Finite Group Databasebeta

← #13139935

Subgroups — #13139935

Order 1728.

489 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 131 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#11721826C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
3C2×#3748⋊C3C211yesmaximal, characteristic
4#11720307C212yesmaximal, complemented
5#11720306C212yesmaximal, complemented
6#10591139—31nomaximal
7#10591131—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3—21no—
10#88552—21no—
11C4⋊#1084C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13C4×#1084—21no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16#3748⋊C3C414yescomplemented
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C2×C4⋊(C3⋊Dic3)D311yescharacteristic
19C2×#3748D311yescharacteristic
20C3×#926—31no—
21C3×#926—31no—
22C3×#926—61no—
23C2×C3×C4⋊Dic3—31no—
24C2×C3×#208—31no—
25#63576—32no—
26#63575—32no—
27#63574—32no—
28#63573—32no—
29C23×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C412yes—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)Q812yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084—21no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—42no—
38(C3⋊(C3⋊C8))⋊C3C2×C414yescomplemented
39#5121—91no—
40#4995—91no—
41#4964—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3—21no—
46C22×C3×Dic3—61no—
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)C2×D312yes—
49#3748Dic314yescomplemented
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C3×#217—62no—
52C3×#217—61no—
53C3×#217—61no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3—61no—
61C3×C4⋊Dic3—61no—
62C3×C4×Dic3—62no—
63C3×C4×Dic3—61no—
64C3×C4×Dic3—61no—
65C3×#208—34no—
66C2×C3×C3⋊C8—32no—
67C22×H(3,𝔽3)—21no—
68C22×H(3,𝔽3)—21no—
69C22×H(3,𝔽3)C4⋊C412yes—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)C4⋊C412yes—
72C4×H(3,𝔽3)C22×C411yescharacteristic
73C4×H(3,𝔽3)C2×Q811yescharacteristic
74#1084—41no—
75#1084—41no—
76(C3⋊Dic3)⋊C3—42no—
77C2×C3×M16—91no—
78C3×#117—91no—
79C2×C3×C4⋊C4—91no—
80#926—31no—
81#926—31no—
82#926—61no—
83C2×C4⋊Dic3—31no—
84C2×C4⋊Dic3—91no—
85C2×#208—31no—
86C2×#208—91no—
87#818—92no—
88#817—92no—
89#808—92no—
90#808—92no—
91#807—92no—
92#807—92no—
93C23×C32—21no—
94C23×C32D1211yescharacteristic
95C23×C32#6711yescharacteristic
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4Dic612yes—
101C2×C32×C4Dic612yes—
102C2×C32×C4C4×D312yes—
103C2×C32×C4D1211yescharacteristic
104C2×C32×C4C2×Dic312yes—
105C2×C32×C4#6711yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—42no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—122no—
118C3⋊(C3⋊C8)C2×Dic314yescomplemented
119C3×C3⋊C8—34no—
120#428—271noSylow
121C2×H(3,𝔽3)—22no—
122C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
123C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
124C2×H(3,𝔽3)#13711yescharacteristic
125C22×C3×C4—31no—
126C22×C3×C4—31no—
127C22×C3×C4—61no—
128C22×C3×C4—181no—
129C22×C3×C4D3211yescharacteristic
130C22×Dic3—61no—
131C22×Dic3—181no—
132C3×M16—94no—
133C2×C3×C8—92no—
134C3×C4⋊C4—92no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4—181no—
138C3×C22⋊C4—181no—
139C3×C42—181no—
140#217—61no—
141#217—61no—
142#217—62no—
143C4⋊Dic3—31no—
144C4⋊Dic3—31no—
145C4⋊Dic3—31no—
146C4⋊Dic3—31no—
147C4⋊Dic3—32no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—61no—
151C4⋊Dic3—92no—
152C4⋊Dic3—181no—
153C4×Dic3—61no—
154C4×Dic3—61no—
155C4×Dic3—62no—
156#208—34no—
157#208—94no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—92no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32—41no—
167C22×C32—41no—
168C22×C32Dic3⋊C412yes—
169C22×C32C4⋊Dic312yes—
170C22×C32C2×D1211yescharacteristic
171C22×C32C2×#6711yescharacteristic
172C32×C4—22no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4Dic3⋊C412yes—
176C32×C4C4⋊Dic312yes—
