Finite Group Databasebeta

← #10592742⋊C3 #13142880

Subgroups — #10592742⋊C3 #13142880

Order 1728.

438 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 102 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)×C22⋊Q8C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
3#63662⋊C3C212yesmaximal, complemented
4#63657⋊C3C211yesmaximal, characteristic
5#63635⋊C3C212yesmaximal, complemented
6#10592742C311yesmaximal, characteristic, complemented
7C3×#5043—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3—21no—
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C22⋊C4×H(3,𝔽3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C32×C22⋊Q8—21no—
18C32×C22⋊Q8D311yescharacteristic
19C32×C22⋊Q8C2×C311yescharacteristic
20C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
21C2×C3×C4⋊Dic3—31no—
22#63662C2×C312yescomplemented
23#63657C2×C311yescharacteristic
24#63635C2×C312yescomplemented
25C3×#835—32no—
26C3×#830—31no—
27C3×#808—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29Q8×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—42no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38C3×#483—91no—
39#5043—31no—
40#5043—91no—
41C2×C32×Q8—21no—
42C2×C32×Q8C2×D311yescharacteristic
43C2×C32×Q8C22×C311yescharacteristic
44C22×C32×C4—21no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C22×C311yescharacteristic
47C22×C3⋊Dic3—21no—
48C22×C3×Dic3—61no—
49C32×C4⋊C4—22no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C4⋊C4C22×C311yescharacteristic
55C32×C22⋊C4—22no—
56C32×C22⋊C4—21no—
57C32×C22⋊C4—21no—
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)C22×C312yes—
60C2×C3⋊(C3⋊C8)C22×C312yes—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—61no—
63C2×C3×C3⋊C8—32no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)—21no—
66C22×H(3,𝔽3)SD1612yes—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)—41no—
70C4×H(3,𝔽3)—41no—
71C4×H(3,𝔽3)#4112yes—
72(C3⋊Dic3)⋊C3—42no—
73C3×C22⋊Q8—31no—
74C3×C22⋊Q8—31no—
75C3×C22⋊Q8—61no—
76C3×C22⋊Q8C3×D311yescharacteristic
77C2×C3×C4⋊C4—91no—
78C2×C4⋊Dic3—31no—
79C2×C4⋊Dic3—91no—
80C3×#106—92no—
81C3×#103—92no—
82C3×C22⋊C8—91no—
83#835—32no—
84#835—92no—
85#830—31no—
86#830—91no—
87#808—32no—
88#808—92no—
89C23×C32—21no—
90C23×C32#6711yescharacteristic
91C23×C32C3×D411yescharacteristic
92C32×Q8—22no—
93C32×Q8—22no—
94C32×Q8—42no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—42no—
106C2×C32×C4#6711yescharacteristic
107C2×C32×C4C3×D411yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C23×C311yescharacteristic
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—42no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—122no—
114C3⋊(C3⋊C8)—22no—
115C3×C3⋊C8—62no—
116#483—91noSylow
117C2×H(3,𝔽3)—22no—
118C2×H(3,𝔽3)#12311yescharacteristic
119C2×H(3,𝔽3)C2×SD1611yescharacteristic
120C2×H(3,𝔽3)#13711yescharacteristic
121C2×C3×Q8—31no—
122C2×C3×Q8—31no—
123C2×C3×Q8—61no—
124C2×C3×Q8C2×C3×D311yescharacteristic
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—61no—
128C22×C3×C4—181no—
129C22×C3×C4C2×C3×D311yescharacteristic
130C22×Dic3—61no—
131C22×Dic3—181no—
132C2×C3×C8—92no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—61no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—62no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—92no—
141C3×C4⋊C4—181no—
142C3×C4⋊C4C2×C3×D311yescharacteristic
143C3×C22⋊C4—21no—
144C3×C22⋊C4—61no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—62no—
147C4⋊Dic3—32no—
148C4⋊Dic3—61no—
149C4⋊Dic3—92no—
150C4⋊Dic3—181no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—92no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32#21512yes—
