Finite Group Databasebeta

← #13048778

Subgroups — #13048778

Order 1296.

511 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 59 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×#6235C211yesmaximal, characteristic, complemented
2#10619977C211yesmaximal, characteristic, complemented
3C2×C3⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4#10619904C211yesmaximal, characteristic, complemented
5#10619903C211yesmaximal, characteristic, complemented
6#10619901C211yesmaximal, characteristic, complemented
7#10619900C211yesmaximal, characteristic, complemented
8#89029—31nomaximal
9#89028—31nomaximal
10#88671—31nomaximal
11C22×C3×H(3,𝔽3)C2211yescharacteristic
12C2×C3×#272C2211yescharacteristic
13C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C3×#1084C2211yescharacteristic
15C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#6312D311yescharacteristic, complemented
19C2×C3⋊(C3⋊Dic3)D311yescharacteristic, complemented
20C3×#638—31no—
21C2×C3×C3⋊Dic3—31no—
22C32×#67—31no—
23C32×#67—31no—
24C32×#67—61no—
25#6293—31no—
26#6293—31no—
27#6291—31no—
28#6290—31no—
29#6289—31no—
30Dic3×C3⋊D3—31no—
31D3×C3⋊Dic3—31no—
32D3×C3⋊Dic3—31no—
33#6235D311yescharacteristic, complemented
34#638⋊C3—31no—
35#638⋊C3—61no—
36C2×(C3⋊Dic3)⋊C3—31no—
37C2×(C3⋊Dic3)⋊C3—61no—
38#6200—31no—
39#6199—31no—
40((C3⋊D3)⋊C3)⋊C4—31no—
41#6196—31no—
42C2×C3×H(3,𝔽3)—21no—
43C2×C3×H(3,𝔽3)C2311yescharacteristic
44C3×#272—21no—
45C3⋊((C3⋊D3)⋊C3)—21no—
46#3830—91no—
47#3805—91no—
48#3805—93no—
49#3804—91no—
50#3804—93no—
51C22×C33—21no—
52C22×C33C2×D311yescharacteristic
53C22×C33C2×D311yescharacteristic
54C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C3×C3⋊D3—31no—
56C2×C32×D3—31no—
57C2×C32×D3—31no—
58C2×C32×D3—61no—
59C3⋊(C3⋊Dic3)C2×D311yescharacteristic
60C3⋊(C3⋊Dic3)C2×D311yescharacteristic
61C3⋊(C3⋊Dic3)C2×D311yescharacteristic
62C3×C3⋊Dic3—31no—
63C3×C3⋊Dic3—31no—
64C3×C3⋊Dic3—31no—
65C32×Dic3—31no—
66C32×Dic3—31no—
67C32×Dic3—61no—
68C22×H(3,𝔽3)—21no—
69C22×H(3,𝔽3)—21no—
70C22×H(3,𝔽3)—41no—
71C22×H(3,𝔽3)C2×D311yescharacteristic
72C2×#272C2×D311yescharacteristic
73C2×(C3⋊D3)⋊C3—31no—
74C2×(C3⋊D3)⋊C3—61no—
75#1084C2×D311yescharacteristic
76(C3⋊Dic3)⋊C3—31no—
77(C3⋊Dic3)⋊C3—31no—
78(C3⋊Dic3)⋊C3—31no—
79(C3⋊Dic3)⋊C3—61no—
80(C3⋊Dic3)⋊C3—61no—
81(C3⋊Dic3)⋊C3—61no—
82C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
83C32×D4—91no—
84#638—31no—
85#638—31no—
86#638—61no—
87#638—93no—
88#638—91no—
89C2×C3⋊Dic3—31no—
90C2×C3⋊Dic3—31no—
91C2×C3⋊Dic3—61no—
92C2×C3⋊Dic3—93no—
93C2×C3⋊Dic3—91no—
94C4×C3⋊D3—91no—
95#634—91no—
96C3×#67—33no—
97C3×#67—33no—
98C3×#67—31no—
99C3×#67—31no—
100C3×#67—61no—
101C3×#67—63no—
102C3×#67—93no—
103C3×#67—91no—
