Finite Group Databasebeta

← #13048714

Subgroups — #13048714

Order 1296.

533 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 74 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⋊#1084C211yesmaximal, characteristic, complemented
2#10619977C211yesmaximal, characteristic, complemented
3C3×#638⋊C3C211yesmaximal, characteristic, complemented
4#10619884C211yesmaximal, characteristic, complemented
5#10619883C211yesmaximal, characteristic, complemented
6((C3⋊D3)⋊C3)⋊Dic3C211yesmaximal, characteristic, complemented
7#10619881C211yesmaximal, characteristic, complemented
8#89046—31nomaximal
9#89028—31nomaximal
10#88672—33nomaximal
11C22×C3×H(3,𝔽3)C2211yescharacteristic
12C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
14C3⋊#1084C2211yescharacteristic
15C3⋊#1084C2211yescharacteristic
16C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#6312D311yescharacteristic, complemented
19C3×#638—31no—
20C3×#638D311yescharacteristic, complemented
21C2×C3×C3⋊Dic3—31no—
22C2×C3×C3⋊Dic3—31no—
23C2×C3×C3⋊Dic3—61no—
24C32×#67—31no—
25#6296—31no—
26#6295—31no—
27#6294—31no—
28#6294—31no—
29#6293—31no—
30#6291—31no—
31#6289—31no—
32D3×C3⋊Dic3—31no—
33C2×#1084—33no—
34C2×#1084—63no—
35#638⋊C3—33no—
36#638⋊C3D313yescomplemented
37#6198—33no—
38((C3⋊D3)⋊C3)⋊C4—33no—
39((C3⋊D3)⋊C3)⋊C4—33no—
40#6196—33no—
41C2×C3×H(3,𝔽3)—21no—
42C2×C3×H(3,𝔽3)C2311yescharacteristic
43C3⋊((C3⋊D3)⋊C3)—21no—
44C3×(C3⋊D3)⋊C3—21no—
45#3830—91no—
46#3805—91no—
47#3804—93no—
48#3804—91no—
49#3804—91no—
50C22×C33—21no—
51C22×C33C2×D311yescharacteristic
52C22×C33C2×D311yescharacteristic
53C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C3×C3⋊D3—31no—
55C2×C3×C3⋊D3C2×D311yescharacteristic
56C2×C32×D3—31no—
57C3⋊(C3⋊Dic3)C2×D311yescharacteristic
58C3×C3⋊Dic3—31no—
59C3×C3⋊Dic3—31no—
60C3×C3⋊Dic3—31no—
61C3×C3⋊Dic3—31no—
62C3×C3⋊Dic3—31no—
63C3×C3⋊Dic3—61no—
64C3×C3⋊Dic3—61no—
65C3×C3⋊Dic3C2×D311yescharacteristic
66C32×Dic3—31no—
67C22×H(3,𝔽3)—23no—
68C22×H(3,𝔽3)C2×D313yes—
69C2×(C3⋊D3)⋊C3—33no—
70C2×(C3⋊D3)⋊C3C2×D313yes—
71#1084—33no—
72#1084—33no—
73#1084—63no—
74#1084—63no—
75(C3⋊Dic3)⋊C3—33no—
76(C3⋊Dic3)⋊C3C2×D313yes—
77C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
78C32×D4—91no—
79#638—31no—
80#638—33no—
81#638—91no—
82#638—91no—
83#638—181no—
84#638C3⋊D311yescharacteristic, complemented
85C2×C3⋊Dic3—31no—
86C2×C3⋊Dic3—31no—
87C2×C3⋊Dic3—61no—
88C2×C3⋊Dic3—91no—
89C4×C3⋊D3—91no—
90#634—91no—
91C3×#67—33no—
92C3×#67—31no—
93C3×#67—91no—
94C3×#67—91no—
95C3×#67—91no—
96C3×#67—181no—
97C2×C3×Dic3—93no—
98C2×C3×Dic3—91no—
99C2×C3×Dic3—91no—
100C2×C3×Dic3—183no—
101C2×C3×Dic3—181no—
102#627—93no—
