Finite Group Databasebeta

← #13048716

Subgroups — #13048716

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
1#10619994C211yesmaximal, characteristic, complemented
2C2×C3⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3C3×#638⋊C3C211yesmaximal, characteristic, complemented
4#10619886C211yesmaximal, characteristic, complemented
5#10619885C211yesmaximal, characteristic, complemented
6((C3⋊D3)⋊C3)⋊Dic3C211yesmaximal, characteristic, complemented
7#10619881C211yesmaximal, characteristic, complemented
8#89046—31nomaximal
9#89029—31nomaximal
10#88671—33nomaximal
11C22×C3×H(3,𝔽3)C2211yescharacteristic
12C2×C3⋊#272C2211yescharacteristic
13C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
14C3⋊#1084C2211yescharacteristic
15C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×C3⋊(C3⋊Dic3)D311yescharacteristic, complemented
19C3×#638—31no—
20C3×#638—31no—
21C3×#638—61no—
22C3×#638D311yescharacteristic, complemented
23C2×C3×C3⋊Dic3—31no—
24C32×#67—31no—
25#6296—31no—
26#6295—31no—
27#6294—31no—
28#6294—31no—
29#6293—31no—
30#6290—31no—
31Dic3×C3⋊D3—31no—
32D3×C3⋊Dic3—31no—
33#6235—33no—
34#6235—63no—
35#638⋊C3D313yescomplemented
36C2×(C3⋊Dic3)⋊C3—33no—
37#6200—33no—
38#6199—33no—
39((C3⋊D3)⋊C3)⋊C4—33no—
40#6196—33no—
41C2×C3×H(3,𝔽3)—21no—
42C2×C3×H(3,𝔽3)C2311yescharacteristic
43C3⋊#272—21no—
44C3×(C3⋊D3)⋊C3—21no—
45#3830—91no—
46#3805—93no—
47#3805—91no—
48#3804—91no—
49#3804—91no—
50C22×C33—21no—
51C22×C33C2×D311yescharacteristic
52C22×C33C2×D311yescharacteristic
53C2×C3×C3⋊D3—31no—
54C2×C3×C3⋊D3—31no—
55C2×C3×C3⋊D3—61no—
56C2×C3×C3⋊D3C2×D311yescharacteristic
57C2×C32×D3—31no—
58C3⋊(C3⋊Dic3)C2×D311yescharacteristic
59C3⋊(C3⋊Dic3)C2×D311yescharacteristic
60C3×C3⋊Dic3—31no—
61C3×C3⋊Dic3—31no—
62C3×C3⋊Dic3—31no—
63C3×C3⋊Dic3—31no—
64C3×C3⋊Dic3—61no—
65C3×C3⋊Dic3C2×D311yescharacteristic
66C32×Dic3—31no—
67C22×H(3,𝔽3)—23no—
68C22×H(3,𝔽3)C2×D313yes—
69C2×#272—33no—
70C2×#272—63no—
71C2×(C3⋊D3)⋊C3C2×D313yes—
72#1084—33no—
73#1084—63no—
74(C3⋊Dic3)⋊C3—33no—
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—31no—
81#638—61no—
82#638—91no—
83#638C3⋊D311yescharacteristic, complemented
84C2×C3⋊Dic3—31no—
85C2×C3⋊Dic3—33no—
86C2×C3⋊Dic3—91no—
87C2×C3⋊Dic3—91no—
88C2×C3⋊Dic3—181no—
89C4×C3⋊D3—91no—
90#634—91no—
91C3×#67—33no—
92C3×#67—31no—
93C3×#67—91no—
94C3×#67—93no—
95C3×#67—91no—
96C3×#67—91no—
97C3×#67—183no—
98C3×#67—181no—
99C2×C3×Dic3—91no—
100C2×C3×Dic3—91no—
101C2×C3×Dic3—181no—
102#627—93no—
103#627—91no—
104#627—91no—
105#627—91no—
106#626—91no—
107#626—91no—
108#625—93no—
109#625—91no—
110#624—91no—
