Finite Group Databasebeta

← #13138950

Subgroups — #13138950

Order 1728.

454 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×#88529C211yesmaximal, characteristic, complemented
2#64110⋊C3C211yesmaximal, characteristic, complemented
3#11721068C212yesmaximal, complemented
4#63646⋊C3C211yesmaximal, characteristic, complemented
5#11720325C212yesmaximal, complemented
6#10590929—31nomaximal
7#10590926—31nomaximal
8C2×C4⋊#272C2212yes—
9#3828⋊C3C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88529—21no—
12#88529—21no—
13#3748⋊C3C2211yescharacteristic
14#3748⋊C3C2212yes—
15#3743⋊C3—21no—
16C8×(C3⋊D3)⋊C3—21no—
17#88437—22no—
18#88436—22no—
19#64110D311yescharacteristic, complemented
20C2×C3×D24—31no—
21C2×C3×D24—31no—
22C2×C3×D24—61no—
23C3×#901—31no—
24#63830—32no—
25#63823—32no—
26#63646D311yescharacteristic, complemented
27C3×#819—31no—
28#63558—32no—
29#63558—32no—
30C22×#272—22no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C4⋊#272—22no—
33C4⋊#272—22no—
34#638⋊C3—21no—
35C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
36C4×(C3⋊D3)⋊C3—21no—
37#634⋊C3D411yescharacteristic
38C8×H(3,𝔽3)D411yescharacteristic
39C8×H(3,𝔽3)D411yescharacteristic
40(C3⋊(C3⋊C8))⋊C3—22no—
41(C3⋊(C3⋊C8))⋊C3—21no—
42#5160—91no—
43#5142—91no—
44#4897—91no—
45#3828C2×D311yescharacteristic
46C3×#235—31no—
47C2×C3×D12—32no—
48C2×C3×D12—32no—
49C2×C3×D12—62no—
50C2×C32×C8—21no—
51C2×C32×C8C2×D311yescharacteristic
52C2×C32×C8C2×D311yescharacteristic
53#3748C2×D311yescharacteristic
54#3748C2×D312yes—
55#3743—21no—
56C8×C3⋊D3—21no—
57C3×#208—31no—
58C3×#208—32no—
59C3×D24—62no—
60C3×D24—62no—
61C3×D24—61no—
62C3×D24—61no—
63C3×D24—61no—
64C3×D24—61no—
65C3×ΓL(1,𝔽25)—61no—
66C3×D3×C8—61no—
67#3716—62no—
68#3715—62no—
69#3714—62no—
70#3713—62no—
71C22×H(3,𝔽3)#4111yescharacteristic
72C2×#272—42no—
73C2×#272—82no—
74C2×(C3⋊D3)⋊C3—21no—
75C4×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)C2×D411yescharacteristic
77(C3⋊Dic3)⋊C3—21no—
78C2×C3×D8—91no—
79C3×#131—91no—
80#949—92no—
81#932—92no—
82#908—92no—
83C2×D24—31no—
84C2×D24—31no—
85C2×D24—61no—
86#901—31no—
87#901—91no—
88C3×#108—91no—
89#833—92no—
90#823—92no—
91#823—92no—
92#819—31no—
93#819—91no—
94C22×C3×D3—62no—
95C22×C3×D3—62no—
96C22×C3×D3—122no—
97C2×C32×C4—21no—
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100#638—21no—
101C4⋊(C3⋊D3)D1211yescharacteristic
102C4×C3⋊D3—21no—
103#634D1211yescharacteristic
104C3×#67—61no—
105C3×D12—31no—
106C3×D12—64no—
107C3×D12—62no—
108C3×D12—62no—
109C3×D12—62no—
110C3×D12—62no—
111C3×D12—122no—
112C3×C4×D3—61no—
113C3×Dic6—31no—
114C32×C8—21no—
115C32×C8—21no—
116C32×C8D1211yescharacteristic
117C32×C8D1211yescharacteristic
118C32×C8#6711yescharacteristic
119C32×C8#6711yescharacteristic
120C3⋊(C3⋊C8)—22no—
121C3⋊(C3⋊C8)—21no—
122C3×C3⋊C8—62no—
123C3×C3⋊C8—61no—
124#472—271noSylow
125C2×H(3,𝔽3)—21no—
126C2×H(3,𝔽3)#12111yescharacteristic
127#272—82no—
128(C3⋊D3)⋊C3—41no—
129C3×#41—91no—
130C2×C3×D4—92no—
131#235—31no—
132#235—91no—
133C2×D12—32no—
134C2×D12—32no—
135C2×D12—62no—
136C3×D8—181no—
137C3×D8—181no—
138C3×M16—91no—
139C3×M16—92no—
140C3×M16—181no—
141C2×C3×C8—31no—
142C2×C3×C8—31no—
143C2×C3×C8—61no—
144C2×C3×C8—181no—
145C2×C3×C8D3211yescharacteristic
146#215—182no—
147#214—182no—
148#213—182no—
149#213—182no—
150#208—32no—
151#208—31no—
152#208—92no—
153#208—91no—
154D24—61no—
155D24—61no—
156D24—61no—
157D24—61no—
158D24—62no—
159D24—62no—
160D24—182no—
161#204—182no—
162ΓL(1,𝔽25)—61no—
163ΓL(1,𝔽25)—181no—
164D3×C8—61no—
