Finite Group Databasebeta

← #13138993

Subgroups — #13138993

Order 1728.

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

normal subgroups only

This group has 410 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×#88530C211yesmaximal, characteristic, complemented
2#64110⋊C3C211yesmaximal, characteristic
3#11721074C212yesmaximal
4#63646⋊C3C211yesmaximal, characteristic, complemented
5#11720327C212yesmaximal, complemented
6#10590933—31nomaximal
7#10590930—31nomaximal
8C2×#6229C2212yes—
9#3828⋊C3C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88530—21no—
12#88530—21no—
13#3748⋊C3C2211yescharacteristic
14#3748⋊C3C2212yes—
15#3743⋊C3—21no—
16C8×(C3⋊D3)⋊C3—21no—
17#88439—22no—
18#88438—22no—
19#64110D311yescharacteristic
20C2×C3×Dic12—31no—
21C2×C3×Dic12—31no—
22C2×C3×Dic12—61no—
23C3×#901—31no—
24#63833—32no—
25#63827—32no—
26#63646D311yescharacteristic, complemented
27C3×#819—31no—
28#63560—32no—
29#63560—32no—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—22no—
32#6229—22no—
33#6229—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#5192—91no—
43#5143—91no—
44#4896—91no—
45#3828C2×D311yescharacteristic
46C3×#235—31no—
47C2×C3×Dic6—32no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—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×Dic12—62no—
60C3×Dic12—62no—
61C3×Dic12—61no—
62C3×Dic12—61no—
63C3×Dic12—61no—
64C3×Dic12—61no—
65C3×ΓL(1,𝔽25)—61no—
66C3×D3×C8—61no—
67#3719—62no—
68#3718—62no—
69#3717—62no—
70#3715—62no—
71C22×H(3,𝔽3)#4111yescharacteristic
72C2×(C3⋊D3)⋊C3—21no—
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)C2×D411yescharacteristic
75#1084—42no—
76(C3⋊Dic3)⋊C3—21no—
77C2×C3×Q16—91no—
78C3×#131—91no—
79#951—92no—
80#942—92no—
81#909—92no—
82C2×Dic12—31no—
83C2×Dic12—31no—
84C2×Dic12—61no—
85#901—31no—
86#901—91no—
87C3×#108—91no—
88#836—92no—
89#824—92no—
90#824—92no—
91#819—31no—
92#819—91no—
93C2×C32×C4—21no—
94C2×C32×C4C22×D311yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
96#638—21no—
97C4⋊(C3⋊D3)D1211yescharacteristic
98C4×C3⋊D3—21no—
99#634D1211yescharacteristic
100C3×#67—61no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—122no—
104C3×D12—31no—
105C3×C4×D3—61no—
106C3×Dic6—31no—
107C3×Dic6—64no—
108C3×Dic6—62no—
109C3×Dic6—62no—
110C3×Dic6—62no—
111C3×Dic6—62no—
112C3×Dic6—122no—
113C32×C8—21no—
114C32×C8—21no—
115C32×C8D1211yescharacteristic
116C32×C8D1211yescharacteristic
117C32×C8#6711yescharacteristic
118C32×C8#6711yescharacteristic
119C3⋊(C3⋊C8)—22no—
120C3⋊(C3⋊C8)—21no—
121C3×C3⋊C8—62no—
122C3×C3⋊C8—61no—
123#473—271noSylow
124C2×H(3,𝔽3)—21no—
125C2×H(3,𝔽3)#12111yescharacteristic
126(C3⋊D3)⋊C3—41no—
127C3×#41—91no—
128C2×C3×Q8—92no—
129#235—31no—
130#235—91no—
131C2×Dic6—32no—
132C2×Dic6—32no—
133C2×Dic6—62no—
134C3×Q16—181no—
135C3×Q16—181no—
136C3×M16—91no—
137C3×M16—92no—
138C3×M16—181no—
139C2×C3×C8—31no—
140C2×C3×C8—31no—
141C2×C3×C8—61no—
142C2×C3×C8—181no—
143C2×C3×C8D3211yescharacteristic
144#216—182no—
145#216—182no—
146#215—182no—
147#214—182no—
148#208—32no—
149#208—31no—
150#208—92no—
151#208—91no—
152Dic12—61no—
153Dic12—61no—
154Dic12—61no—
155Dic12—61no—
156Dic12—62no—
157Dic12—62no—
158Dic12—182no—
159#204—182no—
160ΓL(1,𝔽25)—61no—
161ΓL(1,𝔽25)—181no—