104C2×C3×Dic3—93no—
105C2×C3×Dic3—91no—
106#627—91no—
107#627—91no—
108#627—93no—
109#627—93no—
110#626—91no—
111#626—93no—
112#625—91no—
113#625—93no—
114#624—91no—
115#624—93no—
116D3×Dic3—91no—
117D3×Dic3—93no—
118D3×Dic3—91no—
119D3×Dic3—91no—
120D3×Dic3—93no—
121D3×Dic3—93no—
122C2×C33—21no—
123C2×C33—21no—
124C2×C33—21no—
125C2×C33—41no—
126C2×C33C22×D311yescharacteristic
127C2×C33C22×D311yescharacteristic
128C3⋊(C3⋊D3)—21no—
129C3×C3⋊D3—61no—
130C32×D3—61no—
131C32×D3—61no—
132C32×D3—121no—
133C2×H(3,𝔽3)—22no—
134C2×H(3,𝔽3)—21no—
135C2×H(3,𝔽3)—21no—
136C2×H(3,𝔽3)—21no—
137C2×H(3,𝔽3)—41no—
138C2×H(3,𝔽3)—41no—
139C2×H(3,𝔽3)—42no—
140C2×H(3,𝔽3)C22×D311yescharacteristic
141#272—21no—
142(C3⋊D3)⋊C3—61no—
143(C3⋊D3)⋊C3—121no—
144#237—273no—
145#237—271no—
146#237—271no—
147#235—271no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—33no—
152C22×C32—33no—
153C22×C32—41no—
154C22×C32—63no—
155C22×C32—91no—
156C22×C32D3211yescharacteristic
157C22×C32D3211yescharacteristic
158C22×C32D3211yescharacteristic
159C2×C3⋊D3—31no—
160C2×C3⋊D3—31no—
161C2×C3⋊D3—61no—
162C2×C3⋊D3—93no—
163C2×C3×D3—33no—
164C2×C3×D3—33no—
165C2×C3×D3—31no—
166C2×C3×D3—31no—
167C2×C3×D3—61no—
168C2×C3×D3—63no—
169C2×C3×D3—93no—
170C2×C3×D3—91no—
171C32×C4—91no—
172C3⋊Dic3—31no—
173C3⋊Dic3—31no—
174C3⋊Dic3—31no—
175C3⋊Dic3—31no—
176C3⋊Dic3—31no—
177C3⋊Dic3—31no—
178C3⋊Dic3—61no—
179C3⋊Dic3—61no—
180C3⋊Dic3—61no—
181C3⋊Dic3—93no—
182C3⋊Dic3—93no—
183C3⋊Dic3—93no—
184C3×Dic3—33no—
185C3×Dic3—33no—
186C3×Dic3—31no—
187C3×Dic3—31no—
188C3×Dic3—61no—
189C3×Dic3—63no—
190C3×Dic3—93no—
191C3×Dic3—93no—
192C3×Dic3—93no—
193C3×Dic3—91no—
194C3×Dic3—91no—
195C3×Dic3—91no—
196C33—21no—
197C33#23511yescharacteristic, complemented
198C33#23711yescharacteristic, complemented
199H(3,𝔽3)—21no—
200H(3,𝔽3)—21no—
201H(3,𝔽3)—41no—
202H(3,𝔽3)#23711yescharacteristic, complemented
203C3×D4—93no—
204C3×D4—91no—
205C3×D4—271no—
206C2×C3×C4—271no—
207#67—31no—
208#67—31no—
209#67—61no—
210#67—93no—
211#67—91no—
212#67—271no—
213#67—273no—
214#67—271no—
215#67—271no—
216#67—541no—
217C2×Dic3—93no—
218C2×Dic3—91no—
219C2×Dic3—271no—
220C2×Dic3—271no—
221C2×Dic3—273no—
222C2×Dic3—271no—
223C2×Dic3—541no—
224D12—271no—
225C4×D3—271no—
226C4×D3—273no—
227C4×D3—271no—
228C4×D3—271no—
229C4×D3—271no—
230Dic6—271no—
231Dic6—271no—
232Dic6—273no—
233Dic6—271no—
234C2×C32—21no—
235C2×C32—22no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—33no—
242C2×C32—33no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—42no—
246C2×C32—41no—
247C2×C32—63no—
248C2×C32—63no—
249C2×C32—63no—
250C2×C32—123no—
251C2×C32—181no—
252C2×C32C2×D3211yescharacteristic
253C2×C32C2×D3211yescharacteristic
254C2×C32C2×D3211yescharacteristic
