Finite Group Databasebeta

← #10592514⋊C3 #13141926

Subgroups — #10592514⋊C3 #13141926

Order 1728.

378 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 86 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#64155⋊C3C211yesmaximal, characteristic
2#64101⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic
6#63644⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592514C311yesmaximal, characteristic, complemented
9C3×#4815—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#3752⋊C3—21no—
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64155C2×C311yescharacteristic
22#64101C2×C311yescharacteristic
23C3×#946—31no—
24C3×#892—31no—
25C32×#103—21no—
26C32×#103D311yescharacteristic, complemented
27C32×#103C2×C311yescharacteristic, complemented
28#63662C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic
30#63644C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C3×#835—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#817—31no—
35C3×#807—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37Q8×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47C8×H(3,𝔽3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49C3×#485—91no—
50#4815—31no—
51#4815—91no—
52C2×C32×Q8—21no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C32×Q8C22×C311yescharacteristic
55C2×C4×C3⋊D3C22×C311yescharacteristic
56C2×C3×C4×D3—31no—
57C2×C32×C8—21no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C22×C311yescharacteristic
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C22×C311yescharacteristic
63#3752—21no—
64#3752—21no—
65C4⋊(C3⋊Dic3)C22×C311yescharacteristic
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67#3750—21no—
68C4×C3⋊Dic3—21no—
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C3×(C2×D3)⋊C4—61no—
71C3×(C2×D3)⋊C4—61no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×Dic3⋊C4—61no—
75C3×C4×Dic3—61no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—21no—
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88C3×C22⋊Q8—91no—
89C3×#117—91no—
90#946—31no—
91#946—91no—
92#892—31no—
93#892—91no—
94C3×#107—91no—
95C3×#106—91no—
96C3×#103—31no—
97C3×#103—31no—
98C3×#103—61no—
99C3×#103—91no—
100C3×#103C3×D311yescharacteristic
101C3×C22⋊C8—91no—
102#835—31no—
103#835—91no—
104(C2×D3)⋊C8—31no—
105(C2×D3)⋊C8—91no—
106#817—31no—
107#817—91no—
108#807—31no—
109#807—91no—
110C22×C3⋊D3C3×D411yescharacteristic
111C22×C3×D3—31no—
112C32×Q8—22no—
113C32×Q8—21no—
114C32×Q8—21no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—41no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C23×C311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3C3×D411yescharacteristic
130C4×C3⋊D3—21no—
131C4×C3⋊D3—21no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C32×C8—21no—
138C32×C8—21no—
139C32×C8—41no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142#485—91noSylow
143C2×H(3,𝔽3)#12311yescharacteristic
144C2×H(3,𝔽3)#13511yescharacteristic
145C2×H(3,𝔽3)#13711yescharacteristic
146(C3⋊D3)⋊C3—41no—
147C2×C3×Q8—31no—
148C2×C3×Q8—31no—
149C2×C3×Q8—61no—
150C2×C3×Q8C2×C3×D311yescharacteristic
151C22×C3×C4—91no—
152C2×C4×D3—31no—
153C2×C4×D3—91no—
154C2×C3×C8—31no—
155C2×C3×C8—31no—
156C2×C3×C8—61no—
157C2×C3×C8—91no—
158C2×C3×C8C2×C3×D311yescharacteristic
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—181no—
165C3×C4⋊C4C2×C3×D311yescharacteristic
166C3×C22⋊C4—181no—
167C3×C22⋊C4—181no—
168C3×C42—181no—
169(C2×D3)⋊C4—61no—
170(C2×D3)⋊C4—61no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—181no—
173C4⋊Dic3—31no—
174C4⋊Dic3—31no—
175C4⋊Dic3—91no—
176C4⋊Dic3—91no—
177Dic3⋊C4—61no—
178Dic3⋊C4—181no—
179C4×Dic3—61no—
180C4×Dic3—181no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32D3×D411yescharacteristic
185C22×C32C2×C3×D411yescharacteristic
186C2×C3⋊D3—21no—
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—42no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—81no—
