Finite Group Databasebeta

← #10592528⋊C3 #13142050

Subgroups — #10592528⋊C3 #13142050

Order 1728.

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

normal subgroups only

This group has 353 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
1#64079⋊C3C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic
4#63645⋊C3C211yesmaximal, characteristic
5#63644⋊C3C211yesmaximal, characteristic
6#63643⋊C3C211yesmaximal, characteristic
7#63643⋊C3C211yesmaximal, characteristic
8#10592528C311yesmaximal, characteristic, complemented
9C3×#4829—31nomaximal
10C2×#634⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18#3750⋊C3—21no—
19#3750⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3—21no—
21#64079C2×C311yescharacteristic
22C4×#634C2×C311yescharacteristic
23C3×#870—31no—
24C3×C4×Dic6—31no—
25C32×C4⋊C8—21no—
26C32×C4⋊C8D311yescharacteristic
27C32×C4⋊C8C2×C311yescharacteristic
28#63645C2×C311yescharacteristic
29#63644C2×C311yescharacteristic
30#63643C2×C311yescharacteristic
31#63643C2×C311yescharacteristic
32C3×#818—31no—
33C3×#817—31no—
34C3×#816—31no—
35C3×#816—31no—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43#634⋊C3—21no—
44#634⋊C3Q812yes—
45C8×H(3,𝔽3)—21no—
46C8×H(3,𝔽3)—21no—
47C3×#479—91no—
48#4829—31no—
49#4829—91no—
50C2×#634C22×C311yescharacteristic
51C2×C3×Dic6—31no—
52C2×C32×C8—21no—
53C2×C32×C8—21no—
54C2×C32×C8C2×D311yescharacteristic
55C2×C32×C8C2×D311yescharacteristic
56C2×C32×C8C22×C311yescharacteristic
57C2×C32×C8C22×C311yescharacteristic
58C32×C42—21no—
59C32×C42C2×D311yescharacteristic
60C32×C42C22×C311yescharacteristic
61C4⋊(C3⋊Dic3)C22×C311yescharacteristic
62C4⋊(C3⋊Dic3)C22×C311yescharacteristic
63C4⋊(C3⋊Dic3)C22×C311yescharacteristic
64#3750—21no—
65#3750—21no—
66#3750—21no—
67C4×C3⋊Dic3—21no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×Dic3⋊C4—61no—
72C3×Dic3⋊C4—61no—
73C3×Dic3⋊C4—61no—
74C3×C4×Dic3—61no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)—21no—
77C4×H(3,𝔽3)—21no—
78C4×H(3,𝔽3)C2×Q811yescharacteristic
79C4×H(3,𝔽3)#4111yescharacteristic
80(C3⋊Dic3)⋊C3—22no—
81(C3⋊Dic3)⋊C3—41no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84C3×#125—91no—
85C3×C4×Q8—91no—
86#870—31no—
87#870—91no—
88C4×Dic6—31no—
89C4×Dic6—91no—
90C3×#107—91no—
91C3×#106—91no—
92C3×C4⋊C8—31no—
93C3×C4⋊C8—31no—
94C3×C4⋊C8—61no—
95C3×C4⋊C8C3×D311yescharacteristic
96C3×#103—91no—
97C3×#103—91no—
98#818—31no—
99#818—91no—
100#817—31no—
101#817—91no—
102#816—31no—
103#816—31no—
104#816—91no—
105#816—91no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4D1211yescharacteristic
110C2×C32×C4D1211yescharacteristic
111C2×C32×C4C3×D411yescharacteristic
112C2×C32×C4C3×D411yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C23×C311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—21no—
119#634—21no—
120#634C3×Q812yes—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C3×Dic6—32no—
126C3×Dic6—61no—
127C32×C8—21no—
128C32×C8—21no—
129C32×C8—21no—
130C32×C8—21no—
131C32×C8—41no—
132C32×C8—41no—
133#479—91noSylow
134C2×H(3,𝔽3)C22⋊Q811yescharacteristic
135C2×H(3,𝔽3)#13511yescharacteristic
136C2×H(3,𝔽3)#13711yescharacteristic
137C2×C3×Q8—91no—
138C2×Dic6—31no—
139C2×Dic6—91no—
140C2×C3×C8—31no—
141C2×C3×C8—31no—
142C2×C3×C8—31no—
143C2×C3×C8—31no—
144C2×C3×C8—61no—
145C2×C3×C8—61no—
146C2×C3×C8C2×C3×D311yescharacteristic
147C2×C3×C8C2×C3×D311yescharacteristic
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—181no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4—181no—
154C3×C42—31no—
155C3×C42—31no—
156C3×C42—61no—
157C3×C42—181no—
158C3×C42C2×C3×D311yescharacteristic
159C4⋊Dic3—31no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4⋊Dic3—91no—
