Finite Group Databasebeta

← #13138769

Subgroups — #13138769

Order 1728.

541 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 131 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×#3743⋊C3C211yesmaximal, characteristic
2C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
3C2×#88430C211yesmaximal, characteristic
4C8×#1084C211yesmaximal, characteristic, complemented
5#63642⋊C3C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720302C211yesmaximal, characteristic, complemented
8#10590664—31nomaximal
9Dic3×ΓL(1,𝔽25)—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4C412yes—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4×#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#3743⋊C3C414yescomplemented
19#88447C412yes—
20#88430C414yescomplemented
21C2×#3743D311yescharacteristic
22C2×C3×ΓL(1,𝔽25)—31no—
23C4×#624—31no—
24C4×D3×Dic3—31no—
25C2×#3712—31no—
26C2×#3711—31no—
27#63642D311yescharacteristic, complemented
28C3×#815—31no—
29C3×C8×Dic3—31no—
30C3×C8×Dic3—31no—
31C3×C8×Dic3—61no—
32#63555—31no—
33#63554—31no—
34Dic3×C3⋊C8—31no—
35Dic3×C3⋊C8—31no—
36C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2×C412yes—
39C2×(C3⋊Dic3)⋊C3C2×C412yes—
40C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
41C4×(C3⋊D3)⋊C3C2×C412yes—
42C4×(C3⋊D3)⋊C3C2×C412yes—
43((C3⋊D3)⋊C3)⋊C4—24no—
44C8×H(3,𝔽3)C2×C412yescomplemented
45#6180C2×C412yescomplemented
46(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
47(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
48Dic3×M16—91no—
49#4707—91no—
50C4×ΓL(1,𝔽25)—91no—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×C3×C4×D3—31no—
53C2×#624—32no—
54C2×D3×Dic3—32no—
55C2×C32×C8—21no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C2×D311yescharacteristic
58C4×C3⋊Dic3C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61#3743Dic314yescomplemented
62C3×C4×Dic3—31no—
63C3×C4×Dic3—31no—
64C3×C4×Dic3—31no—
65C3×C4×Dic3—61no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—61no—
71C3×ΓL(1,𝔽25)—34no—
72Dic32—32no—
73Dic32—32no—
74#3712—34no—
75#3711—34no—
76C22×H(3,𝔽3)C22×C411yescharacteristic
77C2×(C3⋊D3)⋊C3—22no—
78C2×(C3⋊D3)⋊C3C4212yes—
79C4×H(3,𝔽3)C22×C411yescharacteristic
80C4×H(3,𝔽3)C22×C411yescharacteristic
81#1084M1614yescomplemented
82(C3⋊Dic3)⋊C3—22no—
83(C3⋊Dic3)⋊C3C4212yes—
84C2×C3×M16—91no—
85C2×C4×Dic3—91no—
86C2×#208—91no—
87C2×ΓL(1,𝔽25)—31no—
88C2×ΓL(1,𝔽25)—91no—
89C2×ΓL(1,𝔽25)—91no—
90C2×ΓL(1,𝔽25)—91no—
91C42×D3—91no—
92C42×D3—91no—
93C3×#97—91no—
94C3×C4×C8—91no—
95#815—31no—
96#815—91no—
97#815—91no—
98#815—91no—
99C8×Dic3—31no—
100C8×Dic3—31no—
101C8×Dic3—61no—
102C8×Dic3—91no—
103C8×Dic3—91no—
104#803—91no—
105C4×C3⋊C8—91no—
106C22×C3⋊D3C4×D311yescharacteristic
107C22×C3×D3—31no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3C4×D312yes—
112C2×C3⋊Dic3C4×D311yescharacteristic
113C4×C3⋊D3C4×D312yes—
114C4×C3⋊D3C2×Dic312yes—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—62no—
120C3×C4×D3—32no—
121C3×C4×D3—32no—
122#624—64no—
123D3×Dic3—64no—
124C32×C8—22no—
125C32×C8C4×D312yescomplemented
126C32×C8C2×Dic312yescomplemented
127C3⋊(C3⋊C8)C4×D312yescomplemented
128C3⋊(C3⋊C8)C2×Dic312yescomplemented
129C3×C3⋊C8—32no—
130C3×C3⋊C8—32no—
131C3×C3⋊C8—32no—
132C3×C3⋊C8—32no—
133C3×C3⋊C8—62no—
134C4×M16—271noSylow
135C2×H(3,𝔽3)C2×C4211yescharacteristic
136C2×H(3,𝔽3)C2×M1612yes—
137(C3⋊D3)⋊C3—22no—
138C22×C3×C4—91no—
139C22×Dic3—92no—
140C2×C4×D3—31no—
141C2×C4×D3—91no—
142C2×C4×D3—92no—
143C2×C4×D3—92no—
144C3×M16—94no—
145C2×C3×C8—31no—
146C2×C3×C8—31no—
147C2×C3×C8—61no—
148C2×C3×C8—91no—
149C2×C3×C8—91no—
150C2×C3×C8—91no—
151C2×C3×C8D3211yescharacteristic
152C3×C42—91no—
153C3×C42—91no—
154C4×Dic3—31no—
155C4×Dic3—31no—
156C4×Dic3—31no—
157C4×Dic3—61no—
158C4×Dic3—91no—
159C4×Dic3—92no—
160C4×Dic3—92no—
161C4×Dic3—92no—
162#208—94no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—31no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8—31no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—91no—
170ΓL(1,𝔽25)—34no—
