Finite Group Databasebeta

← #10592762⋊C3 #13142951

Subgroups — #10592762⋊C3 #13142951

Order 1728.

371 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 96 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic
3#63637⋊C3C211yesmaximal, characteristic
4#63637⋊C3C211yesmaximal, characteristic
5#63635⋊C3C211yesmaximal, characteristic
6#63634⋊C3C211yesmaximal, characteristic
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
8#10592762C311yesmaximal, characteristic, complemented
9C3×#5063—31nomaximal
10C2×#634⋊C3C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C32×#125—21no—
22C32×#125D311yescharacteristic
23C32×#125C2×C311yescharacteristic
24C4×#634C2×C311yescharacteristic
25C3×C4×Dic6—31no—
26#63637C2×C311yescharacteristic
27#63637C2×C311yescharacteristic
28#63635C2×C311yescharacteristic
29#63634C2×C311yescharacteristic
30C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
31C3×#810—31no—
32C3×#810—31no—
33C3×#808—31no—
34C3×#807—31no—
35C3×C4⋊(C3⋊C8)—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43#634⋊C3—21no—
44#634⋊C3Q812yes—
45(C3⋊(C3⋊C8))⋊C3—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47C3×#479—91no—
48#5063—31no—
49#5063—91no—
50C2×#634C22×C311yescharacteristic
51C2×C3×Dic6—31no—
52C32×C4⋊C4—22no—
53C32×C4⋊C4—22no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C22×C311yescharacteristic
63C32×C4⋊C4C22×C311yescharacteristic
64C32×C42—21no—
65C32×C42C2×D311yescharacteristic
66C32×C42C22×C311yescharacteristic
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68#3750—21no—
69C4×C3⋊Dic3—21no—
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×C4⋊Dic3—31no—
73C3×Dic3⋊C4—61no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)—21no—
80C4×H(3,𝔽3)—41no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)C2×Q811yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84(C3⋊Dic3)⋊C3—22no—
85(C3⋊Dic3)⋊C3—41no—
86C3×#125—31no—
87C3×#125—31no—
88C3×#125—61no—
89C3×#125C3×D311yescharacteristic
90C3×C4×Q8—91no—
91C4×Dic6—31no—
92C4×Dic6—91no—
93C3×#107—91no—
94C3×#106—91no—
95C3×C4⋊C8—91no—
96C3×#103—91no—
97C3×#103—91no—
98#810—31no—
99#810—31no—
100#810—91no—
101#810—91no—
102#808—31no—
103#808—91no—
104#807—31no—
105#807—91no—
106C4⋊(C3⋊C8)—31no—
107C4⋊(C3⋊C8)—91no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4C3×D411yescharacteristic
120C2×C32×C4C3×D411yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C23×C311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125#634—21no—
126#634C3×Q812yes—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×Dic6—32no—
130C3×Dic6—61no—
131C3⋊(C3⋊C8)—21no—
132C3⋊(C3⋊C8)—21no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8—61no—
135#479—91noSylow
136C2×H(3,𝔽3)C22⋊Q811yescharacteristic
137C2×H(3,𝔽3)#13511yescharacteristic
138C2×H(3,𝔽3)#13711yescharacteristic
139C2×C3×Q8—91no—
140C2×Dic6—31no—
141C2×Dic6—91no—
142C2×C3×C8—91no—
143C2×C3×C8—91no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—31no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—61no—
151C3×C4⋊C4—61no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4—61no—
154C3×C4⋊C4—62no—
155C3×C4⋊C4—62no—
156C3×C4⋊C4—61no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4C2×C3×D311yescharacteristic
161C3×C4⋊C4C2×C3×D311yescharacteristic
162C3×C42—31no—
163C3×C42—31no—
164C3×C42—61no—
165C3×C42—181no—
166C3×C42C2×C3×D311yescharacteristic
167C4⋊Dic3—31no—
168C4⋊Dic3—91no—
169Dic3⋊C4—61no—
170Dic3⋊C4—181no—
171C4×Dic3—61no—
172C4×Dic3—181no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—91no—
177C22×C32—21no—
178C22×C32C2×#6711yescharacteristic
179C22×C32C2×C3×D411yescharacteristic
180C32×C4—21no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—41no—
187C32×C4—41no—
188C32×C4—42no—
189C32×C4—42no—
190C32×C4—41no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4D3×Q811yescharacteristic
195C32×C4#23911yescharacteristic
196C32×C4C2×C3×Q811yescharacteristic
197C32×C4C3×#4111yescharacteristic
198C3⋊Dic3—22no—
199C3⋊Dic3—41no—
200C3×Dic3—62no—
201C3×Dic3—121no—
202#125(C3⋊D3)⋊C311yescharacteristic
203C4×Q8—91no—
204#107—91no—
205#106—91no—
206C4⋊C8—91no—
207#103—91no—
208#103—91no—
209H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
210C3×Q8—92no—
211C3×Q8—181no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4C3×#6711yescharacteristic
232C2×C3×C4C3×#6711yescharacteristic
233C2×C3×C4C22×C3×D311yescharacteristic
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—181no—
237C2×Dic3—181no—
238Dic6—32no—
239Dic6—61no—
240Dic6—92no—
241Dic6—181no—
242C3×C8—181no—
243C3×C8—181no—
244C3⋊C8—61no—
245C3⋊C8—61no—
246C3⋊C8—181no—
247C3⋊C8—181no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32#94611yescharacteristic
252C2×C32#95011yescharacteristic
253C2×C32#95111yescharacteristic
254C2×C32C3×C22⋊Q811yescharacteristic
255C2×C32C3×#13511yescharacteristic
256C2×C32C3×#13711yescharacteristic
257C2×Q8—91no—
258C2×C8—91no—
259C2×C8—91no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—91no—
263C4⋊C4—181no—
264C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
265C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
266C42—181no—
267C42C2×(C3⋊D3)⋊C311yescharacteristic
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—61no—
271C22×C3C2×C3×#6711yescharacteristic
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—61no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—121no—
287C3×C4—121no—
288C3×C4—121no—
289C3×C4—121no—
290C3×C4—121no—
291C3×C4—121no—
292C3×C4—122no—
293C3×C4—122no—
294C3×C4—182no—
295C3×C4—361no—
296C3×C4C3×D3×Q811yescharacteristic
297C3×C4C3×#23911yescharacteristic
298Dic3—62no—
299Dic3—121no—
300Dic3—182no—
301Dic3—361no—
302C32—21no—
303C32#506311yescharacteristic, complemented
304C32C3×#47911yescharacteristic, complemented
305Q8—92no—
306Q8—181no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4#638⋊C311yescharacteristic
312C2×C4#638⋊C311yescharacteristic
313C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
314C8—181no—
315C8—181no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—61no—
325C2×C3C3×#94611yescharacteristic
326C2×C3C3×#95011yescharacteristic
327C2×C3C3×#95111yescharacteristic
328C22C2×#638⋊C311yescharacteristic
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—182no—
334C4—361no—
335C4Q8×(C3⋊D3)⋊C311yescharacteristic
336C4#8872211yescharacteristic
337C3—31no—
338C3—31no—
339C3—61no—
340C3C3×#506311yescharacteristic
341C2#64155⋊C311yescharacteristic
342C2#64160⋊C311yescharacteristic
343C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#125C2
socleC22×C3C2×C3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#479
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
8. #10592762C3semidirect
209. H(3,𝔽3)#479semidirect
303. C32#5063semidirect
304. C32C3×#479semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
C4×#634⋊C32
#63637⋊C32
#63635⋊C32
#63634⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927623
C3×#50633