Finite Group Databasebeta

← #13139028

Subgroups — #13139028

Order 1728.

783 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 107 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×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic, complemented
3C2×#88434C211yesmaximal, characteristic, complemented
4#63647⋊C3C212yesmaximal, complemented
5#11720317C212yesmaximal, complemented
6D3×(C2×D3)⋊C8—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2C411yescharacteristic
8C2×C4×#272C2211yescharacteristic
9C2×C4×(C3⋊D3)⋊C3C2212yes—
10C2×#6199C411yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4C412yes—
12C4×((C3⋊D3)⋊C3)⋊C2—22no—
13C4×((C3⋊D3)⋊C3)⋊C2—22no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×#6180C2211yescharacteristic, complemented
16C8×#272—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18#88434—22no—
19C2×C4×D32—32no—
20C2×C3×D3×C8—32no—
21C2×C3×D3×C8—61no—
22C2×D3×C3⋊C8—32no—
23#63647D312yescomplemented
24C3×(C2×D3)⋊C8—32no—
25#63557—32no—
26(C2×D3)⋊(C3⋊C8)—32no—
27C22×#272C2×C411yescharacteristic
28C22×(C3⋊D3)⋊C3C2×C412yes—
29C2×((C3⋊D3)⋊C3)⋊C2—22no—
30C2×((C3⋊D3)⋊C3)⋊C2—22no—
31C2×((C3⋊D3)⋊C3)⋊C2C814yescomplemented
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084C2×C411yescharacteristic
34C4×#272D412yes—
35C4×#272D412yes—
36C2×(C3⋊Dic3)⋊C3C2×C412yes—
37C4×(C3⋊D3)⋊C3—22no—
38C4×(C3⋊D3)⋊C3—22no—
39#6199—22no—
40((C3⋊D3)⋊C3)⋊C4—24no—
41C8×H(3,𝔽3)—21no—
42#6180—21no—
43(C3⋊(C3⋊C8))⋊C3—22no—
44C2×(C2×D3)⋊C8—91no—
45D3×C22⋊C8—92no—
46C22×D32—32no—
47C2×C4×C3⋊D3C2×D312yes—
48C2×C3×C4×D3—32no—
49C2×C3×C4×D3—32no—
50C2×C3×C4×D3—61no—
51C2×#624—32no—
52C2×D3×Dic3—32no—
53C2×D3×Dic3—32no—
54C4×D32—64no—
55C4×D32—64no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D312yescomplemented
58C2×C3⋊(C3⋊C8)C2×D312yescomplemented
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—61no—
62C3×D3×C8—64no—
63C3×D3×C8—64no—
64D3×C3⋊C8—64no—
65C22×H(3,𝔽3)C22×C411yescharacteristic
66C2×#272C22⋊C412yes—
67C2×#272C2×C812yes—
68C2×#272M1612yes—
69C2×(C3⋊D3)⋊C3—22no—
70C2×(C3⋊D3)⋊C3—22no—
71C2×(C3⋊D3)⋊C3C2×C814yescomplemented
72((C3⋊D3)⋊C3)⋊C2—28no—
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)C2×D411yescharacteristic
75#1084C22⋊C412yes—
76(C3⋊Dic3)⋊C3—22no—
77C22×C4×D3—91no—
78C22×C4×D3—92no—
79C22×C3×C8—91no—
80C22×C3⋊C8—91no—
81C2×D3×C8—32no—
82C2×D3×C8—61no—
83C2×D3×C8—92no—
84C3×C22⋊C8—92no—
85#830—92no—
86(C2×D3)⋊C8—32no—
87(C2×D3)⋊C8—92no—
88(C2×D3)⋊C8—92no—
89(C2×D3)⋊C8—92no—
90C22×C3⋊D3C4×D312yes—
91C22×C3×D3—32no—
92C22×C3×D3—32no—
93C22×C3×D3—61no—
94C2×D32—38no—
95C2×D32—64no—
96C2×D32—64no—
97C2×C32×C4—21no—
98C2×C32×C4C22×D312yes—
99C2×C3⋊Dic3C4×D312yes—