255C3⋊D3—61no—
256C3⋊D3—61no—
257C3⋊D3—121no—
258C3⋊D3—183no—
259C3×D3—63no—
260C3×D3—63no—
261C3×D3—61no—
262C3×D3—61no—
263C3×D3—121no—
264C3×D3—123no—
265C3×D3—183no—
266C3×D3—181no—
267#41—811noSylow
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—61no—
273C22×C3—93no—
274C22×C3—91no—
275C22×C3—121no—
276C22×C3—271no—
277C22×C3((C3⋊D3)⋊C3)⋊C213yes—
278C22×C3#111311yescharacteristic
279C2×D3—31no—
280C2×D3—31no—
281C2×D3—61no—
282C2×D3—93no—
283C2×D3—91no—
284C2×D3—271no—
285C2×D3—541no—
286C3×C4—93no—
287C3×C4—91no—
288C3×C4—271no—
289C3×C4—271no—
290C3×C4—271no—
291Dic3—31no—
292Dic3—31no—
293Dic3—61no—
294Dic3—93no—
295Dic3—93no—
296Dic3—93no—
297Dic3—91no—
298Dic3—91no—
299Dic3—91no—
300Dic3—271no—
301Dic3—271no—
302Dic3—271no—
303Dic3—541no—
304Dic3—541no—
305Dic3—541no—
306C32—21no—
307C32—21no—
308C32—21no—
309C32—33no—
310C32—33no—
311C32—41no—
312C32—63no—
313C32#380411yescharacteristic
314C32#380411yescharacteristic, complemented
315C32#380511yescharacteristic, complemented
316Q8—811no—
317D4—91no—
318D4—271no—
319D4—811no—
320C2×C4—271no—
321C2×C4—811no—
322C2×C4—811no—
323C2×C3—23no—
324C2×C3—21no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3—62no—
330C2×C3—61no—
331C2×C3—61no—
332C2×C3—61no—
333C2×C3—121no—
334C2×C3—121no—
335C2×C3—122no—
336C2×C3—121no—
337C2×C3—183no—
338C2×C3—181no—
339C2×C3—541no—
340C2×C3C2×((C3⋊D3)⋊C3)⋊C213yes—
341C2×C3C2×#111311yescharacteristic
342D3—61no—
343D3—61no—
344D3—121no—
345D3—183no—
346D3—181no—
347D3—541no—
348D3—1081no—
349C22—91no—
350C22—271no—
351C22#6643211yescharacteristic
352C4—91no—
353C4—271no—
354C4—271no—
355C4—271no—
356C3—31no—
357C3—31no—
358C3—61no—
359C3—61no—
360C3—61no—
361C3—121no—
362C3#8867113yescomplemented
363C3#8904611yescharacteristic
364C2—21no—
365C2—181no—
366C2—541no—
367C2C2×#6643211yescharacteristic
subgroupG/H
centerC2C2×#66432
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×#1113
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC22#66432
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. C3×#6235C2semidirect
2. #10619977C2semidirect
3. C2×C3⋊((C3⋊Dic3)⋊C3)C2semidirect
4. #10619904C2semidirect
5. #10619903C2semidirect
6. #10619901C2semidirect
7. #10619900C2semidirect
18. #6312D3semidirect
19. C2×C3⋊(C3⋊Dic3)D3semidirect
33. #6235D3semidirect
82. C3×H(3,𝔽3)#41semidirect
197. C33#235semidirect
198. C33#237semidirect
202. H(3,𝔽3)#237semidirect
314. C32#3804semidirect
315. C32#3805semidirect
362. C3#88671semidirect

Maximal subgroups

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

subgroupindex
C3×#62352
#106199772
C2×C3⋊((C3⋊Dic3)⋊C3)2
#106199042
#106199032
#106199012
#106199002
#890293
#890283
#886713