Finite Group Databasebeta

← #13137429

Subgroups — #13137429

Order 1728.

518 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 87 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×#6229C211yesmaximal, characteristic, complemented
2#11721385C211yesmaximal, characteristic
3C2×#88438C212yesmaximal, complemented
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
5#11720308C212yesmaximal, complemented
6#10590367—31nomaximal
7#10590364—31nomaximal
8C2×#6229C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)—21no—
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14#88540—21no—
15C4×#1084—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#88438—24no—
18#64083D311yescharacteristic
19C3×#874—31no—
20C3×C4×Dic6—31no—
21C3×C4×Dic6—31no—
22C3×C4×Dic6—61no—
23C2×#3717—32no—
24C2×#3715—32no—
25C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
26C3×C4⋊(C3⋊C8)—31no—
27#63564—32no—
28#63562—32no—
29C22×(C3⋊D3)⋊C3—22no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—21no—
34C2×#1084—21no—
35#6229—21no—
36#6229D412yescomplemented
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4⋊((C3⋊D3)⋊C3)—22no—
39C4⋊((C3⋊D3)⋊C3)—42no—
40C4⋊((C3⋊D3)⋊C3)—42no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42#5075—91no—
43#5027—91no—
44#4848—91no—
45C2×C4⋊(C3⋊D3)—21no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)C2×D312yes—
48C2×C3×D12—32no—
49C2×C3×D12—61no—
50C2×C3×D12—61no—
51C2×C3×Dic6—31no—
52C2×C3×Dic6—31no—
53C2×C3×Dic6—61no—
54C32×C42—21no—
55C32×C42C2×D311yescharacteristic
56C32×C42C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—61no—
61C3×Dic3⋊C4—62no—
62C3×Dic3⋊C4—61no—
63C3×Dic3⋊C4—61no—
64C3×C4×Dic3—62no—
65C3×C4×Dic3—61no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—32no—
68#3717—64no—
69#3715—64no—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C2×(C3⋊D3)⋊C3—42no—
72C2×(C3⋊D3)⋊C3—42no—
73C2×(C3⋊D3)⋊C3—42no—
74C4×H(3,𝔽3)—21no—
75C4×H(3,𝔽3)SD1612yescomplemented
76C4×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)#4111yescharacteristic
78#1084—22no—
79#1084—41no—
80C3×#127—91no—
81C3×C4×Q8—91no—
82C2×#215—92no—
83C2×#214—92no—
84C2×#204—92no—
85#874—31no—
86#874—91no—
87C4×Dic6—31no—
88C4×Dic6—31no—
89C4×Dic6—61no—
90C3×C4⋊C8—91no—
91#832—92no—
92#821—92no—
93#809—92no—
94C4⋊(C3⋊C8)—31no—
95C4⋊(C3⋊C8)—91no—
96C22×C3⋊D3—22no—
97C22×C3×D3—62no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4D1211yescharacteristic
102C2×C32×C4D1211yescharacteristic
103C2×C32×C4#6711yescharacteristic
104C2×C32×C4#6711yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107C4⋊(C3⋊D3)—22no—
108C4⋊(C3⋊D3)—22no—
109C4⋊(C3⋊D3)—42no—
110C4⋊(C3⋊D3)—42no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C3×D12—62no—
118C3×D12—62no—
119C3×D12—122no—
120C3×D12—122no—
121C3×Dic6—32no—
122C3×Dic6—32no—
123C3×Dic6—62no—
124C3×Dic6—62no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127C3⋊(C3⋊C8)—22no—
128C3×C3⋊C8—62no—
129#460—271noSylow
130C2×H(3,𝔽3)#12111yescharacteristic
131C2×H(3,𝔽3)C2×SD1611yescharacteristic
132C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
133(C3⋊D3)⋊C3—82no—
134C2×C3×Q8—91no—
135C2×C3×D4—92no—
136C2×C3×D4—181no—
137C2×C3×D4—181no—
138C2×D12—32no—
139C2×D12—61no—
140C2×D12—61no—
141C2×D12—92no—
142C2×D12—181no—
143C2×D12—181no—
144C2×Dic6—31no—
145C2×Dic6—31no—
146C2×Dic6—61no—
147C2×C3×C8—92no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—181no—
150C3×C42—31no—
151C3×C42—31no—
152C3×C42—61no—
153C3×C42—181no—
154C3×C42D3211yescharacteristic
155#215—184no—
156#214—184no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—61no—
160Dic3⋊C4—61no—
161Dic3⋊C4—61no—
162Dic3⋊C4—62no—
163C4×Dic3—61no—
164C4×Dic3—61no—
165C4×Dic3—62no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—92no—
168#204—184no—
169C22×C32—21no—
170C22×C32C2×D1211yescharacteristic
171C22×C32C2×#6711yescharacteristic
172C2×C3⋊D3—42no—
173C2×C3⋊D3—42no—
174C2×C3⋊D3—42no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C32×C4—21no—
