
90#440—32no—
91C4×Q16—32no—
92C4×D8—32no—
93C2×#107—31no—
94#417—31no—
95#417—31no—
96#405—31no—
97C3×#41—32no—
98C3×#41—34no—
99C3×#41—62no—
100C3×#41C4×D312yes—
101C2×C3×Q8—31no—
102C2×C3×Q8C22×D311yescharacteristic
103C2×C3×D4—31no—
104C2×C3×D4—31no—
105C2×C3×D4—61no—
106C22×C3×C4—31no—
107C22×C3×C4—61no—
108C22×C3×C4—62no—
109C22×C3×C4—61no—
110C22×C3×C4—61no—
111C2×C4×D3—41no—
112#41⋊C3—22no—
113#41⋊C3C2×C3×C412yescomplemented
114C2×SL(2,𝔽3)—21no—
115C2×SL(2,𝔽3)C23×C311yescharacteristic
116GL(2,𝔽3)C2×C3×C412yescomplemented
117#226C2×C3×C412yescomplemented
118C3×Q16—32no—
119C3×Q16—32no—
120C3×SD16—34no—
121C3×SD16—32no—
122C3×SD16—32no—
123C3×D8—34no—
124C3×M16—62no—
125C2×C3×C8—31no—
126C2×C3×C8—31no—
127C2×C3×C8—34no—
128C2×C3×C8—32no—
129C3×C4⋊C4—32no—
130C3×C4⋊C4—31no—
131C3×C4⋊C4—32no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—81no—
136C3×C4⋊C4S411yescharacteristic
137C3×C22⋊C4—32no—
138C3×C22⋊C4—62no—
139C3×C42—32no—
140C3×C42—62no—
141(C2×D3)⋊C4—42no—
142C4⋊Dic3—41no—
143C4×Dic3—42no—
144C22×C32—41no—
145C2×C3×D3—42no—
146C2×C3×D3—81no—
147C2×C3×D3—81no—
148C32×C4—42no—
149C32×C4—82no—
150C3×Dic3—42no—
151C3×Dic3—82no—
152C2×#41—31no—
153C2×#41C2×C3×D311yescharacteristic
154#135—34no—
155#135—34no—
156C2×Q16—31no—
157C2×SD16—31no—
158C2×SD16—31no—
159C2×D8—31no—
160C2×M16—31no—
161C22×C8—31no—
162C4×Q8—32no—
163C4×Q8C2×C3×D312yes—
164C4×D4—32no—
165C4×D4—32no—
166C4×D4—62no—
167#117—31no—
168#117—61no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172#107—32no—
173#107—32no—
174#103—32no—
175#103—32no—
176#102—32no—
177#102—32no—
178#97—32no—
179C4×C8—32no—
180C23×C3—31no—
181C23×C3—61no—
182C22×D3—41no—
183C3×Q8—31no—
184C3×Q8—32no—
185C3×Q8—61no—
186C3×Q8C2×C4×D311yescharacteristic
187C3×D4—32no—
188C3×D4—32no—
189C3×D4—61no—
190C3×D4—61no—
191C3×D4—64no—
192C2×C3×C4—31no—
193C2×C3×C4—32no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—61no—
199C2×C3×C4—64no—
200C2×C3×C4—62no—
201C2×C3×C4—81no—
202C2×C3×C4—82no—
203C2×C3×C4—241no—
204C2×C3×C4—242no—
205C2×C3×C4—241no—
206C2×C3×C4C2×S412yes—
207C2×C3×C4C2×S411yescharacteristic
208C2×Dic3—41no—
209C2×Dic3—42no—
210C4×D3—44no—
211SL(2,𝔽3)—21no—
212SL(2,𝔽3)C22×C3×C411yescharacteristic
213C3×C8—32no—
214C3×C8—32no—
215C3×C8—62no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219C3×D3—82no—
220#41—32no—
221#41—34no—
222#41—62no—
223#41C3×C4×D312yes—
224C2×Q8—31no—
225C2×Q8C22×C3×D311yescharacteristic
226C2×D4—31no—
227C2×D4—31no—
228C2×D4—61no—
229C22×C4—31no—
230C22×C4—61no—
231C22×C4—62no—
232C22×C4—61no—
233C22×C4—61no—
234Q16—32no—
235Q16—32no—
236SD16—34no—
237SD16—32no—
238SD16—32no—