100C4×C3⋊D3—22no—
101C4×C3⋊D3—22no—
102C2×C3×Dic3—32no—
103C2×C3×Dic3—32no—
104C2×C3×Dic3—61no—
105C3×C4×D3—34no—
106C3×C4×D3—34no—
107C3×C4×D3—62no—
108C3×C4×D3—62no—
109C3×C4×D3—62no—
110C3×C4×D3—62no—
111#624—64no—
112D3×Dic3—64no—
113D3×Dic3—64no—
114C32×C8—22no—
115C32×C8—22no—
116C3⋊(C3⋊C8)—22no—
117C3×C3⋊C8—62no—
118C3×C3⋊C8—62no—
119C3×C3⋊C8—62no—
120C2×C22⋊C8—271noSylow
121C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
122C2×H(3,𝔽3)C22×C811yescharacteristic
123C2×H(3,𝔽3)C2×M1611yescharacteristic
124#272C22⋊C814yescomplemented
125(C3⋊D3)⋊C3—24no—
126C23×D3—91no—
127C23×D3—92no—
128C22×C3×C4—92no—
129C22×C3×C4—91no—
130C22×Dic3—91no—
131C22×Dic3—92no—
132C2×C4×D3—32no—
133C2×C4×D3—32no—
134C2×C4×D3—61no—
135C2×C4×D3—92no—
136C2×C4×D3—92no—
137C2×C4×D3—92no—
138C2×C4×D3—92no—
139C2×C4×D3—184no—
140C2×C4×D3—184no—
141C2×C4×D3—182no—
142C2×C4×D3—182no—
143C2×C3×C8—32no—
144C2×C3×C8—61no—
145C2×C3×C8—92no—
146C2×C3×C8—91no—
147C2×C3×C8—182no—
148C2×C3×C8D3211yescharacteristic
149C2×C3⋊C8—32no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—61no—
152C2×C3⋊C8—92no—
153C2×C3⋊C8—182no—
154D3×C8—64no—
155D3×C8—64no—
156D3×C8—184no—
157C22×C32—21no—
158C22×C32C2×C4×D312yes—
159C2×C3⋊D3—22no—
160C2×C3⋊D3—22no—
161C2×C3⋊D3D3×C814yescomplemented
162C2×C3×D3—34no—
163C2×C3×D3—34no—
164C2×C3×D3—34no—
165C2×C3×D3—34no—
166C2×C3×D3—62no—
167C2×C3×D3—62no—
168C2×C3×D3—62no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171D32—616no—
172C32×C4—21no—
173C32×C4—21no—
174C32×C4D3×D412yes—
175C32×C4D3×D412yes—
176C3⋊Dic3—22no—
177C3×Dic3—34no—
178C3×Dic3—62no—
179C3×Dic3—62no—
180C23×C4—271no—
181C22×C8—91no—
182C22×C8—271no—
183C22⋊C8—92no—
184C22⋊C8—272no—
185H(3,𝔽3)C2×C22⋊C811yescharacteristic, complemented, Sylow
186C23×C3—92no—
187C23×C3—91no—
188C22×D3—32no—
189C22×D3—32no—
190C22×D3—61no—
191C22×D3—92no—
192C22×D3—94no—
193C22×D3—98no—
194C22×D3—184no—
195C22×D3—184no—
196C22×D3—182no—
197C22×D3—182no—
198C2×C3×C4—32no—
199C2×C3×C4—61no—
200C2×C3×C4—91no—
201C2×C3×C4—92no—
202C2×C3×C4—92no—
203C2×C3×C4—92no—
204C2×C3×C4—182no—
205C2×C3×C4—182no—
206C2×C3×C4C2×D3211yescharacteristic
207C2×Dic3—32no—
208C2×Dic3—32no—
209C2×Dic3—61no—
210C2×Dic3—92no—
211C2×Dic3—184no—
212C2×Dic3—182no—
213C4×D3—34no—
214C4×D3—34no—
215C4×D3—62no—
216C4×D3—62no—
217C4×D3—62no—
218C4×D3—62no—
219C4×D3—182no—
220C4×D3—182no—
221C4×D3—184no—
222C4×D3—184no—
223C4×D3—184no—
224C3×C8—21no—
225C3×C8—62no—
226C3×C8—62no—
227C3×C8—182no—
228C3×C8—181no—
229C3⋊C8—62no—
230C3⋊C8—62no—
231C3⋊C8—62no—
232C3⋊C8—182no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32D3×C22⋊C412yes—
237C2×C32C2×D3×C812yes—
238C2×C32D3×M1612yes—