111#624—91no—
112D3×Dic3—93no—
113D3×Dic3—91no—
114D3×Dic3—93no—
115D3×Dic3—91no—
116D3×Dic3—91no—
117D3×Dic3—91no—
118C2×C33—21no—
119C2×C33—21no—
120C2×C33—21no—
121C2×C33—41no—
122C2×C33C22×D311yescharacteristic
123C2×C33C22×D311yescharacteristic
124C3×C3⋊D3—21no—
125C3×C3⋊D3—61no—
126C3×C3⋊D3—61no—
127C3×C3⋊D3—121no—
128C32×D3—61no—
129C2×H(3,𝔽3)—23no—
130C2×H(3,𝔽3)—23no—
131C2×H(3,𝔽3)—43no—
132C2×H(3,𝔽3)C22×D313yes—
133#272—63no—
134#272—123no—
135(C3⋊D3)⋊C3—23no—
136#237—271no—
137#237—271no—
138#237—273no—
139#235—271no—
140C22×C32—21no—
141C22×C32—23no—
142C22×C32—31no—
143C22×C32—31no—
144C22×C32—61no—
145C22×C32—61no—
146C22×C32—61no—
147C22×C32—91no—
148C22×C32—121no—
149C22×C32D3211yescharacteristic
150C22×C32D3213yes—
151C22×C32C2×C3⋊D311yescharacteristic
152C2×C3⋊D3—31no—
153C2×C3⋊D3—31no—
154C2×C3⋊D3—61no—
155C2×C3⋊D3C2×C3⋊D311yescharacteristic
156C2×C3×D3—33no—
157C2×C3×D3—31no—
158C2×C3×D3—91no—
159C2×C3×D3—93no—
160C2×C3×D3—91no—
161C2×C3×D3—91no—
162C2×C3×D3—183no—
163C2×C3×D3—181no—
164C32×C4—91no—
165C3⋊Dic3—31no—
166C3⋊Dic3—31no—
167C3⋊Dic3—31no—
168C3⋊Dic3—31no—
169C3⋊Dic3—33no—
170C3⋊Dic3—33no—
171C3⋊Dic3—61no—
172C3⋊Dic3—91no—
173C3⋊Dic3—91no—
174C3⋊Dic3—181no—
175C3⋊Dic3—181no—
176C3⋊Dic3C2×C3⋊D311yescharacteristic
177C3×Dic3—33no—
178C3×Dic3—31no—
179C3×Dic3—91no—
180C3×Dic3—91no—
181C3×Dic3—91no—
182C3×Dic3—93no—
183C3×Dic3—91no—
184C3×Dic3—91no—
185C3×Dic3—91no—
186C3×Dic3—91no—
187C3×Dic3—183no—
188C3×Dic3—181no—
189C3×Dic3—181no—
190C3×Dic3—181no—
191C33—21no—
192C33#23511yescharacteristic, complemented
193C33#23711yescharacteristic, complemented
194H(3,𝔽3)—23no—
195H(3,𝔽3)#23713yescomplemented
196C3×D4—93no—
197C3×D4—91no—
198C3×D4—271no—
199C2×C3×C4—271no—
200#67—31no—
201#67—91no—
202#67—91no—
203#67—93no—
204#67—91no—
205#67—181no—
206#67—183no—
207#67—271no—
208#67—271no—
209#67—273no—
210C2×Dic3—91no—
211C2×Dic3—91no—
212C2×Dic3—181no—
213C2×Dic3—273no—
214C2×Dic3—271no—
215C2×Dic3—271no—
216C2×Dic3—273no—
217D12—271no—
218C4×D3—271no—
219C4×D3—271no—
220C4×D3—271no—
221C4×D3—271no—
222C4×D3—273no—
223Dic6—271no—
224Dic6—271no—
225Dic6—271no—
226Dic6—273no—
227C2×C32—21no—
228C2×C32—23no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—23no—
232C2×C32—31no—
233C2×C32—31no—
234C2×C32—43no—
235C2×C32—41no—
236C2×C32—61no—
237C2×C32—61no—
238C2×C32—61no—
239C2×C32—62no—
240C2×C32—61no—
241C2×C32—62no—
242C2×C32—61no—
243C2×C32—121no—
244C2×C32—121no—
245C2×C32—122no—
246C2×C32—181no—
247C2×C32C2×D3211yescharacteristic
248C2×C32C2×D3213yes—
249C2×C32C22×C3⋊D311yescharacteristic
250C3⋊D3—21no—
