Finite Group Databasebeta

← C3×#10619977 #14518953

Subgroups — C3×#10619977 #14518953

Order 1944.

1521 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1521 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C22×C32×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C3×C3⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4C3×#6312C311yesmaximal, characteristic, complemented
5C32×#638—31nomaximal
6#10619977C313yesmaximal, direct product factor
7C3×#638⋊C3—33nomaximal
8C2×C32×H(3,𝔽3)—21no—
9C2×C32×H(3,𝔽3)C2211yescharacteristic
10C3×C3⋊((C3⋊D3)⋊C3)—21no—
11C22×C34—21no—
12C22×C34D311yescharacteristic, complemented
13C22×C34C2×C311yescharacteristic, complemented
14C2×C3×C3⋊(C3⋊D3)C2×C311yescharacteristic, complemented
15C2×C32×C3⋊D3—31no—
16C3×C3⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
17C32×C3⋊Dic3—31no—
18C22×C3×H(3,𝔽3)—23no—
19C22×C3×H(3,𝔽3)—23no—
20C22×C3×H(3,𝔽3)—29no—
21C22×C3×H(3,𝔽3)D313yescomplemented
22C22×C3×H(3,𝔽3)C2×C313yescomplemented
23C2×C3⋊((C3⋊D3)⋊C3)C2×C313yescomplemented
24C2×C3×(C3⋊D3)⋊C3—33no—
25C3⋊((C3⋊Dic3)⋊C3)C2×C313yescomplemented
26C3×(C3⋊Dic3)⋊C3—33no—
27C32×H(3,𝔽3)D411yescharacteristic, complemented, Sylow
28#6312C3211yescharacteristic, complemented
29C3×#638—31no—
30C3×#638—33no—
31C3×#638—33no—
32C3×#638—93no—
33C32×#67—93no—
34C32×#67—91no—
35#638⋊C3—39no—
36C2×C34—22no—
37C2×C34—21no—
38C2×C34—21no—
39C2×C34—21no—
40C2×C34C2×D311yescharacteristic
41C2×C34C22×C311yescharacteristic
42C3×C3⋊(C3⋊D3)—21no—
43C32×C3⋊D3—61no—
44C2×C3×H(3,𝔽3)—26no—
45C2×C3×H(3,𝔽3)—26no—
46C2×C3×H(3,𝔽3)—218no—
47C2×C3×H(3,𝔽3)—23no—
48C2×C3×H(3,𝔽3)—23no—
49C2×C3×H(3,𝔽3)—23no—
50C2×C3×H(3,𝔽3)—23no—
51C2×C3×H(3,𝔽3)—29no—
52C2×C3×H(3,𝔽3)C2×D313yes—
53C2×C3×H(3,𝔽3)C22×C313yes—
54C3⋊((C3⋊D3)⋊C3)—23no—
55C3×(C3⋊D3)⋊C3—63no—
56C22×C33—26no—
57C22×C33—21no—
58C22×C33—23no—
59C22×C33—23no—
60C22×C33—21no—
61C22×C33—23no—
62C22×C33—23no—
63C22×C33—33no—
64C22×C33—33no—
65C22×C33—63no—
66C22×C33—63no—
67C22×C33—66no—
68C22×C33—63no—
69C22×C33C3×D311yescharacteristic
70C22×C33C3×D313yescomplemented
71C22×C33C3×D313yescomplemented
72C22×C33C3⋊D311yescharacteristic, complemented
73C22×C33C2×C3211yescharacteristic, complemented
74C2×C3⋊(C3⋊D3)C2×C3211yescharacteristic, complemented
75C2×C3×C3⋊D3—33no—
76C2×C3×C3⋊D3—31no—
77C2×C3×C3⋊D3—33no—
78C2×C3×C3⋊D3—93no—
79C2×C32×D3—93no—
80C2×C32×D3—91no—
81C3⋊(C3⋊Dic3)C2×C3211yescharacteristic, complemented
82C3×C3⋊Dic3—33no—
