Finite Group Databasebeta

← #13135152

Subgroups — #13135152

Order 1728.

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

normal subgroups only

This group has 391 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#11720574C211yesmaximal, characteristic, complemented
3#63657⋊C3C211yesmaximal, characteristic, complemented
4#10589151—31nomaximal
5#10589149—31nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C22×#1084—21no—
8C2×#638⋊C3—21no—
9C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
10#3752⋊C3—21no—
11C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic, complemented
12C2×(C3⋊(C3⋊C8))⋊C3—21no—
13#64139D311yescharacteristic
14C3×#930—31no—
15#63657D311yescharacteristic, complemented
16C3×C3⋊#95—31no—
17C3×C3⋊#95—31no—
18C3×C3⋊#95—61no—
19C3×#830—31no—
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
25C2×#1084—22no—
26C2×#1084—41no—
27C2×#1084—41no—
28#638⋊C3—41no—
29#638⋊C3—41no—
30C2×(C3⋊Dic3)⋊C3—21no—
31C4⋊((C3⋊D3)⋊C3)—41no—
32(C3⋊(C3⋊C8))⋊C3—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—
40C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
41C2×C3×#67—61no—
42C22×C3×Dic3—62no—
43C22×C3×Dic3—61no—
44C22×C3×Dic3—61no—
45C2×C3×D12—31no—
46#3752—21no—
47C4⋊(C3⋊Dic3)Dic311yescharacteristic, complemented
48C2×C3⋊(C3⋊C8)—21no—
49C3×(C2×D3)⋊C4—61no—
50C3×C4⋊Dic3—31no—
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#1084—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—31no—
71C3⋊#95—61no—
72#830—31no—
73#830—91no—
74C23×C32—21no—
75C23×C32D1211yescharacteristic
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×C4C2×Dic311yescharacteristic
86C2×C32×C4#6711yescharacteristic
87C2×C32×C4#6711yescharacteristic
88#638—41no—
89#638—41no—
90C2×C3⋊Dic3—21no—
91C4⋊(C3⋊D3)—41no—
92C3×#67—121no—
93C3×#67—121no—
94C2×C3×Dic3—64no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—122no—
99C2×C3×Dic3—121no—
100C2×C3×Dic3—121no—
101C2×C3×Dic3—122no—
102C2×C3×Dic3—121no—
103C2×C3×Dic3—121no—
104C3×D12—121no—
105C3⋊(C3⋊C8)—41no—
106C3×C3⋊C8—121no—
107#327—271noSylow
108C2×H(3,𝔽3)—21no—
109C2×H(3,𝔽3)—21no—
110C2×H(3,𝔽3)#9911yescharacteristic
111C2×H(3,𝔽3)#10211yescharacteristic
112C2×H(3,𝔽3)#10411yescharacteristic
113(C3⋊D3)⋊C3—81no—
114C2×C3×D4—91no—
115C2×C3×D4—181no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—61no—
119C22×C3×C4—181no—
120C22×C3×C4D3211yescharacteristic
121C2×#67—61no—
122C2×#67—181no—
123C22×Dic3—61no—
124C22×Dic3—61no—
125C22×Dic3—62no—
126C2×D12—31no—
127C2×D12—91no—
128C2×C3×C8—181no—
129C3×C4⋊C4—91no—
130C3×C22⋊C4—181no—
131(C2×D3)⋊C4—61no—
132(C2×D3)⋊C4—181no—
133C4⋊Dic3—31no—
134C4⋊Dic3—91no—
135C2×C3⋊C8—61no—
136C2×C3⋊C8—181no—
137C22×C32—22no—
138C22×C32—22no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32#20411yescharacteristic
147C22×C32D2411yescharacteristic
148C22×C32(C2×D3)⋊C411yescharacteristic
149C22×C32#21311yescharacteristic
150C22×C32#21411yescharacteristic
151C22×C32#21711yescharacteristic
152C2×C3⋊D3—41no—
153C2×C3⋊D3—41no—
154C2×C3⋊D3—41no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C32×C4—21no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—21no—
162C32×C4—41no—
163C32×C4—41no—
164C3⋊Dic3—41no—
165C3×Dic3—124no—
166C3×Dic3—122no—
167C3×Dic3—122no—
168C3×Dic3—121no—
169#121—91no—
170C22⋊C8—91no—
171#95—91no—
172H(3,𝔽3)#32711yescharacteristic, complemented, Sylow
173C23×C3—31no—
174C23×C3—31no—
175C23×C3—61no—