165D3×C8—181no—
166C22×C32—21no—
167C22×C32#23511yescharacteristic
168C22×C32#23911yescharacteristic
169C2×C3⋊D3—21no—
170C2×C3×D3—61no—
171C2×C3×D3—124no—
172C2×C3×D3—122no—
173C2×C3×D3—122no—
174C2×C3×D3—244no—
175C2×C3×D3—242no—
176C2×C3×D3—242no—
177C32×C4—21no—
178C32×C4—21no—
179C32×C4C2×D1211yescharacteristic
180C32×C4C2×D1211yescharacteristic
181C32×C4D3×D411yescharacteristic
182C32×C4C2×#6711yescharacteristic
183C3⋊Dic3—21no—
184C3×Dic3—61no—
185AGL(1,ℤ/8)—272no—
186C2×D8—91no—
187#131—91no—
188#108—91no—
189#100—272no—
190H(3,𝔽3)#47211yescharacteristic, complemented, Sylow
191C23×C3—182no—
192C22×D3—62no—
193C22×D3—62no—
194C22×D3—122no—
195C3×Q8—91no—
196C3×D4—91no—
197C3×D4—181no—
198C3×D4—182no—
199C3×D4—182no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—61no—
203C2×C3×C4—181no—
204C2×C3×C4C2×D3211yescharacteristic
205#67—61no—
206#67—181no—
207D12—31no—
208D12—62no—
209D12—62no—
210D12—62no—
211D12—62no—
212D12—64no—
213D12—91no—
214D12—122no—
215C4×D3—61no—
216C4×D3—181no—
217Dic6—31no—
218Dic6—91no—
219C3×C8—31no—
220C3×C8—31no—
221C3×C8—31no—
222C3×C8—31no—
223C3×C8—61no—
224C3×C8—61no—
225C3×C8—182no—
226C3×C8—181no—
227C3×C8#62611yescharacteristic
228C3×C8#62611yescharacteristic
229C3⋊C8—62no—
230C3⋊C8—61no—
231C3⋊C8—181no—
232C3⋊C8—182no—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32#89511yescharacteristic
238C2×C32#93011yescharacteristic
239C3⋊D3—41no—
240C3×D3—121no—
241C3×D3—244no—
242C3×D3—242no—
243C3×D3—242no—
244#41—91no—
245C2×D4—92no—
246SD16—542no—
247D8—181no—
248D8—181no—
249D8—542no—
250M16—91no—
251M16—92no—
252M16—181no—
253C2×C8—181no—
254C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—61no—
258C22×C3—181no—
259C22×C3—362no—
260C22×C3—722no—
261C22×C3#379911yescharacteristic
262C2×D3—61no—
263C2×D3—122no—
264C2×D3—122no—
265C2×D3—124no—
266C2×D3—181no—
267C2×D3—244no—
268C2×D3—242no—
269C2×D3—242no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4—181no—
277C3×C4D3×D1211yescharacteristic
278C3×C4C2×#62611yescharacteristic
279Dic3—61no—
280Dic3—181no—
281C32—21no—
282C32#489711yescharacteristic, complemented
283C32#514211yescharacteristic, complemented
284C23—182no—
285Q8—91no—
286D4—91no—
287D4—181no—
288D4—182no—
289D4—182no—
290C2×C4—181no—
291C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C8—182no—
293C8—181no—
294C8#620011yescharacteristic
295C8#620011yescharacteristic
296C2×C3—21no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—61no—
300C2×C3—61no—
301C2×C3—62no—
302C2×C3—61no—
303C2×C3—361no—
304C2×C3—722no—
305C2×C3#6390911yescharacteristic
306D3—121no—
307D3—242no—
308D3—242no—
309D3—244no—
310D3—361no—
311C22—181no—
312C22—362no—
313C22—722no—
314C22#8865211yescharacteristic
315C4—181no—
316C4#8865411yescharacteristic
317C4C2×#620011yescharacteristic
318C3—31no—
319C3—31no—
320C3—61no—
321C3#1059092711yescharacteristic
322C2—21no—
323C2—361no—
324C2—722no—
325C2#1172115611yescharacteristic
subgroupG/H
centerC2#11721156
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3#63909
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#472
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88529C2semidirect
2. #64110⋊C3C2semidirect
3. #11721068C2semidirect
4. #63646⋊C3C2semidirect
5. #11720325C2semidirect
19. #64110D3semidirect
26. #63646D3semidirect
190. H(3,𝔽3)#472semidirect
282. C32#4897semidirect
283. C32#5142semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
#64110⋊C32
#117210682
#63646⋊C32
#117203252
#105909293
#105909263