Finite Group Databasebeta

← #13135167

Subgroups — #13135167

Order 1728.

409 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 63 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#11721830C211yesmaximal, characteristic
2#63658⋊C3C211yesmaximal, characteristic, complemented
3#63657⋊C3C211yesmaximal, characteristic, complemented
4#10589150—31nomaximal
5#10589149—31nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C2×#6235—21no—
8C2×C4⋊#272C411yescharacteristic, complemented
9C22×(C3⋊Dic3)⋊C3—21no—
10#88542—21no—
11C4⋊#1084C411yescharacteristic, complemented
12C2×(C3⋊(C3⋊C8))⋊C3—21no—
13C3×#930—31no—
14C3×#930—31no—
15C3×#930—61no—
16#63658D311yescharacteristic, complemented
17#63657D311yescharacteristic, complemented
18C3×C3⋊#95—31no—
19C3×#830—31no—
20C23×H(3,𝔽3)D411yescharacteristic
21C22×#272—21no—
22C2×C4×H(3,𝔽3)—41no—
23C2×C4×H(3,𝔽3)C2×C411yescharacteristic
24C2×C4×H(3,𝔽3)D411yescharacteristic
25#6235—41no—
26#6235—41no—
27C2×#1084—21no—
28C4⋊#272—41no—
29C2×(C3⋊Dic3)⋊C3—22no—
30C2×(C3⋊Dic3)⋊C3—41no—
31C2×(C3⋊Dic3)⋊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
39C22×C3⋊Dic3—21no—
40C2×C3×#67—62no—
41C2×C3×#67—61no—
42C2×C3×#67—61no—
43C22×C3×Dic3—61no—
44C2×C3×D12—31no—
45C2×C3×D12—31no—
46C2×C3×D12—61no—
47C2×C3⋊(C3⋊C8)—21no—
48C3×(C2×D3)⋊C4—62no—
49C3×(C2×D3)⋊C4—61no—
50C3×(C2×D3)⋊C4—61no—
51C3×C4⋊Dic3—31no—
52C3×C4⋊Dic3—31no—
53C3×C4⋊Dic3—61no—
54C2×C3×C3⋊C8—61no—
55C22×H(3,𝔽3)—21no—
56C22×H(3,𝔽3)—21no—
57C22×H(3,𝔽3)C22⋊C411yescharacteristic
58C22×H(3,𝔽3)D811yescharacteristic
59C22×H(3,𝔽3)SD1611yescharacteristic
60C2×#272—41no—
61C2×#272—41no—
62C2×#272—41no—
63C4×H(3,𝔽3)—21no—
64C4×H(3,𝔽3)—21no—
65#1084—41no—
66(C3⋊Dic3)⋊C3—42no—
67C3×#121—91no—
68#930—31no—
69#930—31no—
70#930—61no—
71C3×C22⋊C8—91no—
72C3×#95—91no—
73C3⋊#95—31no—
74C3⋊#95—91no—
75#830—31no—
76#830—91no—
77C23×C32—21no—
78C23×C32D1211yescharacteristic
79C23×C32#6711yescharacteristic
80C22×C3×D3—62no—
81C22×C3×D3—61no—
82C22×C3×D3—61no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4—42no—
88C2×C32×C4C4×D311yescharacteristic
89C2×C32×C4C4×D311yescharacteristic
90C2×C32×C4D1211yescharacteristic
91C2×C32×C4#6711yescharacteristic
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3—41no—
94C2×C3⋊Dic3—41no—
95C3×#67—122no—
96C3×#67—122no—
97C3×#67—121no—
98C3×#67—121no—
99C3×#67—121no—
100C3×#67—121no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—121no—
106C2×C3×Dic3—121no—
107C3×D12—121no—
108C3×D12—121no—
109C3×D12—122no—
110C3⋊(C3⋊C8)—41no—
111C3×C3⋊C8—121no—
112#327—271noSylow
113C2×H(3,𝔽3)—21no—
114C2×H(3,𝔽3)—21no—
115C2×H(3,𝔽3)#9911yescharacteristic
116C2×H(3,𝔽3)#10211yescharacteristic
117C2×H(3,𝔽3)#10411yescharacteristic
118#272—81no—
119C2×C3×D4—91no—
120C2×C3×D4—181no—
121C22×C3×C4—31no—
122C22×C3×C4—31no—
123C22×C3×C4—61no—
124C22×C3×C4—181no—
125C22×C3×C4D3211yescharacteristic
126C2×#67—61no—
127C2×#67—61no—
128C2×#67—62no—
129C22×Dic3—61no—
130C22×Dic3—181no—
131C2×D12—31no—
132C2×D12—31no—
133C2×D12—61no—
134C2×C3×C8—181no—
135C3×C4⋊C4—91no—
136C3×C22⋊C4—181no—
137(C2×D3)⋊C4—61no—
138(C2×D3)⋊C4—61no—
139(C2×D3)⋊C4—62no—
140C4⋊Dic3—31no—
141C4⋊Dic3—31no—
142C4⋊Dic3—61no—
143C2×C3⋊C8—61no—
144C2×C3⋊C8—181no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—41no—
153C22×C32—41no—
154C22×C32#20411yescharacteristic
155C22×C32D2411yescharacteristic
156C22×C32(C2×D3)⋊C411yescharacteristic
157C22×C32(C2×D3)⋊C411yescharacteristic
158C22×C32#21311yescharacteristic
159C22×C32#21511yescharacteristic
160C2×C3×D3—122no—
161C2×C3×D3—122no—
162C2×C3×D3—122no—
163C2×C3×D3—121no—
164C2×C3×D3—121no—
165C2×C3×D3—121no—
166C2×C3×D3—121no—
167C2×C3×D3—121no—
168C2×C3×D3—121no—
169C32×C4—22no—
170C32×C4—21no—
171C32×C4—21no—
172C32×C4—21no—
173C32×C4—21no—
174C32×C4—41no—
175C3⋊Dic3—42no—
176C3×Dic3—122no—
