Finite Group Databasebeta

← #13137375

Subgroups — #13137375

Order 1728.

495 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 127 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#64081⋊C3C211yesmaximal, characteristic
2C2×#88430C211yesmaximal, characteristic
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
4#11720520C211yesmaximal, characteristic, complemented
5C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10590321—31nomaximal
9#10590319—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#3752⋊C3C412yes—
15#3750⋊C3C412yes—
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88430C414yescomplemented
20((C3⋊D3)⋊C3)⋊C8C414yescomplemented
21#64081D311yescharacteristic
22C3×#872—31no—
23C2×#3712—31no—
24C2×#3711—31no—
25C2×#3710—31no—
26C2×D3×C3⋊C8—31no—
27C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
28C3×#803—31no—
29C3×#803—31no—
30C3×#803—61no—
31C3×C4×C3⋊C8—31no—
32#63555—31no—
33#63554—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2×C412yes—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×(C3⋊Dic3)⋊C3C2×C412yes—
40C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
41C4×(C3⋊D3)⋊C3C2×C412yes—
42C4×(C3⋊D3)⋊C3C2×C412yes—
43#6180C2×C412yes—
44#6180C2×C412yes—
45(C3⋊(C3⋊C8))⋊C3C2×C412yes—
46(C3⋊(C3⋊C8))⋊C3C2×C412yes—
47#4997—91no—
48#4708—91no—
49#4689—91no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C2×C3×C4×D3—31no—
52C32×C42—21no—
53C32×C42C2×D311yescharacteristic
54C32×C42C2×D311yescharacteristic
55#3752Dic312yes—
56#3750Dic312yes—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
60C3×(C2×D3)⋊C4—32no—
61C3×Dic3⋊C4—32no—
62C3×C4×Dic3—31no—
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—61no—
70C2×C3×C3⋊C8—61no—
71#3712—34no—
72#3711—34no—
73#3710—34no—
74D3×C3⋊C8—34no—
75C22×H(3,𝔽3)C22×C411yescharacteristic
76C2×(C3⋊D3)⋊C3—22no—
77C2×(C3⋊D3)⋊C3C4212yes—
78C4×H(3,𝔽3)—22no—
79C4×H(3,𝔽3)C22×C411yescharacteristic
80C4×H(3,𝔽3)C22×C411yescharacteristic
81(C3⋊Dic3)⋊C3—22no—
82(C3⋊Dic3)⋊C3C4212yes—
83C3×#117—91no—
84C2×#208—91no—
85C22×C3⋊C8—91no—
86C2×ΓL(1,𝔽25)—91no—
87C2×ΓL(1,𝔽25)—91no—
88C2×D3×C8—91no—
89C2×D3×C8—91no—
90#872—31no—
91#872—91no—
92C3×#97—91no—
93C3×C4×C8—91no—
94#815—91no—
95#815—91no—
96C8×Dic3—91no—
97C8×Dic3—91no—
98#803—31no—
99#803—31no—
100#803—61no—
101#803—91no—
102C4×C3⋊C8—31no—
103C4×C3⋊C8—91no—
104C4×C3⋊C8—91no—
105C22×C3⋊D3C2×Dic311yescharacteristic
106C22×C3×D3—31no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4C4×D312yes—
110C2×C32×C4C2×Dic312yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3C2×Dic312yes—
114C2×C3⋊Dic3C2×Dic311yescharacteristic
115C4×C3⋊D3C4×D312yes—
116C4×C3⋊D3C4×D312yes—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—32no—
119C3×C4×D3—32no—
120C3×C4×D3—32no—
121C3⋊(C3⋊C8)C4×D312yes—
122C3⋊(C3⋊C8)C4×D312yes—
123C3×C3⋊C8—32no—
124C3×C3⋊C8—32no—
125C3×C3⋊C8—32no—
126C3×C3⋊C8—32no—
127C3×C3⋊C8—32no—
128C3×C3⋊C8—32no—
129C3×C3⋊C8—62no—
130C3×C3⋊C8—62no—
131#405—271noSylow
132C2×H(3,𝔽3)C2×C4211yescharacteristic
133C2×H(3,𝔽3)#13112yes—
134(C3⋊D3)⋊C3—22no—
135C22×C3×C4—91no—
136C2×C4×D3—31no—
137C2×C4×D3—91no—
138C2×C3×C8—91no—
139C2×C3×C8—91no—
140C2×C3×C8—91no—
141C2×C3×C8—91no—
142C3×C4⋊C4—92no—
143C3×C22⋊C4—92no—
144C3×C42—31no—
145C3×C42—31no—
146C3×C42—61no—
147C3×C42—91no—
148C3×C42D3211yescharacteristic
149(C2×D3)⋊C4—32no—
150(C2×D3)⋊C4—92no—
151Dic3⋊C4—32no—
152Dic3⋊C4—92no—
153C4×Dic3—31no—
154C4×Dic3—91no—
155#208—94no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—31no—
158C2×C3⋊C8—31no—
159C2×C3⋊C8—31no—
160C2×C3⋊C8—31no—
161C2×C3⋊C8—31no—
162C2×C3⋊C8—61no—
163C2×C3⋊C8—61no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—91no—