251C3⋊D3—61no—
252C3⋊D3—61no—
253C3⋊D3—121no—
254C3×D3—61no—
255C3×D3—63no—
256C3×D3—181no—
257C3×D3—181no—
258C3×D3—183no—
259C3×D3—181no—
260C3×D3—363no—
261C3×D3—361no—
262#41—811noSylow
263C22×C3—21no—
264C22×C3—33no—
265C22×C3—31no—
266C22×C3—61no—
267C22×C3—63no—
268C22×C3—93no—
269C22×C3—91no—
270C22×C3—271no—
271C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C22×C3D3×C3⋊D311yescharacteristic
273C2×D3—31no—
274C2×D3—91no—
275C2×D3—91no—
276C2×D3—93no—
277C2×D3—91no—
278C2×D3—181no—
279C2×D3—183no—
280C3×C4—93no—
281C3×C4—91no—
282C3×C4—271no—
283C3×C4—271no—
284C3×C4—271no—
285Dic3—31no—
286Dic3—91no—
287Dic3—91no—
288Dic3—93no—
289Dic3—91no—
290Dic3—91no—
291Dic3—91no—
292Dic3—91no—
293Dic3—91no—
294Dic3—181no—
295Dic3—181no—
296Dic3—181no—
297Dic3—183no—
298Dic3—273no—
299Dic3—273no—
300C32—21no—
301C32—23no—
302C32—31no—
303C32—31no—
304C32—61no—
305C32—61no—
306C32—61no—
307C32—121no—
308C32#380411yescharacteristic
309C32#380413yescomplemented
310C32#383011yescharacteristic, complemented
311Q8—811no—
312D4—91no—
313D4—271no—
314D4—811no—
315C2×C4—271no—
316C2×C4—811no—
317C2×C4—811no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—31no—
323C2×C3—33no—
324C2×C3—61no—
325C2×C3—63no—
326C2×C3—61no—
327C2×C3—63no—
328C2×C3—121no—
329C2×C3—123no—
330C2×C3—183no—
331C2×C3—181no—
332C2×C3—541no—
333C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
334C2×C3C2×D3×C3⋊D311yescharacteristic
335D3—61no—
336D3—181no—
337D3—181no—
338D3—183no—
339D3—181no—
340D3—361no—
341D3—363no—
342C22—91no—
343C22—271no—
344C22((C3⋊D3)⋊C3)⋊D311yescharacteristic
345C4—91no—
346C4—271no—
347C4—271no—
348C4—271no—
349C3—21no—
350C3—31no—
351C3—33no—
352C3—61no—
353C3—63no—
354C3#8867111yescharacteristic, complemented
355C3#8902811yescharacteristic
356C2—21no—
357C2—181no—
358C2—541no—
359C2C2×((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. #10619994C2semidirect
2. C2×C3⋊((C3⋊Dic3)⋊C3)C2semidirect
3. C3×#638⋊C3C2semidirect
4. #10619886C2semidirect
5. #10619885C2semidirect
6. ((C3⋊D3)⋊C3)⋊Dic3C2semidirect
7. #10619881C2semidirect
18. C2×C3⋊(C3⋊Dic3)D3semidirect
22. C3×#638D3semidirect
35. #638⋊C3D3semidirect
77. C3×H(3,𝔽3)#41semidirect
83. #638C3⋊D3semidirect
192. C33#235semidirect
193. C33#237semidirect
195. H(3,𝔽3)#237semidirect
309. C32#3804semidirect
310. C32#3830semidirect
354. C3#88671semidirect

Maximal subgroups

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

subgroupindex
#106199942
C2×C3⋊((C3⋊Dic3)⋊C3)2
C3×#638⋊C32
#106198862
#106198852
((C3⋊D3)⋊C3)⋊Dic32
#106198812
#890463
#890293
#886713