83C3×C3⋊Dic3—31no—
84C3×C3⋊Dic3—33no—
85C3×C3⋊Dic3—93no—
86C32×Dic3—93no—
87C32×Dic3—91no—
88C22×H(3,𝔽3)—218no—
89C22×H(3,𝔽3)—29no—
90C22×H(3,𝔽3)—29no—
91C22×H(3,𝔽3)C3×D319yescomplemented
92C2×(C3⋊D3)⋊C3—39no—
93(C3⋊Dic3)⋊C3—39no—
94C34—21no—
95C34#6711yescharacteristic, complemented
96C34C3×D411yescharacteristic, complemented
97C3×H(3,𝔽3)—23no—
98C3×H(3,𝔽3)—23no—
99C3×H(3,𝔽3)—29no—
100C3×H(3,𝔽3)#6713yescomplemented
101C3×H(3,𝔽3)C3×D413yescomplemented
102C32×D4—271no—
103#638—31no—
104#638—33no—
105#638—93no—
106C3×#67—99no—
107C3×#67—93no—
108C3×#67—91no—
109C3×#67—93no—
110C3×#67—273no—
111C2×C33—21no—
112C2×C33—212no—
113C2×C33—26no—
114C2×C33—26no—
115C2×C33—22no—
116C2×C33—26no—
117C2×C33—22no—
118C2×C33—26no—
119C2×C33—26no—
120C2×C33—23no—
121C2×C33—23no—
122C2×C33—21no—
123C2×C33—21no—
124C2×C33—23no—
125C2×C33—21no—
126C2×C33—23no—
127C2×C33—23no—
128C2×C33—21no—
129C2×C33—23no—
130C2×C33—33no—
131C2×C33—33no—
132C2×C33—63no—
133C2×C33—63no—
134C2×C33—63no—
135C2×C33—63no—
136C2×C33—66no—
137C2×C33—66no—
138C2×C33—612no—
139C2×C33—66no—
140C2×C33—66no—
141C2×C33—63no—
142C2×C33C2×C3×D311yescharacteristic
143C2×C33C2×C3×D313yes—
144C2×C33C2×C3×D313yes—
145C2×C33C2×C3⋊D311yescharacteristic
146C2×C33C22×C3211yescharacteristic
147C3⋊(C3⋊D3)—21no—
148C3×C3⋊D3—61no—
149C3×C3⋊D3—63no—
150C3×C3⋊D3—63no—
151C3×C3⋊D3—183no—
152C32×D3—183no—
153C32×D3—181no—
154C2×H(3,𝔽3)—236no—
155C2×H(3,𝔽3)—218no—
156C2×H(3,𝔽3)—218no—
157C2×H(3,𝔽3)—218no—
158C2×H(3,𝔽3)—29no—
159C2×H(3,𝔽3)—29no—
160C2×H(3,𝔽3)—29no—
161C2×H(3,𝔽3)C2×C3×D319yes—
162(C3⋊D3)⋊C3—69no—
163C22×C32—23no—
164C22×C32—21no—
165C22×C32—23no—
166C22×C32—23no—
167C22×C32—29no—
168C22×C32—33no—
169C22×C32—31no—
170C22×C32—39no—
171C22×C32—33no—
172C22×C32—61no—
173C22×C32—69no—
174C22×C32—69no—
175C22×C32—63no—
176C22×C32—618no—
177C22×C32—62no—
178C22×C32—66no—
179C22×C32—63no—
180C22×C32—69no—
181C22×C32—61no—
182C22×C32—63no—
183C22×C32—271no—
184C22×C32(C3⋊D3)⋊C313yescomplemented
185C22×C32C32×D311yescharacteristic
186C22×C32C32×D313yescomplemented
187C22×C32C3×C3⋊D311yescharacteristic
188C22×C32C3×C3⋊D313yescomplemented
189C2×C3⋊D3—31no—
190C2×C3⋊D3—33no—
191C2×C3⋊D3—93no—
192C2×C3×D3—99no—
193C2×C3×D3—93no—
194C2×C3×D3—93no—
195C2×C3×D3—91no—
196C2×C3×D3—273no—
197C32×C4—271no—
198C3⋊Dic3—31no—
199C3⋊Dic3—33no—
200C3⋊Dic3—93no—
201C3×Dic3—99no—
202C3×Dic3—93no—
203C3×Dic3—93no—
204C3×Dic3—91no—
205C3×Dic3—273no—
206C33—26no—
207C33—21no—