177C3×Dic3—122no—
178C3×Dic3—121no—
179C3×Dic3—121no—
180#121—91no—
181C22⋊C8—91no—
182#95—91no—
183H(3,𝔽3)#32711yescharacteristic, complemented, Sylow
184C23×C3—31no—
185C23×C3—31no—
186C23×C3—61no—
187C23×C3—181no—
188C23×C3#62611yescharacteristic
189C22×D3—61no—
190C22×D3—61no—
191C22×D3—62no—
192C3×D4—361no—
193C3×D4—361no—
194C3×D4—361no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—41no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—121no—
203C2×C3×C4—121no—
204C2×C3×C4—122no—
205C2×C3×C4—182no—
206C2×C3×C4—181no—
207C2×C3×C4—361no—
208C2×C3×C4—361no—
209C2×C3×C4#62411yescharacteristic
210C2×C3×C4#62611yescharacteristic
211#67—121no—
212#67—121no—
213#67—121no—
214#67—121no—
215#67—122no—
216#67—122no—
217C2×Dic3—62no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—62no—
221C2×Dic3—121no—
222C2×Dic3—121no—
223C2×Dic3—182no—
224C2×Dic3—361no—
225C2×Dic3—361no—
226D12—121no—
227D12—121no—
228D12—122no—
229C3×C8—361no—
230C3⋊C8—121no—
231C3⋊C8—361no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32#80511yescharacteristic
242C2×C32#80611yescharacteristic
243C2×C32#80611yescharacteristic
244C2×C32#80911yescharacteristic
245C2×C32#82111yescharacteristic
246C2×C32#82511yescharacteristic
247C3×D3—242no—
248C3×D3—241no—
249C3×D3—241no—
250C2×D4—91no—
251C2×D4—181no—
252C22×C4—181no—
253C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C2×C8—181no—
255C4⋊C4—91no—
256C22⋊C4—181no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—31no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—121no—
273C22×C3—121no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3—361no—
277C22×C3#371411yescharacteristic
278C22×C3#371711yescharacteristic
279C22×C3#372211yescharacteristic
280C2×D3—121no—
281C2×D3—121no—
282C2×D3—121no—
283C2×D3—121no—
284C2×D3—121no—
285C2×D3—121no—
286C2×D3—122no—
287C2×D3—122no—
288C2×D3—122no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—61no—
294C3×C4—61no—
295C3×C4—62no—
296C3×C4—121no—
297C3×C4—362no—
298C3×C4—361no—
299Dic3—122no—
300Dic3—121no—
301Dic3—121no—
302Dic3—122no—
303Dic3—362no—
304C32—21no—
305C32#445111yescharacteristic, complemented
306C32#447011yescharacteristic, complemented
307C23—181no—
308C23#620011yescharacteristic
309D4—361no—
310D4—361no—
311D4—361no—
312C2×C4—41no—
313C2×C4—182no—
314C2×C4—181no—
315C2×C4—361no—
316C2×C4—361no—
317C2×C4#619911yescharacteristic
318C2×C4#620011yescharacteristic
319C8—361no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—61no—
329C2×C3—61no—
330C2×C3—61no—
331C2×C3—61no—
332C2×C3—61no—
333C2×C3—61no—
334C2×C3—61no—
335C2×C3—121no—
336C2×C3—121no—
337C2×C3—721no—
338C2×C3#6356411yescharacteristic
339C2×C3#6357011yescharacteristic
340C2×C3#6358111yescharacteristic
341D3—241no—
342D3—241no—
343D3—242no—
344C22—21no—
345C22—21no—
346C22—361no—
347C22—361no—
348C22—361no—
349C22#8843611yescharacteristic
350C22#8843711yescharacteristic
351C22#8845111yescharacteristic
352C4—21no—
353C4—21no—
354C4—362no—
355C4—361no—
356C3—31no—
357C3—31no—
358C3—61no—
359C3#1058915111yescharacteristic
360C2—21no—
361C2—21no—
362C2—721no—
363C2#1172032411yescharacteristic
364C2#1172032811yescharacteristic
365C2#1172033511yescharacteristic
subgroupG/H
centerC22#88451
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC22×C3×C4D32
Fitting, cosocleC22×C4×H(3,𝔽3)C22
socleC22×C3#3722
prime radical#11721830C2
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. #63658⋊C3C2semidirect
3. #63657⋊C3C2semidirect
8. C2×C4⋊#272C4semidirect
11. C4⋊#1084C4semidirect
16. #63658D3semidirect
17. #63657D3semidirect
183. H(3,𝔽3)#327semidirect
305. C32#4451semidirect
306. C32#4470semidirect

Maximal subgroups

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

subgroupindex
#117218302
#63658⋊C32
#63657⋊C32
#105891503
#105891493