Finite Group Databasebeta

← #10592576⋊C3 #13142217

Subgroups — #10592576⋊C3 #13142217

Order 1728.

351 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 90 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#64099⋊C3C211yesmaximal, characteristic
2#64096⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#107C211yesmaximal, characteristic
4#63643⋊C3C211yesmaximal, characteristic
5#63641⋊C3C211yesmaximal, characteristic
6#63637⋊C3C211yesmaximal, characteristic
7#63635⋊C3C211yesmaximal, characteristic
8#10592576C311yesmaximal, characteristic, complemented
9C3×#4877—31nomaximal
10C2×#634⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64099C2×C311yescharacteristic
22#64096C2×C311yescharacteristic
23C3×#890—31no—
24C3×#887—31no—
25C32×#107—21no—
26C32×#107D311yescharacteristic
27C32×#107C2×C311yescharacteristic
28#63643C2×C311yescharacteristic
29#63641C2×C311yescharacteristic
30#63637C2×C311yescharacteristic
31#63635C2×C311yescharacteristic
32C3×#816—31no—
33C3×Dic3⋊C8—31no—
34C3×#810—31no—
35C3×#808—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43#634⋊C3—21no—
44#634⋊C3Q812yes—
45C8×H(3,𝔽3)—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47C3×#479—91no—
48#4877—31no—
49#4877—91no—
50C2×#634C22×C311yescharacteristic
51C2×C3×Dic6—31no—
52C2×C32×C8—21no—
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C22×C311yescharacteristic
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C4⋊C4C22×C311yescharacteristic
60C32×C4⋊C4C22×C311yescharacteristic
61C4⋊(C3⋊Dic3)—21no—
62C4⋊(C3⋊Dic3)C22×C311yescharacteristic
63#3750—21no—
64#3750—21no—
65C4×C3⋊Dic3—21no—
66C4×C3⋊Dic3C22×C311yescharacteristic
67C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×Dic3⋊C4—61no—
71C3×Dic3⋊C4—61no—
72C3×C4×Dic3—31no—
73C3×C4×Dic3—61no—
74C2×C3×C3⋊C8—31no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)—41no—
77C4×H(3,𝔽3)—41no—
78C4×H(3,𝔽3)C2×Q811yescharacteristic
79C4×H(3,𝔽3)#4111yescharacteristic
80(C3⋊Dic3)⋊C3—22no—
81(C3⋊Dic3)⋊C3—21no—
82(C3⋊Dic3)⋊C3—21no—
83(C3⋊Dic3)⋊C3—41no—
84C3×#125—91no—
85C3×C4×Q8—91no—
86#890—31no—
87#890—91no—
88#887—31no—
89#887—91no—
90C3×#107—31no—
91C3×#107—31no—
92C3×#107—61no—
93C3×#107C3×D311yescharacteristic
94C3×#106—91no—
95C3×C4⋊C8—91no—
96C3×#103—91no—
97C3×#103—91no—
98#816—31no—
99#816—91no—
100Dic3⋊C8—31no—
101Dic3⋊C8—91no—
102#810—31no—
103#810—91no—
104#808—31no—
105#808—91no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—41no—
112C2×C32×C4—41no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C23×C311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3C3×D411yescharacteristic
118C2×C3⋊Dic3C3×D411yescharacteristic
119#634—21no—
120#634C3×Q812yes—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C3×Dic6—32no—
126C3×Dic6—61no—
127C32×C8—22no—
128C32×C8—21no—
129C32×C8—21no—
130C3⋊(C3⋊C8)—21no—
131C3×C3⋊C8—61no—
132#479—91noSylow
133C2×H(3,𝔽3)C22⋊Q811yescharacteristic
134C2×H(3,𝔽3)#13511yescharacteristic
135C2×H(3,𝔽3)#13711yescharacteristic
136C2×C3×Q8—91no—
137C2×Dic6—31no—
138C2×Dic6—91no—
139C2×C3×C8—31no—
140C2×C3×C8—31no—
141C2×C3×C8—61no—
142C2×C3×C8—91no—
143C2×C3×C8C2×C3×D311yescharacteristic
144C3×C4⋊C4—31no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—61no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—181no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4—181no—
154C3×C4⋊C4C2×C3×D311yescharacteristic
155C3×C4⋊C4C2×C3×D311yescharacteristic
156C3×C42—91no—
157C3×C42—181no—
158C4⋊Dic3—31no—
159C4⋊Dic3—61no—
160C4⋊Dic3—91no—
161C4⋊Dic3—181no—
162Dic3⋊C4—61no—