176C23×C3—181no—
177C23×C3#62611yescharacteristic
178C22×D3—61no—
179C22×D3—181no—
180C3×D4—361no—
181C3×D4—361no—
182C3×D4—361no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—41no—
188C2×C3×C4—61no—
189C2×C3×C4—61no—
190C2×C3×C4—121no—
191C2×C3×C4—121no—
192C2×C3×C4—122no—
193C2×C3×C4—182no—
194C2×C3×C4—181no—
195C2×C3×C4—361no—
196C2×C3×C4—361no—
197C2×C3×C4D3×Dic311yescharacteristic
198C2×C3×C4#62511yescharacteristic
199#67—121no—
200#67—121no—
201#67—361no—
202#67—361no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—64no—
206C2×Dic3—61no—
207C2×Dic3—121no—
208C2×Dic3—121no—
209C2×Dic3—121no—
210C2×Dic3—121no—
211C2×Dic3—122no—
212C2×Dic3—122no—
213C2×Dic3—181no—
214D12—121no—
215D12—361no—
216C3×C8—361no—
217C3⋊C8—121no—
218C3⋊C8—361no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32#80611yescharacteristic
229C2×C32#82111yescharacteristic
230C2×C32#82511yescharacteristic
231C2×C32#83211yescharacteristic
232C2×C32#83411yescharacteristic
233C2×C32#83711yescharacteristic
234C3⋊D3—81no—
235C3×D3—241no—
236C2×D4—91no—
237C2×D4—181no—
238C22×C4—181no—
239C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
240C2×C8—181no—
241C4⋊C4—91no—
242C22⋊C4—181no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—31no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—62no—
256C22×C3—62no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3—361no—
263C22×C3#371411yescharacteristic
264C22×C3#371611yescharacteristic
265C22×C3(C2×D3)⋊Dic311yescharacteristic
266C2×D3—121no—
267C2×D3—121no—
268C2×D3—121no—
269C2×D3—361no—
270C2×D3—361no—
271C2×D3—361no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—121no—
279C3×C4—121no—
280C3×C4—362no—
281C3×C4—361no—
282Dic3—122no—
283Dic3—122no—
284Dic3—124no—
285Dic3—121no—
286Dic3—361no—
287C32—21no—
288C32#447011yescharacteristic, complemented
289C32#453711yescharacteristic, complemented
290C23—181no—
291C23#620011yescharacteristic
292D4—361no—
293D4—361no—
294D4—361no—
295C2×C4—41no—
296C2×C4—182no—
297C2×C4—181no—
298C2×C4—361no—
299C2×C4—361no—
300C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
301C2×C4#619811yescharacteristic
302C8—361no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—61no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3—62no—
316C2×C3—62no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3—61no—
320C2×C3—721no—
321C2×C3#6356311yescharacteristic
322C2×C3#6356811yescharacteristic
323C2×C3#6358011yescharacteristic
324D3—241no—
325D3—721no—
326C22—21no—
327C22—21no—
328C22—361no—
329C22—361no—
330C22—361no—
331C22#8843611yescharacteristic
332C22#8843811yescharacteristic
333C22#8845011yescharacteristic
334C4—21no—
335C4—21no—
336C4—362no—
337C4—361no—
338C3—31no—
339C3—31no—
340C3—61no—
341C3#1058915011yescharacteristic
342C2—21no—
343C2—21no—
344C2—721no—
345C2#1172030511yescharacteristic
346C2#1172030811yescharacteristic
347C2#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. #11720574C2semidirect
3. #63657⋊C3C2semidirect
9. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
11. C4⋊((C3⋊Dic3)⋊C3)C4semidirect
15. #63657D3semidirect
40. C2×C4⋊(C3⋊D3)Dic3semidirect
47. C4⋊(C3⋊Dic3)Dic3semidirect
172. H(3,𝔽3)#327semidirect
288. C32#4470semidirect
289. C32#4537semidirect

Maximal subgroups

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

subgroupindex
#64139⋊C32
#117205742
#63657⋊C32
#105891513
#105891493