Finite Group Databasebeta

← #10592498⋊C3 #13141909

Subgroups — #10592498⋊C3 #13141909

Order 1728.

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

normal subgroups only

This group has 357 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
1Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#63644⋊C3C211yesmaximal, characteristic
6#63641⋊C3C211yesmaximal, characteristic
7#63634⋊C3C211yesmaximal, characteristic
8#10592498C311yesmaximal, characteristic, complemented
9C3×#4799—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21Q8×C3⋊Dic3C2×C311yescharacteristic
22#64099C2×C311yescharacteristic
23C3×Q8×Dic3—31no—
24C3×#890—31no—
25C32×#103—21no—
26C32×#103D311yescharacteristic
27C32×#103C2×C311yescharacteristic
28#63662C2×C311yescharacteristic
29#63644C2×C311yescharacteristic
30#63641C2×C311yescharacteristic
31#63634C2×C311yescharacteristic
32C3×#835—31no—
33C3×#817—31no—
34C3×Dic3⋊C8—31no—
35C3×#807—31no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)Q812yes—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C8×H(3,𝔽3)—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47C3×#479—91no—
48#4799—31no—
49#4799—91no—
50C2×C32×Q8—21no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×Q8C22×C311yescharacteristic
53C2×C32×C8—21no—
54C2×C32×C8C2×D311yescharacteristic
55C2×C32×C8C22×C311yescharacteristic
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C22×C311yescharacteristic
59C4⋊(C3⋊Dic3)—21no—
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)C22×C311yescharacteristic
62C4⋊(C3⋊Dic3)C22×C311yescharacteristic
63#3750—21no—
64C4×C3⋊Dic3—21no—
65C4×C3⋊Dic3C22×C311yescharacteristic
66C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×C4⋊Dic3—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)—22no—
77C4×H(3,𝔽3)—41no—
78C4×H(3,𝔽3)C2×Q811yescharacteristic
79C4×H(3,𝔽3)#4111yescharacteristic
80(C3⋊Dic3)⋊C3—21no—
81(C3⋊Dic3)⋊C3—21no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84C3×#125—91no—
85C3×C4×Q8—91no—
86Q8×Dic3—31no—
87Q8×Dic3—91no—
88#890—31no—
89#890—91no—
90C3×#107—91no—
91C3×#106—91no—
92C3×C4⋊C8—91no—
93C3×#103—31no—
94C3×#103—31no—
95C3×#103—61no—
96C3×#103—91no—
97C3×#103C3×D311yescharacteristic
98#835—31no—
99#835—91no—
100#817—31no—
101#817—91no—
102Dic3⋊C8—31no—
103Dic3⋊C8—91no—
104#807—31no—
105#807—91no—
106C32×Q8—21no—
107C32×Q8—21no—
108C32×Q8—22no—
109C32×Q8—41no—
110C32×Q8Dic612yes—
111C32×Q8C3×Q812yes—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3C3×D411yescharacteristic
124C2×C3⋊Dic3C3×D411yescharacteristic
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C32×C8—21no—
130C32×C8—21no—
131C32×C8—41no—
132C3⋊(C3⋊C8)—21no—
133C3×C3⋊C8—61no—
134#479—91noSylow
135C2×H(3,𝔽3)C22⋊Q811yescharacteristic
136C2×H(3,𝔽3)#13511yescharacteristic
137C2×H(3,𝔽3)#13711yescharacteristic
138C2×C3×Q8—31no—
139C2×C3×Q8—31no—
140C2×C3×Q8—61no—
141C2×C3×Q8C2×C3×D311yescharacteristic
142C2×C3×C8—31no—
143C2×C3×C8—31no—
144C2×C3×C8—61no—
145C2×C3×C8—91no—
146C2×C3×C8C2×C3×D311yescharacteristic
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4—181no—
154C3×C4⋊C4—181no—
155C3×C4⋊C4C2×C3×D311yescharacteristic
156C3×C42—91no—
157C3×C42—181no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—61no—
161C4⋊Dic3—61no—
162C4⋊Dic3—91no—
163C4⋊Dic3—91no—
164C4⋊Dic3—181no—
165C4⋊Dic3—181no—
166Dic3⋊C4—61no—
167Dic3⋊C4—181no—
168C4×Dic3—31no—
169C4×Dic3—61no—
170C4×Dic3—91no—
171C4×Dic3—181no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—91no—
174C22×C32—21no—
175C22×C32D3×D411yescharacteristic
