Finite Group Databasebeta

← #13135157

Subgroups — #13135157

Order 1728.

365 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 65 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#64139⋊C3C211yesmaximal, characteristic
2#11720573C211yesmaximal, characteristic, complemented
3#63658⋊C3C211yesmaximal, characteristic, complemented
4#10589151—31nomaximal
5#10589150—31nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C2×#638⋊C3—21no—
8C22×(C3⋊Dic3)⋊C3—21no—
9C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
10C2×#6180—21no—
11#3752⋊C3—21no—
12C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic, complemented
13#64139D311yescharacteristic
14C3×#930—31no—
15#63658D311yescharacteristic, complemented
16C3×C3⋊#95—31no—
17C3×#830—31no—
18C3×#830—31no—
19C3×#830—61no—
20C23×H(3,𝔽3)D411yescharacteristic
21C22×(C3⋊D3)⋊C3—21no—
22C2×C4×H(3,𝔽3)—41no—
23C2×C4×H(3,𝔽3)C2×C411yescharacteristic
24C2×C4×H(3,𝔽3)D411yescharacteristic
25#638⋊C3—41no—
26#638⋊C3—41no—
27C2×(C3⋊Dic3)⋊C3—22no—
28C2×(C3⋊Dic3)⋊C3—21no—
29C2×(C3⋊Dic3)⋊C3—41no—
30C2×(C3⋊Dic3)⋊C3—41no—
31C4⋊((C3⋊D3)⋊C3)—41no—
32#6180—41no—
33#4537—91no—
34#4470—91no—
35#4451—91no—
36C22×C32×C4—21no—
37C22×C32×C4C2×D311yescharacteristic
38C22×C32×C4C2×D311yescharacteristic
39C2×#638—21no—
40C22×C3⋊Dic3—21no—
41C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
42C2×C3×#67—61no—
43C22×C3×Dic3—61no—
44C2×C3×D12—31no—
45#3752—21no—
46C4⋊(C3⋊Dic3)Dic311yescharacteristic, complemented
47C3×(C2×D3)⋊C4—61no—
48C3×C4⋊Dic3—31no—
49C2×C3×C3⋊C8—62no—
50C2×C3×C3⋊C8—61no—
51C2×C3×C3⋊C8—61no—
52C22×H(3,𝔽3)—21no—
53C22×H(3,𝔽3)—21no—
54C22×H(3,𝔽3)C22⋊C411yescharacteristic
55C22×H(3,𝔽3)D811yescharacteristic
56C22×H(3,𝔽3)SD1611yescharacteristic
57C2×(C3⋊D3)⋊C3—41no—
58C2×(C3⋊D3)⋊C3—41no—
59C2×(C3⋊D3)⋊C3—41no—
60C4×H(3,𝔽3)—21no—
61C4×H(3,𝔽3)—21no—
62(C3⋊Dic3)⋊C3—42no—
63(C3⋊Dic3)⋊C3—41no—
64C3×#121—91no—
65#930—31no—
66#930—91no—
67C3×C22⋊C8—91no—
68C3×#95—91no—
69C3⋊#95—31no—
70C3⋊#95—91no—
71#830—31no—
72#830—31no—
73#830—61no—
74C23×C32—21no—
75C23×C32#6711yescharacteristic
76C23×C32#6711yescharacteristic
77C22×C3⋊D3—21no—
78C22×C3×D3—61no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—42no—
84C2×C32×C4C4×D311yescharacteristic
85C2×C32×C4D1211yescharacteristic
86C2×C32×C4C2×Dic311yescharacteristic
87C2×C32×C4#6711yescharacteristic
88#638—41no—
89#638—41no—
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—41no—
93C2×C3⋊Dic3—41no—
94C4⋊(C3⋊D3)—41no—
95C3×#67—121no—
96C3×#67—121no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—121no—
100C2×C3×Dic3—121no—
101C3×D12—121no—
102C3×C3⋊C8—122no—
103C3×C3⋊C8—121no—
104C3×C3⋊C8—121no—
105#327—271noSylow
106C2×H(3,𝔽3)—21no—
107C2×H(3,𝔽3)—21no—
108C2×H(3,𝔽3)#9911yescharacteristic
109C2×H(3,𝔽3)#10211yescharacteristic
110C2×H(3,𝔽3)#10411yescharacteristic
111(C3⋊D3)⋊C3—81no—
112C2×C3×D4—91no—
113C2×C3×D4—181no—
114C22×C3×C4—31no—
115C22×C3×C4—31no—
116C22×C3×C4—61no—
117C22×C3×C4—181no—
118C22×C3×C4D3211yescharacteristic
119C2×#67—61no—
120C2×#67—181no—
121C22×Dic3—61no—
122C22×Dic3—181no—
123C2×D12—31no—
124C2×D12—91no—
125C2×C3×C8—181no—
126C3×C4⋊C4—91no—
127C3×C22⋊C4—181no—
128(C2×D3)⋊C4—61no—
129(C2×D3)⋊C4—181no—
130C4⋊Dic3—31no—
131C4⋊Dic3—91no—
132C2×C3⋊C8—61no—
133C2×C3⋊C8—61no—
134C2×C3⋊C8—62no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32(C2×D3)⋊C411yescharacteristic
145C22×C32#21311yescharacteristic
146C22×C32#21311yescharacteristic
147C22×C32#21411yescharacteristic
148C22×C32#21511yescharacteristic
149C22×C32#21711yescharacteristic
150C2×C3⋊D3—41no—
151C2×C3⋊D3—41no—
152C2×C3⋊D3—41no—
153C2×C3×D3—121no—
154C2×C3×D3—121no—
155C2×C3×D3—121no—
156C32×C4—22no—
157C32×C4—21no—
158C32×C4—21no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—41no—
162C3⋊Dic3—42no—
163C3⋊Dic3—41no—
164C3×Dic3—122no—
165C3×Dic3—121no—