179C32×C4—21no—
180C32×C4—22no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—41no—
184C32×C4#20412yescomplemented
185C32×C4#21412yescomplemented
186C32×C4D3×D411yescharacteristic
187C32×C4D3×D411yescharacteristic
188C32×C4#23711yescharacteristic
189C32×C4#23911yescharacteristic
190C3×Dic3—64no—
191C3×Dic3—62no—
192C3×Dic3—62no—
193C3×Dic3—122no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196C2×SD16—272no—
197#127—91no—
198C4×Q8—91no—
199C4⋊C8—91no—
200#102—272no—
201H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
202C23×C3—182no—
203C22×D3—62no—
204C22×D3—182no—
205C3×Q8—92no—
206C3×Q8—181no—
207C3×D4—182no—
208C3×D4—182no—
209C3×D4—362no—
210C3×D4—362no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4#62611yescharacteristic
223C2×C3×C4#62611yescharacteristic
224C2×C3×C4C2×D3211yescharacteristic
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—62no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—62no—
231D12—62no—
232D12—62no—
233D12—122no—
234D12—122no—
235D12—182no—
236D12—182no—
237D12—362no—
238D12—362no—
239Dic6—32no—
240Dic6—32no—
241Dic6—62no—
242Dic6—61no—
243Dic6—61no—
244Dic6—62no—
245C3×C8—182no—
246C3⋊C8—62no—
247C3⋊C8—182no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32#89511yescharacteristic
252C2×C32C2×#20411yescharacteristic
253C2×C32#90811yescharacteristic
254C2×C32#93211yescharacteristic
255C2×C32C2×#21411yescharacteristic
256C2×C32#93811yescharacteristic
257C3⋊D3—82no—
258C3×D3—242no—
259C2×Q8—91no—
260C2×D4—92no—
261C2×D4—181no—
262C2×D4—181no—
263SD16—544no—
264C2×C8—92no—
265C4⋊C4—91no—
266C4⋊C4—181no—
267C42—181no—
268C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—61no—
272C22×C3—362no—
273C22×C3—362no—
274C22×C3—362no—
275C22×C3C2×#62611yescharacteristic
276C2×D3—122no—
277C2×D3—122no—
278C2×D3—122no—
279C2×D3—362no—
280C2×D3—362no—
281C2×D3—362no—
282C3×C4—21no—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—32no—
288C3×C4—32no—
289C3×C4—61no—
290C3×C4—61no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—62no—
294C3×C4—121no—
295C3×C4—182no—
296C3×C4—361no—
297C3×C4#371612yes—
298C3×C4#379611yescharacteristic
299C3×C4#380211yescharacteristic
300Dic3—62no—
301Dic3—62no—
302Dic3—64no—
303Dic3—121no—
304Dic3—121no—
305Dic3—122no—
306C32—21no—
307C32#484811yescharacteristic, complemented
308C32#507511yescharacteristic, complemented
309C23—182no—
310Q8—92no—
311Q8—181no—
312D4—182no—
313D4—182no—
314D4—362no—
315D4—362no—
316C2×C4—181no—
317C2×C4—181no—
318C2×C4#620011yescharacteristic
319C2×C4#620011yescharacteristic
320C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
321C8—182no—
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—722no—
332C2×C3C2×#371611yescharacteristic
333C2×C3#6383011yescharacteristic
334C2×C3#6391611yescharacteristic
335D3—242no—
336D3—722no—
337C22—362no—
338C22—362no—
339C22—362no—
340C22C2×#620011yescharacteristic
341C4—21no—
342C4—182no—
343C4—361no—
344C4#8843812yescomplemented
345C4#8864911yescharacteristic
346C4#8865511yescharacteristic
347C3—31no—
348C3—31no—
349C3—61no—
350C3#1059037711yescharacteristic
351C2—722no—
352C2#1172107111yescharacteristic
353C2C2×#8843811yescharacteristic
354C2#1172115911yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#11721385C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
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
1. C4×#6229C2semidirect
3. C2×#88438C2semidirect
4. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
5. #11720308C2semidirect
25. C4⋊(C3⋊(C3⋊C8))D3semidirect
36. #6229D4semidirect
75. C4×H(3,𝔽3)SD16semidirect
184. C32×C4#204semidirect
185. C32×C4#214semidirect
201. H(3,𝔽3)#460semidirect
307. C32#4848semidirect
308. C32#5075semidirect
344. C4#88438semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
#117213852
C2×#884382
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203082
#105903673
#105903643