176C22×C32C2×C3×D411yescharacteristic
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—21no—
180C32×C4—21no—
181C32×C4—42no—
182C32×C4—41no—
183C32×C4—41no—
184C32×C4—81no—
185C32×C4C2×Dic611yescharacteristic
186C32×C4#23711yescharacteristic
187C32×C4C2×C3×Q811yescharacteristic
188C32×C4C3×#4111yescharacteristic
189C3⋊Dic3—21no—
190C3⋊Dic3—21no—
191C3⋊Dic3—41no—
192C3⋊Dic3—41no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197#125—91no—
198C4×Q8—91no—
199#107—91no—
200#106—91no—
201C4⋊C8—91no—
202#103—91no—
203#103(C3⋊D3)⋊C311yescharacteristic
204H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
205C3×Q8—21no—
206C3×Q8—32no—
207C3×Q8—32no—
208C3×Q8—61no—
209C3×Q8—61no—
210C3×Q8—62no—
211C3×Q8—121no—
212C3×Q8C3×Dic612yes—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—121no—
225C2×C3×C4—121no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4C22×C3×D311yescharacteristic
229C2×Dic3—31no—
230C2×Dic3—31no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—181no—
236C2×Dic3—181no—
237C3×C8—21no—
238C3×C8—61no—
239C3×C8—61no—
240C3×C8—121no—
241C3×C8—181no—
242C3⋊C8—61no—
243C3⋊C8—181no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32#87811yescharacteristic
248C2×C32#91611yescharacteristic
249C2×C32#91811yescharacteristic
250C2×C32C3×C22⋊Q811yescharacteristic
251C2×C32C3×#13511yescharacteristic
252C2×C32C3×#13711yescharacteristic
253C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
254C2×C8—91no—
255C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
256C4⋊C4—91no—
257C4⋊C4—91no—
258C4⋊C4—181no—
259C4⋊C4—181no—
260C4⋊C4—181no—
261C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
262C42—91no—
263C42—181no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—61no—
267C22×C3C3×D3×D411yescharacteristic
268C3×C4—22no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—41no—
274C3×C4—62no—
275C3×C4—62no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—122no—
279C3×C4—121no—
280C3×C4—121no—
281C3×C4—181no—
282C3×C4—181no—
283C3×C4—241no—
284C3×C4—361no—
285C3×C4—361no—
286C3×C4C2×C3×Dic611yescharacteristic
287C3×C4C3×#23711yescharacteristic
288Dic3—61no—
289Dic3—61no—
290Dic3—121no—
291Dic3—121no—
292Dic3—181no—
293Dic3—181no—
294Dic3—361no—
295Dic3—361no—
296C32—21no—
297C32#479911yescharacteristic, complemented
298C32C3×#47911yescharacteristic, complemented
299Q8—21no—
300Q8#634⋊C312yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—91no—
304C2×C4—91no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
308C8—21no—
309C8—181no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—61no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3C3×#87811yescharacteristic
320C2×C3C3×#91611yescharacteristic
321C2×C3C3×#91811yescharacteristic
322C22D4×(C3⋊D3)⋊C311yescharacteristic
323C4—22no—
324C4—41no—
325C4—181no—
326C4—181no—
327C4—361no—
328C4—361no—
329C4C2×#634⋊C311yescharacteristic
330C4#3830⋊C311yescharacteristic
331C3—31no—
332C3—31no—
333C3—61no—
334C3C3×#479911yescharacteristic
335C2#64087⋊C311yescharacteristic
336C2#64125⋊C311yescharacteristic
337C2#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×H(3,𝔽3)C2×D4
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#479
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
8. #10592498C3semidirect
204. H(3,𝔽3)#479semidirect
297. C32#4799semidirect
298. C32C3×#479semidirect

Maximal subgroups

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

subgroupindex
Q8×(C3⋊Dic3)⋊C32
#64099⋊C32
H(3,𝔽3)×#1032
#63662⋊C32
#63644⋊C32
#63641⋊C32
#63634⋊C32
#105924983
C3×#47993