166#121—91no—
167C22⋊C8—91no—
168#95—91no—
169H(3,𝔽3)#32711yescharacteristic, complemented, Sylow
170C23×C3—31no—
171C23×C3—31no—
172C23×C3—61no—
173C23×C3—181no—
174C23×C3#62511yescharacteristic
175C22×D3—61no—
176C22×D3—181no—
177C3×D4—361no—
178C3×D4—361no—
179C3×D4—361no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—41no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—121no—
188C2×C3×C4—121no—
189C2×C3×C4—122no—
190C2×C3×C4—181no—
191C2×C3×C4—182no—
192C2×C3×C4—361no—
193C2×C3×C4—361no—
194C2×C3×C4D3×Dic311yescharacteristic
195C2×C3×C4#62611yescharacteristic
196#67—121no—
197#67—121no—
198#67—361no—
199#67—361no—
200C2×Dic3—62no—
201C2×Dic3—61no—
202C2×Dic3—121no—
203C2×Dic3—121no—
204C2×Dic3—181no—
205C2×Dic3—182no—
206C2×Dic3—361no—
207C2×Dic3—361no—
208D12—121no—
209D12—361no—
210C3×C8—361no—
211C3⋊C8—121no—
212C3⋊C8—121no—
213C3⋊C8—122no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32#80511yescharacteristic
224C2×C32#80611yescharacteristic
225C2×C32#80911yescharacteristic
226C2×C32#83211yescharacteristic
227C2×C32#83411yescharacteristic
228C2×C32#83711yescharacteristic
229C3⋊D3—81no—
230C3×D3—241no—
231C2×D4—91no—
232C2×D4—181no—
233C22×C4—181no—
234C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C2×C8—181no—
236C4⋊C4—91no—
237C22⋊C4—181no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—31no—
241C22×C3—31no—
242C22×C3—31no—
243C22×C3—31no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—61no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3—61no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—121no—
254C22×C3—121no—
255C22×C3—361no—
256C22×C3—361no—
257C22×C3—361no—
258C22×C3#371311yescharacteristic
259C22×C3#371511yescharacteristic
260C22×C3(C2×D3)⋊Dic311yescharacteristic
261C2×D3—121no—
262C2×D3—121no—
263C2×D3—121no—
264C2×D3—361no—
265C2×D3—361no—
266C2×D3—361no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4—61no—
273C3×C4—62no—
274C3×C4—121no—
275C3×C4—361no—
276C3×C4—362no—
277Dic3—122no—
278Dic3—121no—
279Dic3—361no—
280Dic3—362no—
281C32—21no—
282C32#445111yescharacteristic, complemented
283C32#453711yescharacteristic, complemented
284C23—181no—
285C23#619811yescharacteristic
286D4—361no—
287D4—361no—
288D4—361no—
289C2×C4—41no—
290C2×C4—182no—
291C2×C4—181no—
292C2×C4—361no—
293C2×C4—361no—
294C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
295C2×C4#620011yescharacteristic
296C8—361no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—61no—
310C2×C3—61no—
311C2×C3—61no—
312C2×C3—121no—
313C2×C3—121no—
314C2×C3—721no—
315C2×C3#6356211yescharacteristic
316C2×C3#6356911yescharacteristic
317C2×C3#6358011yescharacteristic
318D3—241no—
319D3—721no—
320C22—21no—
321C22—21no—
322C22—361no—
323C22—361no—
324C22—361no—
325C22#8843111yescharacteristic
326C22#8843211yescharacteristic
327C22#8845011yescharacteristic
328C4—21no—
329C4—21no—
330C4—362no—
331C4—361no—
332C3—31no—
333C3—31no—
334C3—61no—
335C3#1058914911yescharacteristic
336C2—21no—
337C2—21no—
338C2—721no—
339C2#1172031811yescharacteristic
340C2#1172032211yescharacteristic
341C2#1172033311yescharacteristic
subgroupG/H
centerC22#88450
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC22×C3×C4D32
Fitting, cosocleC22×C4×H(3,𝔽3)C22
socleC22×C3(C2×D3)⋊Dic3
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#327
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11720573C2semidirect
3. #63658⋊C3C2semidirect
9. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
12. C4⋊((C3⋊Dic3)⋊C3)C4semidirect
15. #63658D3semidirect
41. C2×C4⋊(C3⋊D3)Dic3semidirect
46. C4⋊(C3⋊Dic3)Dic3semidirect
169. H(3,𝔽3)#327semidirect
282. C32#4451semidirect
283. C32#4537semidirect

Maximal subgroups

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

subgroupindex
#64139⋊C32
#117205732
#63658⋊C32
#105891513
#105891503