163C4⋊Dic3—91no—
164C4⋊Dic3—91no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167Dic3⋊C4—61no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171C4×Dic3—61no—
172C4×Dic3—181no—
173C22×C32—21no—
174C22×C32C2×D1211yescharacteristic
175C22×C32C2×C3×D411yescharacteristic
176C32×C4—22no—
177C32×C4—21no—
178C32×C4—21no—
179C32×C4—21no—
180C32×C4—21no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—41no—
184C32×C4#23711yescharacteristic
185C32×C4D3×Q811yescharacteristic
186C32×C4C2×C3×Q811yescharacteristic
187C32×C4C3×#4111yescharacteristic
188C3⋊Dic3—22no—
189C3⋊Dic3—41no—
190C3⋊Dic3—41no—
191C3⋊Dic3—41no—
192C3×Dic3—62no—
193C3×Dic3—121no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196#125—91no—
197C4×Q8—91no—
198#107—91no—
199#106—91no—
200C4⋊C8(C3⋊D3)⋊C311yescharacteristic
201#103—91no—
202#103—91no—
203H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
204C3×Q8—92no—
205C3×Q8—181no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4C3×D1211yescharacteristic
220C2×C3×C4C3×D1211yescharacteristic
221C2×C3×C4C22×C3×D311yescharacteristic
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229C2×Dic3—181no—
230Dic6—32no—
231Dic6—61no—
232Dic6—92no—
233Dic6—181no—
234C3×C8—21no—
235C3×C8—21no—
236C3×C8—61no—
237C3×C8—61no—
238C3×C8—61no—
239C3×C8—61no—
240C3×C8—121no—
241C3×C8—121no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32#89711yescharacteristic
246C2×C32#90411yescharacteristic
247C2×C32#90911yescharacteristic
248C2×C32C3×C22⋊Q811yescharacteristic
249C2×C32C3×#13511yescharacteristic
250C2×C32C3×#13711yescharacteristic
251C2×Q8—91no—
252C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
253C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
254C4⋊C4—91no—
255C4⋊C4—91no—
256C4⋊C4—91no—
257C4⋊C4—181no—
258C4⋊C4—181no—
259C4⋊C4—181no—
260C42—181no—
261C42C2×(C3⋊D3)⋊C311yescharacteristic
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—61no—
265C22×C3C2×C3×D1211yescharacteristic
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4—62no—
277C3×C4—61no—
278C3×C4—61no—
279C3×C4—121no—
280C3×C4—182no—
281C3×C4—361no—
282C3×C4—361no—
283C3×C4—361no—
284C3×C4C3×#23711yescharacteristic
285C3×C4C3×D3×Q811yescharacteristic
286Dic3—62no—
287Dic3—121no—
288Dic3—121no—
289Dic3—121no—
290Dic3—182no—
291Dic3—361no—
292Dic3—361no—
293Dic3—361no—
294C32—21no—
295C32#482911yescharacteristic, complemented
296C32C3×#47911yescharacteristic, complemented
297Q8—92no—
298Q8—181no—
299C2×C4—181no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4—181no—
303C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
304C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
305C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
306C8—21no—
307C8—21no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—61no—
315C2×C3—61no—
316C2×C3—61no—
317C2×C3C3×#89711yescharacteristic
318C2×C3C3×#90411yescharacteristic
319C2×C3C3×#90911yescharacteristic
320C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
321C4—21no—
322C4—21no—
323C4—182no—
324C4—361no—
325C4—361no—
326C4—361no—
327C4#3830⋊C311yescharacteristic
328C4Q8×(C3⋊D3)⋊C311yescharacteristic
329C3—31no—
330C3—31no—
331C3—61no—
332C3C3×#482911yescharacteristic
333C2#64106⋊C311yescharacteristic
334C2#64113⋊C311yescharacteristic
335C2#64118⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#479
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
8. #10592528C3semidirect
203. H(3,𝔽3)#479semidirect
295. C32#4829semidirect
296. C32C3×#479semidirect

Maximal subgroups

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

subgroupindex
#64079⋊C32
C4×#634⋊C32
H(3,𝔽3)×C4⋊C82
#63645⋊C32
#63644⋊C32
#63643⋊C32
#105925283
C3×#48293