162C22×C32C3×SD1612yes—
163C22×C32C2×#6711yescharacteristic
164C22×C32C2×C3×D411yescharacteristic
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—22no—
168C32×C4—41no—
169C32×C4—42no—
170C32×C4—42no—
171C32×C4—41no—
172C32×C4—41no—
173C32×C4—41no—
174C32×C4—41no—
175C32×C4#23712yes—
176C32×C4C3×#4112yes—
177C3⋊Dic3—42no—
178C3×Dic3—122no—
179C22⋊Q8(C3⋊D3)⋊C311yescharacteristic
180C2×C4⋊C4—91no—
181#106—92no—
182#103—92no—
183C22⋊C8—91no—
184H(3,𝔽3)#48311yescharacteristic, complemented, Sylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3C3×#6711yescharacteristic
189C3×Q8—22no—
190C3×Q8—62no—
191C3×Q8—62no—
192C3×Q8—122no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—62no—
201C2×C3×C4—62no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—122no—
211C2×C3×C4—182no—
212C2×C3×C4—362no—
213C2×C3×C4C3×#6711yescharacteristic
214C2×C3×C4C22×C3×D311yescharacteristic
215C2×Dic3—62no—
216C2×Dic3—122no—
217C2×Dic3—182no—
218C2×Dic3—362no—
219C3×C8—182no—
220C3⋊C8—62no—
221C3⋊C8—182no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—42no—
228C2×C32#93511yescharacteristic
229C2×C32C2×#21511yescharacteristic
230C2×C32#95111yescharacteristic
231C2×C32C3×#12311yescharacteristic
232C2×C32C2×C3×SD1611yescharacteristic
233C2×C32C3×#13711yescharacteristic
234C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
235C22×C4—181no—
236C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
237C2×C8—92no—
238C4⋊C4—21no—
239C4⋊C4—92no—
240C4⋊C4—181no—
241C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
242C22⋊C4—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—32no—
248C22×C3—32no—
249C22×C3—61no—
250C22×C3—61no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—62no—
255C22×C3—121no—
256C22×C3—121no—
257C22×C3C3×#21512yes—
258C22×C3C2×C3×#6711yescharacteristic
259C3×C4—21no—
260C3×C4—32no—
261C3×C4—32no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—62no—
267C3×C4—121no—
268C3×C4—121no—
269C3×C4—121no—
270C3×C4—121no—
271C3×C4—121no—
272C3×C4—122no—
273C3×C4—122no—
274C3×C4—362no—
275C3×C4C3×#23712yes—
276Dic3—122no—
277Dic3—362no—
278C32—21no—
279C32#504311yescharacteristic, complemented
280C32C3×#48311yescharacteristic, complemented
281C23#638⋊C311yescharacteristic
282Q8—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—182no—
287C2×C4—362no—
288C2×C4#638⋊C311yescharacteristic
289C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
290C8—182no—
291C2×C3—22no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—62no—
299C2×C3—62no—
300C2×C3—61no—
301C2×C3—61no—
302C2×C3—61no—
303C2×C3—122no—
304C2×C3C3×#93511yescharacteristic
305C2×C3C2×C3×#21511yescharacteristic
306C2×C3C3×#95111yescharacteristic
307C22—21no—
308C22—21no—
309C22#3755⋊C312yes—
310C22C2×#638⋊C311yescharacteristic
311C4—21no—
312C4—41no—
313C4—41no—
314C4—362no—
315C4#3830⋊C312yes—
316C3—31no—
317C3—31no—
318C3—61no—
319C3C3×#504311yescharacteristic
320C2—22no—
321C2#64144⋊C311yescharacteristic
322C2C2×#3755⋊C311yescharacteristic
323C2#64160⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC23×H(3,𝔽3)D4
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#483
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #63662⋊C3C2semidirect
5. #63635⋊C3C2semidirect
6. #10592742C3semidirect
22. #63662C2×C3semidirect
24. #63635C2×C3semidirect
184. H(3,𝔽3)#483semidirect
279. C32#5043semidirect
280. C32C3×#483semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
C2×C4⋊((C3⋊Dic3)⋊C3)2
#63662⋊C32
#63657⋊C32
#63635⋊C32
#105927423
C3×#50433