200C32×C4#23511yescharacteristic
201C32×C4#23711yescharacteristic
202C32×C4C3×#4111yescharacteristic
203C32×C4C3×#4111yescharacteristic
204C3⋊Dic3—21no—
205C3⋊Dic3—41no—
206C3⋊Dic3—41no—
207C3×Dic3—61no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C22⋊Q8—91no—
211#117—91no—
212#107—91no—
213#106—91no—
214#103—91no—
215#103(C3⋊D3)⋊C311yescharacteristic, complemented
216C22⋊C8—91no—
217H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
218C23×C3—91no—
219C22×D3—31no—
220C22×D3—91no—
221C3×Q8—21no—
222C3×Q8—21no—
223C3×Q8—61no—
224C3×Q8—61no—
225C3×Q8—61no—
226C3×Q8—61no—
227C3×Q8—62no—
228C3×Q8—121no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—91no—
239C2×C3×C4—121no—
240C2×C3×C4—121no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4C22×C3×D311yescharacteristic
246C2×Dic3—31no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—91no—
250C2×Dic3—181no—
251C2×Dic3—181no—
252C4×D3—61no—
253C4×D3—61no—
254C4×D3—181no—
255C4×D3—181no—
256C3×C8—21no—
257C3×C8—61no—
258C3×C8—61no—
259C3×C8—121no—
260C3×C8—181no—
261C3⋊C8—61no—
262C3⋊C8—181no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32#88311yescharacteristic
267C2×C32#91611yescharacteristic
268C2×C32#91811yescharacteristic
269C2×C32C3×#12311yescharacteristic
270C2×C32C3×#13511yescharacteristic
271C2×C32C3×#13711yescharacteristic
272C3⋊D3—41no—
273C3×D3—121no—
274C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
275C22×C4—91no—
276C2×C8—91no—
277C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4—181no—
281C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
282C22⋊C4—181no—
283C22⋊C4—181no—
284C42—181no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—61no—
288C22×C3—181no—
289C22×C3—181no—
290C22×C3—181no—
291C22×C3C3×D3×D411yescharacteristic
292C2×D3—61no—
293C2×D3—61no—
294C2×D3—61no—
295C2×D3—181no—
296C2×D3—181no—
297C2×D3—181no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—61no—
305C3×C4—61no—
306C3×C4—121no—
307C3×C4—121no—
308C3×C4—121no—
309C3×C4—121no—
310C3×C4—122no—
311C3×C4—181no—
312C3×C4—241no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4C3×#23511yescharacteristic
316C3×C4C3×#23711yescharacteristic
317Dic3—61no—
318Dic3—121no—
319Dic3—121no—
320Dic3—181no—
321Dic3—361no—
322Dic3—361no—
323C32—21no—
324C32#481511yescharacteristic, complemented
325C32C3×#48511yescharacteristic, complemented
326C23—91no—
327Q8—21no—
328Q8—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—91no—
332C2×C4—181no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
337C8—21no—
338C8—181no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—31no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—61no—
348C2×C3—361no—
349C2×C3C3×#88311yescharacteristic
350C2×C3C3×#91611yescharacteristic
351C2×C3C3×#91811yescharacteristic
352D3—121no—
353D3—361no—
354C22—181no—
355C22—181no—
356C22—181no—
357C22D4×(C3⋊D3)⋊C311yescharacteristic
358C4—41no—
359C4—41no—
360C4—181no—
361C4—361no—
362C4—361no—
363C4#3828⋊C311yescharacteristic
364C4#3830⋊C311yescharacteristic
365C3—31no—
366C3—31no—
367C3—61no—
368C3C3×#481511yescharacteristic
369C2—361no—
370C2#64092⋊C311yescharacteristic
371C2#64125⋊C311yescharacteristic
372C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#485
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#103C2semidirect
4. #63662⋊C3C2semidirect
6. #63644⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592514C3semidirect
26. C32×#103D3semidirect
27. C32×#103C2×C3semidirect
28. #63662C2×C3semidirect
30. #63644C2×C3semidirect
31. #63634C2×C3semidirect
215. #103(C3⋊D3)⋊C3semidirect
217. H(3,𝔽3)#485semidirect
324. C32#4815semidirect
325. C32C3×#485semidirect

Maximal subgroups

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

subgroupindex
#64155⋊C32
#64101⋊C32
H(3,𝔽3)×#1032
#63662⋊C32
#63647⋊C32
#63644⋊C32
#63634⋊C32
#105925143
C3×#48153