162D3×C8—61no—
163D3×C8—181no—
164C22×C32—21no—
165C22×C32#23511yescharacteristic
166C22×C32#23911yescharacteristic
167C2×C3⋊D3—21no—
168C2×C3×D3—61no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4C2×D1211yescharacteristic
172C32×C4C2×D1211yescharacteristic
173C32×C4D3×D411yescharacteristic
174C32×C4C2×#6711yescharacteristic
175C3⋊Dic3—21no—
176C3×Dic3—61no—
177C3×Dic3—124no—
178C3×Dic3—122no—
179C3×Dic3—122no—
180#137—272no—
181C2×Q16—91no—
182#131—91no—
183#108—91no—
184#101—272no—
185H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
186C3×Q8—91no—
187C3×Q8—182no—
188C3×Q8—182no—
189C3×D4—91no—
190C3×D4—181no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—61no—
194C2×C3×C4—181no—
195C2×C3×C4—182no—
196C2×C3×C4C2×D3211yescharacteristic
197#67—61no—
198#67—181no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—122no—
202D12—31no—
203D12—91no—
204C4×D3—61no—
205C4×D3—181no—
206Dic6—31no—
207Dic6—62no—
208Dic6—62no—
209Dic6—62no—
210Dic6—62no—
211Dic6—64no—
212Dic6—91no—
213Dic6—122no—
214C3×C8—31no—
215C3×C8—31no—
216C3×C8—31no—
217C3×C8—31no—
218C3×C8—61no—
219C3×C8—61no—
220C3×C8—182no—
221C3×C8—181no—
222C3×C8#62611yescharacteristic
223C3×C8#62611yescharacteristic
224C3⋊C8—62no—
225C3⋊C8—61no—
226C3⋊C8—181no—
227C3⋊C8—182no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32#89511yescharacteristic
233C2×C32#93011yescharacteristic
234C3⋊D3—41no—
235C3×D3—121no—
236#41—91no—
237C2×Q8—92no—
238Q16—181no—
239Q16—181no—
240Q16—542no—
241SD16—542no—
242M16—91no—
243M16—92no—
244M16—181no—
245C2×C8—181no—
246C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—61no—
250C22×C3—181no—
251C22×C3#379911yescharacteristic
252C2×D3—61no—
253C2×D3—181no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—31no—
257C3×C4—31no—
258C3×C4—61no—
259C3×C4—61no—
260C3×C4—181no—
261C3×C4—362no—
262C3×C4D3×D1211yescharacteristic
263C3×C4C2×#62611yescharacteristic
264Dic3—61no—
265Dic3—122no—
266Dic3—122no—
267Dic3—124no—
268Dic3—181no—
269C32—21no—
270C32#489611yescharacteristic, complemented
271C32#514311yescharacteristic, complemented
272Q8—91no—
273Q8—182no—
274Q8—182no—
275D4—91no—
276D4—181no—
277C2×C4—181no—
278C2×C4—182no—
279C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
280C8—182no—
281C8—181no—
282C8#620011yescharacteristic
283C8#620011yescharacteristic
284C2×C3—21no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—61no—
288C2×C3—61no—
289C2×C3—62no—
290C2×C3—61no—
291C2×C3—361no—
292C2×C3#6390911yescharacteristic
293D3—121no—
294D3—361no—
295C22—181no—
296C22#8865211yescharacteristic
297C4—181no—
298C4—362no—
299C4#8865411yescharacteristic
300C4C2×#620011yescharacteristic
301C3—31no—
302C3—31no—
303C3—61no—
304C3#1059093211yescharacteristic
305C2—21no—
306C2—361no—
307C2#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
prime radical#3828⋊C3C22
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#473
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×#88530C2semidirect
4. #63646⋊C3C2semidirect
5. #11720327C2semidirect
26. #63646D3semidirect
185. H(3,𝔽3)#473semidirect
270. C32#4896semidirect
271. C32#5143semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
#64110⋊C32
#117210742
#63646⋊C32
#117203272
#105909333
#105909303