Finite Group Databasebeta

← #13137388

Subgroups — #13137388

Order 1728.

458 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3C2×#88431C212yesmaximal
4C4⋊#6180C211yesmaximal, characteristic, complemented
5#11720313C212yesmaximal, complemented
6#10590379—31nomaximal
7#10590365—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×C4×(C3⋊D3)⋊C3—21no—
10C2×#634⋊C3C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2212yes—
13#3752⋊C3—21no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#88431—24no—
18C4×C4⋊(C3⋊D3)D311yescharacteristic
19#64078D311yescharacteristic, complemented
20C3×C4×D12—31no—
21C3×#869—31no—
22C2×#3717—32no—
23C2×#3716—32no—
24C3×C4⋊(C3⋊C8)—31no—
25C3×C4⋊(C3⋊C8)—31no—
26C3×C4⋊(C3⋊C8)—61no—
27#63567—32no—
28#63566—32no—
29C22×(C3⋊D3)⋊C3—21no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C4⋊((C3⋊D3)⋊C3)—21no—
36C4⋊((C3⋊D3)⋊C3)D412yes—
37C4×(C3⋊D3)⋊C3—42no—
38#634⋊C3—22no—
39#634⋊C3—22no—
40#6180—22no—
41#5083—91no—
42#5019—91no—
43#4841—91no—
44C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
45C2×C4×C3⋊D3—21no—
46C2×#634C2×D312yes—
47C2×C3×D12—31no—
48C2×C3×C4×D3—61no—
49C2×C3×Dic6—32no—
50C32×C42—21no—
51C32×C42C2×D311yescharacteristic
52C32×C42C2×D311yescharacteristic
53#3752—21no—
54C4⋊(C3⋊Dic3)—21no—
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)C2×D311yescharacteristic
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—61no—
60C3×C4⋊Dic3—61no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—62no—
64#3717—64no—
65#3716—64no—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C2×(C3⋊D3)⋊C3—22no—
68C2×(C3⋊D3)⋊C3—41no—
69C2×(C3⋊D3)⋊C3—41no—
70C4×H(3,𝔽3)—21no—
71C4×H(3,𝔽3)SD1612yescomplemented
72C4×H(3,𝔽3)C2×D411yescharacteristic
73C4×H(3,𝔽3)#4111yescharacteristic
74(C3⋊Dic3)⋊C3—42no—
75(C3⋊Dic3)⋊C3—41no—
76C3×C4⋊Q8—91no—
77C3×C4×D4—91no—
78C2×#215—92no—
79C2×#214—92no—
80C2×#204—92no—
81C4×D12—31no—
82C4×D12—91no—
83#869—31no—
84#869—91no—
85C3×C4⋊C8—91no—
86#835—92no—
87#816—92no—
88#810—92no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—31no—
91C4⋊(C3⋊C8)—61no—
92C22×C3⋊D3—21no—
93C22×C3×D3—61no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4#6711yescharacteristic
98C2×C32×C4#6711yescharacteristic
99C2×C32×C4#6711yescharacteristic
100C2×C32×C4#6711yescharacteristic
101C2×C32×C4C22×D311yescharacteristic
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—21no—
105C4⋊(C3⋊D3)—21no—
106C4⋊(C3⋊D3)D1212yes—
107C4×C3⋊D3—42no—
108#634—22no—
109#634—22no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—62no—
112C3×D12—32no—
113C3×D12—61no—
114C3×C4×D3—122no—
115C3×Dic6—62no—
116C3×Dic6—62no—
117C3×C3⋊C8—62no—
118C3×C3⋊C8—62no—
119C3×C3⋊C8—122no—
120#461—271noSylow
121C2×H(3,𝔽3)#12111yescharacteristic
122C2×H(3,𝔽3)C2×SD1611yescharacteristic
123C2×H(3,𝔽3)#13711yescharacteristic
124(C3⋊D3)⋊C3—42no—
125C2×C3×Q8—92no—
126C2×C3×D4—91no—
127C22×C3×C4—181no—
128C2×D12—31no—
129C2×D12—91no—
130C2×C4×D3—61no—
131C2×C4×D3—181no—
132C2×Dic6—32no—
133C2×Dic6—92no—
134C2×C3×C8—92no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—181no—
137C3×C4⋊C4—181no—
138C3×C22⋊C4—181no—
139C3×C42—31no—
140C3×C42—31no—
141C3×C42—61no—
142C3×C42D3211yescharacteristic
143#215—184no—
144#214—184no—
145(C2×D3)⋊C4—61no—
146(C2×D3)⋊C4—181no—
147C4⋊Dic3—31no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—91no—
151C4⋊Dic3—181no—
152C4⋊Dic3—181no—
153C2×C3⋊C8—32no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—62no—
156#204—184no—
157C22×C32—21no—
158C22×C32C2×#6711yescharacteristic
159C22×C32C2×#6711yescharacteristic
160C2×C3⋊D3—22no—
161C2×C3⋊D3—41no—