208C33—23no—
209C33—23no—
210C33—21no—
211C33—23no—
212C33—23no—
213C33—33no—
214C33—33no—
215C33—63no—
216C33—63no—
217C33—66no—
218C33—63no—
219C33C3×#6711yescharacteristic
220C33C3×#6713yescomplemented
221C33C3×#6713yescomplemented
222C33#63811yescharacteristic, complemented
223C33C32×D411yescharacteristic, complemented
224H(3,𝔽3)—218no—
225H(3,𝔽3)—29no—
226H(3,𝔽3)—29no—
227H(3,𝔽3)C3×#6719yescomplemented
228C3×D4—273no—
229C3×D4—271no—
230#67—93no—
231#67—91no—
232#67—273no—
233C2×C32—22no—
234C2×C32—26no—
235C2×C32—23no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—23no—
239C2×C32—21no—
240C2×C32—26no—
241C2×C32—26no—
242C2×C32—218no—
243C2×C32—23no—
244C2×C32—23no—
245C2×C32—23no—
246C2×C32—23no—
247C2×C32—29no—
248C2×C32—33no—
249C2×C32—31no—
250C2×C32—39no—
251C2×C32—33no—
252C2×C32—61no—
253C2×C32—69no—
254C2×C32—69no—
255C2×C32—63no—
256C2×C32—63no—
257C2×C32—69no—
258C2×C32—61no—
259C2×C32—63no—
260C2×C32—62no—
261C2×C32—66no—
262C2×C32—618no—
263C2×C32—618no—
264C2×C32—636no—
265C2×C32—64no—
266C2×C32—612no—
267C2×C32—66no—
268C2×C32—618no—
269C2×C32—62no—
270C2×C32—66no—
271C2×C32—618no—
272C2×C32—62no—
273C2×C32—66no—
274C2×C32—63no—
275C2×C32—69no—
276C2×C32—61no—
277C2×C32—63no—
278C2×C32—541no—
279C2×C32C2×(C3⋊D3)⋊C313yes—
280C2×C32C2×C32×D313yes—
281C2×C32C2×C32×D311yescharacteristic
282C2×C32C2×C3×C3⋊D311yescharacteristic
283C2×C32C2×C3×C3⋊D313yes—
284C3⋊D3—61no—
285C3⋊D3—63no—
286C3⋊D3—183no—
287C3×D3—189no—
288C3×D3—183no—
289C3×D3—181no—
290C3×D3—183no—
291C3×D3—543no—
292C22×C3—23no—
293C22×C3—21no—
294C22×C3—33no—
295C22×C3—33no—
296C22×C3—63no—
297C22×C3—63no—
298C22×C3—69no—
299C22×C3—273no—
300C22×C3—271no—
301C22×C3C3×(C3⋊D3)⋊C313yescomplemented
302C22×C3C3⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
303C22×C3C32×C3⋊D311yescharacteristic
304C2×D3—93no—
305C2×D3—91no—
306C2×D3—273no—
307C3×C4—273no—
308C3×C4—271no—
309Dic3—93no—
310Dic3—91no—
311Dic3—273no—
312C32—21no—
313C32—23no—
314C32—23no—
315C32—23no—
316C32—29no—
317C32—33no—
318C32—31no—
319C32—39no—
320C32—33no—
321C32—61no—
322C32—63no—
323C32—69no—
324C32—69no—
325C32—618no—
326C32—62no—
327C32—66no—
328C32—63no—
329C32—69no—
330C32—61no—
331C32—63no—
332C32#638⋊C313yescomplemented
333C32C32×#6711yescharacteristic
334C32C32×#6713yescomplemented
335C32C3×#63811yescharacteristic
336C32C3×#63813yescomplemented
337D4—271noSylow
338C2×C3—26no—
339C2×C3—22no—
340C2×C3—21no—
341C2×C3—23no—