103#627—91no—
104#627—91no—
105#627—91no—
106#626—93no—
107#626—91no—
108#626—91no—
109#625—91no—
110#624—93no—
111#624—91no—
112#624—91no—
113D3×Dic3—93no—
114D3×Dic3—91no—
115D3×Dic3—91no—
116D3×Dic3—91no—
117D3×Dic3—91no—
118C2×C33—22no—
119C2×C33—21no—
120C2×C33—21no—
121C2×C33—21no—
122C2×C33C22×D311yescharacteristic
123C2×C33C22×D311yescharacteristic
124C3⋊(C3⋊D3)—21no—
125C3×C3⋊D3—21no—
126C3×C3⋊D3—61no—
127C32×D3—61no—
128C2×H(3,𝔽3)—26no—
129C2×H(3,𝔽3)—23no—
130C2×H(3,𝔽3)—23no—
131C2×H(3,𝔽3)C22×D313yes—
132(C3⋊D3)⋊C3—23no—
133(C3⋊D3)⋊C3—63no—
134#237—271no—
135#237—271no—
136#237—273no—
137#235—271no—
138C22×C32—21no—
139C22×C32—23no—
140C22×C32—31no—
141C22×C32—31no—
142C22×C32—61no—
143C22×C32—61no—
144C22×C32—61no—
145C22×C32—91no—
146C22×C32—121no—
147C22×C32D3211yescharacteristic
148C22×C32D3213yes—
149C22×C32C2×C3⋊D311yescharacteristic
150C2×C3⋊D3—31no—
151C2×C3⋊D3—33no—
152C2×C3⋊D3—91no—
153C2×C3⋊D3—181no—
154C2×C3⋊D3C2×C3⋊D311yescharacteristic
155C2×C3×D3—33no—
156C2×C3×D3—31no—
157C2×C3×D3—91no—
158C2×C3×D3—91no—
159C2×C3×D3—91no—
160C2×C3×D3—181no—
161C32×C4—91no—
162C3⋊Dic3—31no—
163C3⋊Dic3—31no—
164C3⋊Dic3—31no—
165C3⋊Dic3—31no—
166C3⋊Dic3—31no—
167C3⋊Dic3—33no—
168C3⋊Dic3—61no—
169C3⋊Dic3—61no—
170C3⋊Dic3—91no—
171C3⋊Dic3—181no—
172C3⋊Dic3C2×C3⋊D311yescharacteristic
173C3×Dic3—33no—
174C3×Dic3—31no—
175C3×Dic3—91no—
176C3×Dic3—91no—
177C3×Dic3—93no—
178C3×Dic3—93no—
179C3×Dic3—91no—
180C3×Dic3—91no—
181C3×Dic3—91no—
182C3×Dic3—91no—
183C3×Dic3—91no—
184C3×Dic3—183no—
185C3×Dic3—183no—
186C3×Dic3—181no—
187C3×Dic3—181no—
188C3×Dic3—181no—
189C33—21no—
190C33#23711yescharacteristic, complemented
191C33#23711yescharacteristic, complemented
192H(3,𝔽3)—23no—
193H(3,𝔽3)#23713yescomplemented
194C3×D4—93no—
195C3×D4—91no—
196C3×D4—271no—
197C2×C3×C4—271no—
198#67—31no—
199#67—91no—
200#67—91no—
201#67—91no—
202#67—181no—
203#67—273no—
204#67—271no—
205#67—271no—
206#67—273no—
207C2×Dic3—91no—
208C2×Dic3—93no—
209C2×Dic3—91no—
210C2×Dic3—181no—
211C2×Dic3—183no—
212C2×Dic3—271no—
213C2×Dic3—271no—
214C2×Dic3—273no—
215D12—271no—
216C4×D3—271no—
217C4×D3—271no—
218C4×D3—273no—
219C4×D3—271no—
220C4×D3—271no—
221Dic6—271no—
222Dic6—271no—
223Dic6—271no—
224Dic6—273no—
225C2×C32—21no—
226C2×C32—22no—
227C2×C32—26no—
228C2×C32—23no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—23no—
232C2×C32—31no—
233C2×C32—31no—
234C2×C32—61no—
235C2×C32—61no—
236C2×C32—61no—
237C2×C32—61no—
238C2×C32—62no—
239C2×C32—61no—
240C2×C32—121no—
241C2×C32—121no—
242C2×C32—121no—
243C2×C32—122no—
244C2×C32—181no—
245C2×C32C2×D3211yescharacteristic
246C2×C32C2×D3213yes—
247C2×C32C22×C3⋊D311yescharacteristic