177C32×C4C2×Dic611yescharacteristic
178C32×C4C2×Dic611yescharacteristic
179C32×C4C2×C4×D311yescharacteristic
180C32×C4C22×Dic311yescharacteristic
181C3⋊Dic3—42no—
182C3×Dic3—122no—
183C3×Dic3—122no—
184C3×Dic3—122no—
185C3×Dic3—121no—
186C3×Dic3—121no—
187C3×Dic3—121no—
188C3×Dic3—121no—
189C2×M16—91no—
190#117—91no—
191C2×C4⋊C4—91no—
192#107—272no—
193#106—272no—
194H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
195C23×C3—31no—
196C23×C3—31no—
197C23×C3—61no—
198C23×C3#62611yescharacteristic
199C2×C3×C4—32no—
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—32no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—182no—
212C2×C3×C4—181no—
213C2×C3×C4—181no—
214C2×C3×C4—362no—
215C2×C3×C4D3×Dic312yes—
216C2×C3×C4#62611yescharacteristic
217C2×C3×C4#62712yes—
218C2×C3×C4C2×D3211yescharacteristic
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C2×Dic3—122no—
227C2×Dic3—182no—
228C2×Dic3—362no—
229C3×C8—94no—
230C3⋊C8—34no—
231C3⋊C8—94no—
232C2×C32—22no—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—42no—
238C2×C32#90811yescharacteristic
239C2×C32#90911yescharacteristic
240C2×C32C2×Dic3⋊C411yescharacteristic
241C2×C32C2×C4⋊Dic311yescharacteristic
242C2×C32#93211yescharacteristic
243C2×C32#94211yescharacteristic
244C22×C4—181no—
245C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
246M16—94no—
247C2×C8—92no—
248C4⋊C4—91no—
249C4⋊C4—92no—
250C4⋊C4—91no—
251C4⋊C4—181no—
252C22⋊C4—181no—
253C42—181no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—32no—
259C22×C3—32no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3—62no—
266C22×C3—121no—
267C22×C3—121no—
268C22×C3Dic3⋊Dic312yes—
269C22×C3C2×#62611yescharacteristic
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—32no—
275C3×C4—32no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—62no—
279C3×C4—362no—
280C3×C4—361no—
281C3×C4—361no—
282C3×C4Dic3⋊Dic312yes—
283C3×C4C2×D3×Dic311yescharacteristic
284C3×C4C2×#62711yescharacteristic
285Dic3—121no—
286Dic3—121no—
287Dic3—121no—
288Dic3—121no—
289Dic3—122no—
290Dic3—122no—
291Dic3—122no—
292Dic3—362no—
293C32—21no—
294C32#496411yescharacteristic, complemented
295C32#512111yescharacteristic, complemented
296C23#620011yescharacteristic
297C2×C4—181no—
298C2×C4—181no—
299C2×C4—182no—
300C2×C4—362no—
301C2×C4#619612yes—
302C2×C4((C3⋊D3)⋊C3)⋊C412yes—
303C2×C4#620011yescharacteristic
304C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
305C8—94no—
306C2×C3—22no—
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—62no—
314C2×C3—62no—
315C2×C3—61no—
316C2×C3—61no—
317C2×C3—61no—
318C2×C3—122no—
319C2×C3#6383011yescharacteristic
320C2×C3#6383311yescharacteristic
321C2×C3C2×Dic3⋊Dic311yescharacteristic
322C22—21no—
323C22—21no—
324C22#8844812yes—
325C22C2×#620011yescharacteristic
326C4—361no—
327C4—362no—
328C4—361no—
329C4#8844812yes—
330C4C2×#619611yescharacteristic
331C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
332C3—31no—
333C3—31no—
334C3—61no—
335C3#1059113011yescharacteristic
336C2—22no—
337C2#1172106911yescharacteristic
338C2#1172107111yescharacteristic
339C2C2×#8844811yescharacteristic
subgroupG/H
centerC22C2×#6200
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC23×H(3,𝔽3)D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#428
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720307C2semidirect
5. #11720306C2semidirect
16. #3748⋊C3C4semidirect
38. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
49. #3748Dic3semidirect
118. C3⋊(C3⋊C8)C2×Dic3semidirect
194. H(3,𝔽3)#428semidirect
294. C32#4964semidirect
295. C32#5121semidirect

Maximal subgroups

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

subgroupindex
#117218262
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3748⋊C32
#117203072
#117203062
#105911393
#105911313