Finite Group Databasebeta

← #13138899

Subgroups — #13138899

Order 1728.

565 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 123 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×#88527C211yesmaximal, characteristic
2C4×#6199C211yesmaximal, characteristic
3C2×#88435C211yesmaximal, characteristic
4#63642⋊C3C211yesmaximal, characteristic, complemented
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10590667—31nomaximal
9Dic3×ΓL(1,𝔽25)—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#6199C412yes—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#88527C414yescomplemented
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88447C412yes—
20#88435C414yescomplemented
21C2×C3×ΓL(1,𝔽25)—31no—
22C2×C3×ΓL(1,𝔽25)—31no—
23C2×C3×ΓL(1,𝔽25)—61no—
24C4×D3×Dic3—31no—
25C4×D3×Dic3—31no—
26C2×#3711—31no—
27C2×#3711—31no—
28#63642D311yescharacteristic, complemented
29C8×C3⋊Dic3D311yescharacteristic, complemented
30C3×#815—31no—
31C3×C8×Dic3—31no—
32#63555—31no—
33#63554—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C22×#272C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C4×#272C2×C412yes—
40C4×#272C2×C412yes—
41C2×(C3⋊Dic3)⋊C3C2×C412yes—
42C2×(C3⋊Dic3)⋊C3C2×C412yes—
43#6199—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×C3×C4×D3—31no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—61no—
54C2×D3×Dic3—32no—
55C2×D3×Dic3—32no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C2×D311yescharacteristic
59C4×C3⋊Dic3C2×D311yescharacteristic
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×C4×Dic3—31no—
64C3×C4×Dic3—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—
70C3×ΓL(1,𝔽25)—34no—
71C3×ΓL(1,𝔽25)—34no—
72C3×ΓL(1,𝔽25)—64no—
73Dic32—32no—
74Dic32—32no—
75#3711—34no—
76#3711—34no—
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C2×#272—22no—
79C2×#272C4212yes—
80C4×H(3,𝔽3)C22×C411yescharacteristic
81C4×H(3,𝔽3)C22×C411yescharacteristic
82#1084C4212yes—
83(C3⋊Dic3)⋊C3—22no—
84(C3⋊Dic3)⋊C3M1614yescomplemented
85C2×C3×M16—91no—
86C2×C4×Dic3—91no—
87C2×#208—91no—
88C2×ΓL(1,𝔽25)—31no—
89C2×ΓL(1,𝔽25)—31no—
90C2×ΓL(1,𝔽25)—61no—
91C2×ΓL(1,𝔽25)—91no—
92C2×ΓL(1,𝔽25)—91no—
93C42×D3—91no—
94C42×D3—91no—
95C3×#97—91no—
96C3×C4×C8—91no—
97#815—31no—
98#815—91no—
99#815—91no—
100#815—91no—
101C8×Dic3—31no—
102C8×Dic3—91no—
103C8×Dic3—91no—
104C8×Dic3—91no—
105#803—91no—
106C4×C3⋊C8—91no—
107C22×C3×D3—31no—
108C22×C3×D3—31no—
109C22×C3×D3—61no—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3C4×D312yes—
114C2×C3⋊Dic3C4×D312yes—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—61no—
120C3×C4×D3—32no—
121C3×C4×D3—32no—
122C3×C4×D3—32no—
123C3×C4×D3—32no—
124C3×C4×D3—62no—
125C3×C4×D3—62no—
126D3×Dic3—64no—
127D3×Dic3—64no—
128C32×C8—22no—
129C32×C8C4×D312yescomplemented
130C32×C8C4×D312yescomplemented
131C3⋊(C3⋊C8)C4×D312yescomplemented
132C3⋊(C3⋊C8)C4×D312yescomplemented
133C3×C3⋊C8—32no—
134C3×C3⋊C8—32no—
135C3×C3⋊C8—32no—
136C3×C3⋊C8—32no—
137C3×C3⋊C8—62no—
138C4×M16—271noSylow
139C2×H(3,𝔽3)C2×C4211yescharacteristic
140C2×H(3,𝔽3)C2×M1612yes—
141#272—22no—
142C22×C3×C4—91no—
143C22×Dic3—92no—
144C2×C4×D3—31no—
145C2×C4×D3—31no—
146C2×C4×D3—61no—
147C2×C4×D3—92no—
148C2×C4×D3—92no—
149C3×M16—94no—
150C2×C3×C8—31no—
151C2×C3×C8—31no—
152C2×C3×C8—61no—
153C2×C3×C8—91no—
154C2×C3×C8—91no—
155C2×C3×C8—91no—
156C2×C3×C8D3211yescharacteristic
157C3×C42—91no—
158C3×C42—91no—
159C4×Dic3—31no—
160C4×Dic3—31no—
161C4×Dic3—91no—
162C4×Dic3—91no—
163C4×Dic3—92no—
164C4×Dic3—92no—
165C4×Dic3—92no—
166#208—94no—
167C2×C3⋊C8—31no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8—31no—
171C2×C3⋊C8—61no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—91no—
174ΓL(1,𝔽25)—34no—
175ΓL(1,𝔽25)—34no—
176ΓL(1,𝔽25)—64no—
177ΓL(1,𝔽25)—94no—
178ΓL(1,𝔽25)—94no—
179C22×C32—21no—
180C22×C32C2×C4×D311yescharacteristic
181C22×C32C2×C4×D311yescharacteristic
182C2×C3×D3—32no—
183C2×C3×D3—32no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—62no—