163Dic3⋊C4—61no—
164Dic3⋊C4—181no—
165Dic3⋊C4—181no—
166C4×Dic3—31no—
167C4×Dic3—61no—
168C4×Dic3—91no—
169C4×Dic3—181no—
170C2×C3⋊C8—31no—
171C2×C3⋊C8—91no—
172C22×C32—21no—
173C22×C32D3×D411yescharacteristic
174C22×C32C2×C3×D411yescharacteristic
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—41no—
181C32×C4—81no—
182C32×C4—81no—
183C32×C4#23511yescharacteristic
184C32×C4D3×Q811yescharacteristic
185C32×C4C2×C3×Q811yescharacteristic
186C32×C4C3×#4111yescharacteristic
187C3⋊Dic3—22no—
188C3⋊Dic3—21no—
189C3⋊Dic3—21no—
190C3⋊Dic3—41no—
191C3×Dic3—62no—
192C3×Dic3—61no—
193C3×Dic3—61no—
194C3×Dic3—121no—
195#125—91no—
196C4×Q8—91no—
197#107(C3⋊D3)⋊C311yescharacteristic
198#106—91no—
199C4⋊C8—91no—
200#103—91no—
201#103—91no—
202H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
203C3×Q8—92no—
204C3×Q8—181no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4—121no—
217C2×C3×C4—121no—
218C2×C3×C4—181no—
219C2×C3×C4—181no—
220C2×C3×C4C22×C3×D311yescharacteristic
221C2×Dic3—31no—
222C2×Dic3—31no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—91no—
226C2×Dic3—91no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229Dic6—32no—
230Dic6—61no—
231Dic6—92no—
232Dic6—181no—
233C3×C8—21no—
234C3×C8—61no—
235C3×C8—61no—
236C3×C8—62no—
237C3×C8—181no—
238C3⋊C8—61no—
239C3⋊C8—181no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32#89611yescharacteristic
244C2×C32#91211yescharacteristic
245C2×C32#91811yescharacteristic
246C2×C32C3×C22⋊Q811yescharacteristic
247C2×C32C3×#13511yescharacteristic
248C2×C32C3×#13711yescharacteristic
249C2×Q8—91no—
250C2×C8—91no—
251C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
252C4⋊C4—91no—
253C4⋊C4—181no—
254C4⋊C4—181no—
255C4⋊C4—181no—
256C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
257C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
258C42—91no—
259C42—181no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—61no—
263C22×C3C3×D3×D411yescharacteristic
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4—121no—
273C3×C4—121no—
274C3×C4—121no—
275C3×C4—121no—
276C3×C4—181no—
277C3×C4—181no—
278C3×C4—182no—
279C3×C4—241no—
280C3×C4—241no—
281C3×C4—361no—
282C3×C4C3×#23511yescharacteristic
283C3×C4C3×D3×Q811yescharacteristic
284Dic3—62no—
285Dic3—61no—
286Dic3—61no—
287Dic3—121no—
288Dic3—182no—
289Dic3—181no—
290Dic3—181no—
291Dic3—361no—
292C32—21no—
293C32#487711yescharacteristic, complemented
294C32C3×#47911yescharacteristic, complemented
295Q8—92no—
296Q8—181no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—181no—
302C2×C4—181no—
303C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
304C8—21no—
305C8—181no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3C3×#89611yescharacteristic
316C2×C3C3×#91211yescharacteristic
317C2×C3C3×#91811yescharacteristic
318C22D4×(C3⋊D3)⋊C311yescharacteristic
319C4—41no—
320C4—41no—
321C4—181no—
322C4—181no—
323C4—182no—
324C4—361no—
325C4#3828⋊C311yescharacteristic
326C4Q8×(C3⋊D3)⋊C311yescharacteristic
327C3—31no—
328C3—31no—
329C3—61no—
330C3C3×#487711yescharacteristic
331C2#64105⋊C311yescharacteristic
332C2#64121⋊C311yescharacteristic
333C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#479
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
8. #10592576C3semidirect
202. H(3,𝔽3)#479semidirect
293. C32#4877semidirect
294. C32C3×#479semidirect

Maximal subgroups

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

subgroupindex
#64099⋊C32
#64096⋊C32
H(3,𝔽3)×#1072
#63643⋊C32
#63641⋊C32
#63637⋊C32
#63635⋊C32
#105925763
C3×#48773