248C3⋊D3—21no—
249C3⋊D3—61no—
250C3⋊D3—63no—
251C3⋊D3—181no—
252C3⋊D3—361no—
253C3×D3—61no—
254C3×D3—63no—
255C3×D3—181no—
256C3×D3—181no—
257C3×D3—181no—
258C3×D3—361no—
259#41—811noSylow
260C22×C3—21no—
261C22×C3—31no—
262C22×C3—33no—
263C22×C3—61no—
264C22×C3—63no—
265C22×C3—93no—
266C22×C3—91no—
267C22×C3—271no—
268C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C22×C3D3×C3⋊D311yescharacteristic
270C2×D3—31no—
271C2×D3—91no—
272C2×D3—91no—
273C2×D3—91no—
274C2×D3—181no—
275C2×D3—273no—
276C3×C4—93no—
277C3×C4—91no—
278C3×C4—271no—
279C3×C4—271no—
280C3×C4—271no—
281Dic3—31no—
282Dic3—91no—
283Dic3—91no—
284Dic3—93no—
285Dic3—91no—
286Dic3—93no—
287Dic3—91no—
288Dic3—91no—
289Dic3—91no—
290Dic3—91no—
291Dic3—181no—
292Dic3—181no—
293Dic3—181no—
294Dic3—183no—
295Dic3—183no—
296Dic3—273no—
297C32—21no—
298C32—23no—
299C32—31no—
300C32—31no—
301C32—61no—
302C32—61no—
303C32—61no—
304C32—121no—
305C32#380511yescharacteristic
306C32#380513yescomplemented
307C32#383011yescharacteristic, complemented
308Q8—811no—
309D4—91no—
310D4—271no—
311D4—811no—
312C2×C4—271no—
313C2×C4—811no—
314C2×C4—811no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—31no—
319C2×C3—33no—
320C2×C3—41no—
321C2×C3—61no—
322C2×C3—63no—
323C2×C3—62no—
324C2×C3—66no—
325C2×C3—61no—
326C2×C3—63no—
327C2×C3—183no—
328C2×C3—181no—
329C2×C3—541no—
330C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
331C2×C3C2×D3×C3⋊D311yescharacteristic
332D3—61no—
333D3—181no—
334D3—181no—
335D3—181no—
336D3—361no—
337D3—543no—
338C22—91no—
339C22—271no—
340C22((C3⋊D3)⋊C3)⋊D311yescharacteristic
341C4—91no—
342C4—271no—
343C4—271no—
344C4—271no—
345C3—21no—
346C3—31no—
347C3—33no—
348C3—61no—
349C3—63no—
350C3#8867211yescharacteristic, complemented
351C3#8902911yescharacteristic
352C2—21no—
353C2—181no—
354C2—541no—
355C2C2×((C3⋊D3)⋊C3)⋊D311yescharacteristic
subgroupG/H
centerC2C2×((C3⋊D3)⋊C3)⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC22((C3⋊D3)⋊C3)⋊D3
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊#1084C2semidirect
2. #10619977C2semidirect
3. C3×#638⋊C3C2semidirect
4. #10619884C2semidirect
5. #10619883C2semidirect
6. ((C3⋊D3)⋊C3)⋊Dic3C2semidirect
7. #10619881C2semidirect
18. #6312D3semidirect
20. C3×#638D3semidirect
36. #638⋊C3D3semidirect
77. C3×H(3,𝔽3)#41semidirect
84. #638C3⋊D3semidirect
190. C33#237semidirect
191. C33#237semidirect
193. H(3,𝔽3)#237semidirect
306. C32#3805semidirect
307. C32#3830semidirect
350. C3#88672semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#10842
#106199772
C3×#638⋊C32
#106198842
#106198832
((C3⋊D3)⋊C3)⋊Dic32
#106198812
#890463
#890283
#886723