187C2×C3×D3—122no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4C2×C4×D311yescharacteristic
191C32×C4C2×C4×D311yescharacteristic
192C32×C4C2×C4×D311yescharacteristic
193C32×C4C2×C4×D311yescharacteristic
194C3⋊Dic3—22no—
195C3⋊Dic3ΓL(1,𝔽25)14yescomplemented
196C3×Dic3—32no—
197C3×Dic3—32no—
198C3×Dic3—34no—
199C3×Dic3—62no—
200C3×Dic3—62no—
201C2×M16—91no—
202C2×M16—271no—
203C2×C42—271no—
204#97—91no—
205#97—271no—
206C4×C8—91no—
207C4×C8—271no—
208H(3,𝔽3)C4×M1611yescharacteristic, complemented, Sylow
209C23×C3—91no—
210C22×D3—31no—
211C22×D3—31no—
212C22×D3—61no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—61no—
216C2×C3×C4—92no—
217C2×C3×C4—91no—
218C2×C3×C4—92no—
219C2×C3×C4—92no—
220C2×C3×C4—92no—
221C2×C3×C4C2×D3211yescharacteristic
222C2×Dic3—31no—
223C2×Dic3—31no—
224C2×Dic3—32no—
225C2×Dic3—32no—
226C2×Dic3—61no—
227C2×Dic3—92no—
228C2×Dic3—92no—
229C2×Dic3—184no—
230C4×D3—32no—
231C4×D3—32no—
232C4×D3—32no—
233C4×D3—32no—
234C4×D3—62no—
235C4×D3—62no—
236C4×D3—184no—
237C4×D3—184no—
238C3×C8—32no—
239C3×C8—32no—
240C3×C8—62no—
241C3×C8—92no—
242C3×C8—92no—
243C3×C8—92no—
244C3×C8#62412yes—
245C3⋊C8—32no—
246C3⋊C8—32no—
247C3⋊C8—32no—
248C3⋊C8—32no—
249C3⋊C8—62no—
250C3⋊C8—92no—
251C3⋊C8—92no—
252C2×C32—22no—
253C2×C32—21no—
254C2×C32C42×D311yescharacteristic
255C2×C32C42×D311yescharacteristic
256C2×C32C2×ΓL(1,𝔽25)12yes—
257C2×C32D3×M1612yes—
258C3×D3—62no—
259C3×D3—62no—
260C3×D3—122no—
261C22×C4—91no—
262C22×C4—272no—
263M16—94no—
264M16—274no—
265C2×C8—91no—
266C2×C8—91no—
267C2×C8—91no—
268C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C42—91no—
270C42—91no—
271C42—272no—
272C22×C3—31no—
273C22×C3—31no—
274C22×C3—61no—
275C22×C3—92no—
276C22×C3—182no—
277C22×C3C4×D3211yescharacteristic
278C2×D3—32no—
279C2×D3—32no—
280C2×D3—62no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—122no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—61no—
289C3×C4—61no—
290C3×C4—94no—
291C3×C4—92no—
292C3×C4—182no—
293C3×C4C4×D3211yescharacteristic
294C3×C4C2×#62411yescharacteristic
295Dic3—32no—
296Dic3—32no—
297Dic3—34no—
298Dic3—62no—
299Dic3—62no—
300Dic3—94no—
301Dic3—182no—
302C32—21no—
303C32C4×ΓL(1,𝔽25)11yescharacteristic, complemented
304C32#470711yescharacteristic, complemented
305C23—91no—
306C2×C4—92no—
307C2×C4—91no—
308C2×C4—92no—
309C2×C4—92no—
310C2×C4—92no—
311C2×C4—544no—
312C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
313C8—92no—
314C8—92no—
315C8—92no—
316C8#619912yescomplemented
317C2×C3—32no—
318C2×C3—32no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—62no—
322C2×C3—61no—
323C2×C3—182no—
324C2×C3D3×ΓL(1,𝔽25)12yes—
325C2×C3C4×#62411yescharacteristic
326D3—62no—
327D3—62no—
328D3—122no—
329C22—92no—
330C22—182no—
331C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
332C4—94no—
333C4—92no—
334C4—182no—
335C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
336C4C2×#619911yescharacteristic
337C3—31no—
338C3—31no—
339C3—61no—
340C3#1059066411yescharacteristic
341C2—182no—
342C2#1172083512yes—
343C2C4×#619911yescharacteristic
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×#272C2×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. #63642⋊C3C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
6. #11720312C2semidirect
7. #11720310C2semidirect
14. #88527C4semidirect
20. #88435C4semidirect
28. #63642D3semidirect
29. C8×C3⋊Dic3D3semidirect
44. C8×H(3,𝔽3)C2×C4semidirect
45. #6180C2×C4semidirect
46. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
47. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
84. (C3⋊Dic3)⋊C3M16semidirect
129. C32×C8C4×D3semidirect
130. C32×C8C4×D3semidirect
131. C3⋊(C3⋊C8)C4×D3semidirect
132. C3⋊(C3⋊C8)C4×D3semidirect
195. C3⋊Dic3ΓL(1,𝔽25)semidirect
208. H(3,𝔽3)C4×M16semidirect
303. C32C4×ΓL(1,𝔽25)semidirect
304. C32#4707semidirect
316. C8#6199semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
C4×#61992
C2×#884352
#63642⋊C32
C8×(C3⋊Dic3)⋊C32
#117203122
#117203102
#105906673
Dic3×ΓL(1,𝔽25)3