239D8—34no—
240M16—62no—
241C2×C8—31no—
242C2×C8—31no—
243C2×C8—34no—
244C2×C8—32no—
245C4⋊C4—32no—
246C4⋊C4—31no—
247C4⋊C4—32no—
248C4⋊C4—62no—
249C4⋊C4—61no—
250C4⋊C4C3×S411yescharacteristic
251C22⋊C4—32no—
252C22⋊C4—62no—
253C42—32no—
254C42—62no—
255C22×C3—32no—
256C22×C3—41no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—62no—
260C22×C3—81no—
261C22×C3—121no—
262C22×C3—121no—
263C22×C3C22×S411yescharacteristic
264C2×D3—42no—
265C2×D3—81no—
266C2×D3—81no—
267C3×C4—22no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—42no—
271C3×C4—62no—
272C3×C4—62no—
273C3×C4—82no—
274C3×C4—82no—
275C3×C4—122no—
276C3×C4—162no—
277C3×C4C4×S412yes—
278Dic3—42no—
279Dic3—82no—
280C32—41noSylow
281C23—31no—
282C23—61no—
283Q8—31no—
284Q8—32no—
285Q8—61no—
286Q8C2×C3×C4×D311yescharacteristic
287D4—32no—
288D4—32no—
289D4—61no—
290D4—61no—
291D4—64no—
292C2×C4—31no—
293C2×C4—32no—
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—61no—
297C2×C4—64no—
298C2×C4—62no—
299C2×C4—241no—
300C2×C4—242no—
301C2×C4—241no—
302C2×C4C2×C3×S412yes—
303C2×C4C2×C3×S411yescharacteristic
304C8—32no—
305C8—32no—
306C8—62no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—62no—
311C2×C3—81no—
312C2×C3—81no—
313C2×C3—81no—
314C2×C3—122no—
315C2×C3C2×C4×S411yescharacteristic
316C2×C3#592511yescharacteristic
317C2×C3#592711yescharacteristic
318D3—82no—
319C22—32no—
320C22—61no—
321C22—61no—
322C22—62no—
323C22—121no—
324C22—121no—
325C22C22×C3×S411yescharacteristic
326C4—22no—
327C4—31no—
328C4—31no—
329C4—62no—
330C4—62no—
331C4—122no—
332C4C3×C4×S412yes—
333C3—41no—
334C3—81no—
335C3#8536511yescharacteristic, direct product factor
336C2—62no—
337C2—122no—
338C2C2×C3×C4×S411yescharacteristic
339C2C3×#592511yescharacteristic
340C2C3×#592711yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, hypocenterSL(2,𝔽3)C22×C3×C4
FrattiniC22C22×C3×S4
FittingC3×#520D3
prime radicalC2×C3×#985C2
cosocleC2×SL(2,𝔽3)C23×C3
hypercenterC3×C4⋊C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. C3×#5426C2semidirect
4. C3×#5394C2semidirect
5. C3×C4×#226C2semidirect
6. #85365C3direct (N × H)
9. C3×#985C4semidirect
18. C3×#520D3semidirect
20. #520⋊C3C2×C3semidirect
21. #5426C2×C3semidirect
22. #5394C2×C3semidirect
23. C4×#226C2×C3semidirect
33. C3×#41⋊C3C2×C4semidirect
35. C3×GL(2,𝔽3)C2×C4semidirect
36. C3×#226C2×C4semidirect
46. #985C3×C4semidirect
88. #520C3×D3semidirect
113. #41⋊C3C2×C3×C4semidirect
116. GL(2,𝔽3)C2×C3×C4semidirect
117. #226C2×C3×C4semidirect
335. C3#85365direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9852
C3×#520⋊C32
C3×#54262
C3×#53942
C3×C4×#2262
#853653
C3×#29513
C3×#8924