239C3⋊D3—24no—
240C3×D3—38no—
241C3×D3—64no—
242C3×D3—64no—
243C24—271no—
244C22×C4—91no—
245C22×C4—92no—
246C22×C4—272no—
247C22×C4—271no—
248C22×C4—542no—
249C22×C4—542no—
250C2×C8—92no—
251C2×C8—91no—
252C2×C8—182no—
253C2×C8—542no—
254C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
255C22×C3—32no—
256C22×C3—61no—
257C22×C3—94no—
258C22×C3—92no—
259C22×C3—92no—
260C22×C3—92no—
261C22×C3—182no—
262C22×C3—182no—
263C22×C3C4×D3211yescharacteristic
264C2×D3—34no—
265C2×D3—34no—
266C2×D3—34no—
267C2×D3—34no—
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—62no—
272C2×D3—62no—
273C2×D3—94no—
274C2×D3—182no—
275C2×D3—182no—
276C2×D3—1816no—
277C2×D3—188no—
278C3×C4—32no—
279C3×C4—32no—
280C3×C4—61no—
281C3×C4—61no—
282C3×C4—92no—
283C3×C4—182no—
284C3×C4#380211yescharacteristic
285C3×C4#381111yescharacteristic
286Dic3—34no—
287Dic3—62no—
288Dic3—62no—
289Dic3—182no—
290C32—21no—
291C32D3×C22⋊C812yescomplemented
292C23—91no—
293C23—92no—
294C23—274no—
295C23—542no—
296C23—542no—
297C2×C4—91no—
298C2×C4—92no—
299C2×C4—92no—
300C2×C4—92no—
301C2×C4—182no—
302C2×C4—182no—
303C2×C4—544no—
304C2×C4—542no—
305C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C8—21no—
307C8—182no—
308C8—181no—
309C2×C3—32no—
310C2×C3—32no—
311C2×C3—32no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3—94no—
316C2×C3—184no—
317C2×C3D32×C811yescharacteristic
318C2×C3#6379111yescharacteristic
319C2×C3#6392111yescharacteristic
320D3—38no—
321D3—64no—
322D3—64no—
323D3—184no—
324C22—92no—
325C22—92no—
326C22—92no—
327C22—94no—
328C22—182no—
329C22—182no—
330C22—548no—
331C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
332C4—92no—
333C4—182no—
334C4#8865511yescharacteristic
335C4#8867711yescharacteristic
336C3—32no—
337C3—61no—
338C3#1059093511yescharacteristic
339C2—94no—
340C2—184no—
341C2C8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C2#1172083611yescharacteristic
343C2#1172116111yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊C8
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
2. C2×C8×#272C2semidirect
3. C2×#88434C2semidirect
4. #63647⋊C3C2semidirect
5. #11720317C2semidirect
14. C2×C8×H(3,𝔽3)C22semidirect
15. C2×#6180C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
23. #63647D3semidirect
31. C2×((C3⋊D3)⋊C3)⋊C2C8semidirect
57. C2×C32×C8C2×D3semidirect
58. C2×C3⋊(C3⋊C8)C2×D3semidirect
71. C2×(C3⋊D3)⋊C3C2×C8semidirect
124. #272C22⋊C8semidirect
161. C2×C3⋊D3D3×C8semidirect
185. H(3,𝔽3)C2×C22⋊C8semidirect
254. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
291. C32D3×C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×C8×#2722
C2×#884342
#63647⋊C32
#117203172
D3×(C2×D3)⋊C83