166C2×C3⋊C8—94no—
167ΓL(1,𝔽25)—94no—
168ΓL(1,𝔽25)—94no—
169D3×C8—94no—
170D3×C8—94no—
171C22×C32—21no—
172C22×C32C2×C4×D311yescharacteristic
173C22×C32C22×Dic311yescharacteristic
174C2×C3⋊D3—22no—
175C2×C3⋊D3C4×Dic312yes—
176C2×C3×D3—32no—
177C2×C3×D3—62no—
178C32×C4—22no—
179C32×C4—22no—
180C32×C4—21no—
181C32×C4—21no—
182C32×C4—42no—
183C32×C4C2×C4×D311yescharacteristic
184C32×C4C2×C4×D311yescharacteristic
185C32×C4C2×C4×D311yescharacteristic
186C32×C4C2×C4×D311yescharacteristic
187C3⋊Dic3—22no—
188C3⋊Dic3C4×Dic312yes—
189C3×Dic3—32no—
190C3×Dic3—62no—
191C2×M16—271no—
192C22×C8—271no—
193#117—91no—
194#97—91no—
195#97—271no—
196C4×C8—91no—
197C4×C8—271no—
198H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
199C23×C3—91no—
200C22×D3—31no—
201C22×D3—91no—
202C2×C3×C4—32no—
203C2×C3×C4—32no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—62no—
207C2×C3×C4—61no—
208C2×C3×C4—92no—
209C2×C3×C4—92no—
210C2×C3×C4—91no—
211C2×C3×C4—92no—
212C2×C3×C4D3×Dic312yes—
213C2×C3×C4C2×D3211yescharacteristic
214C2×Dic3—31no—
215C2×Dic3—32no—
216C2×Dic3—91no—
217C2×Dic3—92no—
218C4×D3—32no—
219C4×D3—32no—
220C4×D3—92no—
221C4×D3—92no—
222C3×C8—92no—
223C3×C8—92no—
224C3×C8—92no—
225C3×C8—92no—
226C3⋊C8—32no—
227C3⋊C8—32no—
228C3⋊C8—32no—
229C3⋊C8—32no—
230C3⋊C8—32no—
231C3⋊C8—32no—
232C3⋊C8—62no—
233C3⋊C8—62no—
234C3⋊C8—92no—
235C3⋊C8—92no—
236C2×C32—22no—
237C2×C32—21no—
238C2×C32C42×D311yescharacteristic
239C2×C32#90112yes—
240C2×C32C2×C4×Dic311yescharacteristic
241C2×C32#94812yes—
242C3⋊D3—22no—
243C3×D3—62no—
244C22×C4—91no—
245M16—274no—
246C2×C8—91no—
247C2×C8—91no—
248C2×C8—91no—
249C2×C8—91no—
250C2×C8—274no—
251C4⋊C4—92no—
252C22⋊C4—92no—
253C42—91no—
254C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—61no—
258C22×C3—92no—
259C22×C3—182no—
260C22×C3C2×D3×Dic311yescharacteristic
261C2×D3—32no—
262C2×D3—62no—
263C2×D3—92no—
264C2×D3—182no—
265C3×C4—22no—
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—62no—
271C3×C4—62no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—92no—
275C3×C4—122no—
276C3×C4—182no—
277C3×C4C4×D3211yescharacteristic
278C3×C4C4×D3211yescharacteristic
279Dic3—32no—
280Dic3—62no—
281Dic3—92no—
282Dic3—182no—
283C32—21no—
284C32#468911yescharacteristic, complemented
285C32#499711yescharacteristic, complemented
286C23—91no—
287C2×C4—91no—
288C2×C4—92no—
289C2×C4—92no—
290C2×C4—92no—
291C2×C4((C3⋊D3)⋊C3)⋊C412yes—
292C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
293C8—92no—
294C8—92no—
295C8—92no—
296C8—92no—
297C2×C3—32no—
298C2×C3—32no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—62no—
302C2×C3—61no—
303C2×C3—182no—
304C2×C3#6381412yes—
305C2×C3C4×D3×Dic311yescharacteristic
306D3—62no—
307D3—182no—
308C22—92no—
309C22—182no—
310C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
311C4—22no—
312C4—92no—
313C4—182no—
314C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
316C3—31no—
317C3—31no—
318C3—61no—
319C3#1059031811yescharacteristic
320C2—182no—
321C2#1172105412yes—
322C2C4×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#405
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
4. #11720520C2semidirect
5. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
6. #11720312C2semidirect
7. #11720310C2semidirect
19. #88430C4semidirect
20. ((C3⋊D3)⋊C3)⋊C8C4semidirect
27. C4×C3⋊(C3⋊C8)D3semidirect
198. H(3,𝔽3)#405semidirect
284. C32#4689semidirect
285. C32#4997semidirect

Maximal subgroups

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

subgroupindex
#64081⋊C32
C2×#884302
C2×((C3⋊D3)⋊C3)⋊C82
#117205202
C4×(C3⋊(C3⋊C8))⋊C32
#117203122
#117203102
#105903213
#105903193