162C2×C3⋊D3—41no—
163C2×C3×D3—62no—
164C2×C3×D3—121no—
165C2×C3×D3—121no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4#21412yescomplemented
173C32×C4#21512yescomplemented
174C32×C4C2×D1211yescharacteristic
175C32×C4#23511yescharacteristic
176C32×C4D3×D411yescharacteristic
177C32×C4#23711yescharacteristic
178C3⋊Dic3—41no—
179C3⋊Dic3—42no—
180C3×Dic3—122no—
181C3×Dic3—121no—
182C2×SD16—272no—
183C4⋊Q8—91no—
184C4×D4—91no—
185C4⋊C8—91no—
186#103—272no—
187H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
188C23×C3—181no—
189C22×D3—61no—
190C22×D3—181no—
191C3×Q8—182no—
192C3×Q8—182no—
193C3×D4—92no—
194C3×D4—181no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—181no—
205C2×C3×C4—182no—
206C2×C3×C4—362no—
207C2×C3×C4#62511yescharacteristic
208C2×C3×C4#62511yescharacteristic
209C2×C3×C4C2×D3211yescharacteristic
210C2×Dic3—61no—
211C2×Dic3—62no—
212C2×Dic3—181no—
213C2×Dic3—182no—
214D12—32no—
215D12—61no—
216D12—92no—
217D12—181no—
218C4×D3—122no—
219C4×D3—362no—
220Dic6—62no—
221Dic6—62no—
222Dic6—182no—
223Dic6—182no—
224C3×C8—182no—
225C3⋊C8—62no—
226C3⋊C8—62no—
227C3⋊C8—122no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32#93011yescharacteristic
232C2×C32C2×#21411yescharacteristic
233C2×C32#93811yescharacteristic
234C2×C32C2×#21511yescharacteristic
235C2×C32#94211yescharacteristic
236C2×C32#95111yescharacteristic
237C3⋊D3—42no—
238C3×D3—122no—
239C2×Q8—92no—
240C2×D4—91no—
241C22×C4—181no—
242SD16—544no—
243C2×C8—92no—
244C4⋊C4—91no—
245C4⋊C4—181no—
246C4⋊C4—181no—
247C22⋊C4—181no—
248C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
249C22×C3—31no—
250C22×C3—31no—
251C22×C3—61no—
252C22×C3—182no—
253C22×C3—361no—
254C22×C3—361no—
255C22×C3C2×#62511yescharacteristic
256C2×D3—62no—
257C2×D3—121no—
258C2×D3—121no—
259C2×D3—182no—
260C2×D3—361no—
261C2×D3—361no—
262C3×C4—21no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—32no—
268C3×C4—32no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4—61no—
273C3×C4—62no—
274C3×C4—121no—
275C3×C4—361no—
276C3×C4—362no—
277C3×C4#371512yes—
278C3×C4#379511yescharacteristic
279C3×C4D3×D1211yescharacteristic
280Dic3—121no—
281Dic3—122no—
282Dic3—362no—
283Dic3—361no—
284C32—21no—
285C32#501911yescharacteristic, complemented
286C32#508311yescharacteristic, complemented
287C23—181no—
288Q8—182no—
289Q8—182no—
290D4—92no—
291D4—181no—
292C2×C4—181no—
293C2×C4—182no—
294C2×C4—362no—
295C2×C4#619811yescharacteristic
296C2×C4#619811yescharacteristic
297C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C8—182no—
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—362no—
309C2×C3C2×#371511yescharacteristic
310C2×C3#6382711yescharacteristic
311C2×C3#6390811yescharacteristic
312D3—122no—
313D3—362no—
314C22—182no—
315C22—361no—
316C22—361no—
317C22C2×#619811yescharacteristic
318C4—21no—
319C4—361no—
320C4—362no—
321C4#8843112yescomplemented
322C4#8864811yescharacteristic
323C4#8865411yescharacteristic
324C3—31no—
325C3—31no—
326C3—61no—
327C3#1059036311yescharacteristic
328C2—362no—
329C2C2×#8843111yescharacteristic
330C2#1172105811yescharacteristic
331C2#1172114111yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#461
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
4. C4⋊#6180C2semidirect
5. #11720313C2semidirect
19. #64078D3semidirect
71. C4×H(3,𝔽3)SD16semidirect
172. C32×C4#214semidirect
173. C32×C4#215semidirect
187. H(3,𝔽3)#461semidirect
285. C32#5019semidirect
286. C32#5083semidirect
321. C4#88431semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊((C3⋊D3)⋊C3)2
#64078⋊C32
C2×#884312
C4⋊#61802
#117203132
#105903793
#105903653