342C2×C3—21no—
343C2×C3—23no—
344C2×C3—21no—
345C2×C3—33no—
346C2×C3—33no—
347C2×C3—63no—
348C2×C3—63no—
349C2×C3—66no—
350C2×C3—66no—
351C2×C3—618no—
352C2×C3—63no—
353C2×C3—63no—
354C2×C3—69no—
355C2×C3—543no—
356C2×C3—541no—
357C2×C3C2×C3×(C3⋊D3)⋊C313yes—
358C2×C3C2×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
359C2×C3C2×C32×C3⋊D311yescharacteristic
360D3—181no—
361D3—183no—
362D3—543no—
363C22—271no—
364C22C3×C3⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
365C4—271no—
366C3—23no—
367C3—21no—
368C3—33no—
369C3—33no—
370C3—63no—
371C3—63no—
372C3—69no—
373C3C3×#638⋊C313yescomplemented
374C3#1061997711yescharacteristic, direct product factor
375C3C32×#63811yescharacteristic
376C2—21no—
377C2—541no—
378C2C2×C3×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
subgroupG/H
centerC2×C3C2×C3⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C33C22×C32
FrattiniC2×C3C2×C32×C3⋊D3
FittingC22×C32×H(3,𝔽3)C2
socleC2×C33C2×C3×D3
hypercenterC22×C3C3⋊((C3⋊D3)⋊C3)
hypocenterC33C32×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C32×H(3,𝔽3)C2semidirect
2. C2×C3×C3⋊((C3⋊D3)⋊C3)C2semidirect
3. C3×C3⋊((C3⋊Dic3)⋊C3)C2semidirect
4. C3×#6312C3semidirect
6. #10619977C3direct (N × H)
12. C22×C34D3semidirect
13. C22×C34C2×C3semidirect
14. C2×C3×C3⋊(C3⋊D3)C2×C3semidirect
16. C3×C3⋊(C3⋊Dic3)C2×C3semidirect
21. C22×C3×H(3,𝔽3)D3semidirect
22. C22×C3×H(3,𝔽3)C2×C3semidirect
23. C2×C3⋊((C3⋊D3)⋊C3)C2×C3semidirect
25. C3⋊((C3⋊Dic3)⋊C3)C2×C3semidirect
27. C32×H(3,𝔽3)D4semidirect
28. #6312C32semidirect
70. C22×C33C3×D3semidirect
71. C22×C33C3×D3semidirect
72. C22×C33C3⋊D3semidirect
73. C22×C33C2×C32semidirect
74. C2×C3⋊(C3⋊D3)C2×C32semidirect
81. C3⋊(C3⋊Dic3)C2×C32semidirect
91. C22×H(3,𝔽3)C3×D3semidirect
95. C34#67semidirect
96. C34C3×D4semidirect
100. C3×H(3,𝔽3)#67semidirect
101. C3×H(3,𝔽3)C3×D4semidirect
184. C22×C32(C3⋊D3)⋊C3semidirect
186. C22×C32C32×D3semidirect
188. C22×C32C3×C3⋊D3semidirect
220. C33C3×#67semidirect
221. C33C3×#67semidirect
222. C33#638semidirect
223. C33C32×D4semidirect
227. H(3,𝔽3)C3×#67semidirect
301. C22×C3C3×(C3⋊D3)⋊C3semidirect
302. C22×C3C3⋊((C3⋊D3)⋊C3)semidirect
332. C32#638⋊C3semidirect
334. C32C32×#67semidirect
336. C32C3×#638semidirect
364. C22C3×C3⋊((C3⋊D3)⋊C3)semidirect
373. C3C3×#638⋊C3semidirect
374. C3#10619977direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×H(3,𝔽3)2
C2×C3×C3⋊((C3⋊D3)⋊C3)2
C3×C3⋊((C3⋊Dic3)⋊C3)2
C3×#63123
C32×#6383
#106199773
C3×#638⋊C33