171ΓL(1,𝔽25)—94no—
172ΓL(1,𝔽25)—94no—
173ΓL(1,𝔽25)—94no—
174C22×C32—21no—
175C22×C32C2×C4×D311yescharacteristic
176C22×C32C2×C4×D311yescharacteristic
177C2×C3⋊D3—22no—
178C2×C3⋊D3C4×Dic312yes—
179C2×C3×D3—32no—
180C2×C3×D3—62no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4C2×C4×D311yescharacteristic
184C32×C4C2×C4×D311yescharacteristic
185C32×C4C2×C4×D311yescharacteristic
186C32×C4C22×Dic311yescharacteristic
187C3⋊Dic3—22no—
188C3⋊Dic3C4×Dic312yes—
189C3×Dic3—32no—
190C3×Dic3—34no—
191C3×Dic3—34no—
192C3×Dic3—64no—
193C3×Dic3—62no—
194C2×M16—91no—
195C2×M16—271no—
196C2×C42—271no—
197#97—91no—
198#97—271no—
199C4×C8—91no—
200C4×C8—271no—
201H(3,𝔽3)C4×M1611yescharacteristic, complemented, Sylow
202C23×C3—91no—
203C22×D3—31no—
204C22×D3—91no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—61no—
208C2×C3×C4—92no—
209C2×C3×C4—92no—
210C2×C3×C4—91no—
211C2×C3×C4—92no—
212C2×C3×C4—92no—
213C2×C3×C4C2×D3211yescharacteristic
214C2×Dic3—32no—
215C2×Dic3—32no—
216C2×Dic3—31no—
217C2×Dic3—32no—
218C2×Dic3—62no—
219C2×Dic3—91no—
220C2×Dic3—92no—
221C2×Dic3—184no—
222C4×D3—32no—
223C4×D3—32no—
224C4×D3—92no—
225C4×D3—92no—
226C4×D3—184no—
227C4×D3—184no—
228C3×C8—32no—
229C3×C8—32no—
230C3×C8—62no—
231C3×C8—92no—
232C3×C8—92no—
233C3×C8—92no—
234C3×C8D3×Dic312yes—
235C3⋊C8—32no—
236C3⋊C8—32no—
237C3⋊C8—32no—
238C3⋊C8—32no—
239C3⋊C8—62no—
240C3⋊C8—92no—
241C3⋊C8—92no—
242C2×C32—22no—
243C2×C32—21no—
244C2×C32C42×D311yescharacteristic
245C2×C32D3×M1612yes—
246C2×C32D3×M1612yes—
247C2×C32C2×C4×Dic311yescharacteristic
248C3⋊D3—22no—
249C3×D3—62no—
250C22×C4—91no—
251C22×C4—272no—
252M16—94no—
253M16—274no—
254C2×C8—91no—
255C2×C8—91no—
256C2×C8—91no—
257C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C42—91no—
259C42—91no—
260C42—272no—
261C22×C3—31no—
262C22×C3—31no—
263C22×C3—61no—
264C22×C3—92no—
265C22×C3—182no—
266C22×C3C4×D3211yescharacteristic
267C2×D3—32no—
268C2×D3—62no—
269C2×D3—92no—
270C2×D3—182no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—31no—
275C3×C4—61no—
276C3×C4—61no—
277C3×C4—92no—
278C3×C4—94no—
279C3×C4—182no—
280C3×C4C4×D3211yescharacteristic
281C3×C4C2×D3×Dic311yescharacteristic
282Dic3—34no—
283Dic3—34no—
284Dic3—32no—
285Dic3—64no—
286Dic3—62no—
287Dic3—92no—
288Dic3—182no—
289C32—21no—
290C32#470711yescharacteristic, complemented
291C32Dic3×M1611yescharacteristic, complemented
292C23—91no—
293C2×C4—92no—
294C2×C4—91no—
295C2×C4—92no—
296C2×C4—92no—
297C2×C4—92no—
298C2×C4—544no—
299C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C8—92no—
301C8—92no—
302C8—92no—
303C8((C3⋊D3)⋊C3)⋊C412yescomplemented
304C2×C3—32no—
305C2×C3—32no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—62no—
309C2×C3—61no—
310C2×C3—182no—
311C2×C3#6379112yes—
312C2×C3C4×D3×Dic311yescharacteristic
313D3—62no—
314D3—182no—
315C22—92no—
316C22—182no—
317C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
318C4—94no—
319C4—92no—
320C4—182no—
321C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
322C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
323C3—31no—
324C3—31no—
325C3—61no—
326C3#1059066711yescharacteristic
327C2—182no—
328C2#1172083612yes—
329C2C4×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×M16
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C8×#1084C2semidirect
5. #63642⋊C3C2semidirect
6. #11720312C2semidirect
7. #11720302C2semidirect
18. #3743⋊C3C4semidirect
20. #88430C4semidirect
27. #63642D3semidirect
44. C8×H(3,𝔽3)C2×C4semidirect
45. #6180C2×C4semidirect
46. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
47. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
61. #3743Dic3semidirect
81. #1084M16semidirect
125. C32×C8C4×D3semidirect
126. C32×C8C2×Dic3semidirect
127. C3⋊(C3⋊C8)C4×D3semidirect
128. C3⋊(C3⋊C8)C2×Dic3semidirect
201. H(3,𝔽3)C4×M16semidirect
290. C32#4707semidirect
291. C32Dic3×M16semidirect
303. C8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
C4×((C3⋊D3)⋊C3)⋊C42
C2×#884302
C8×#10842
#63642⋊C32
#117203122
#117203022
#105906643
Dic3×ΓL(1,𝔽25)3