Finite Group Databasebeta

← #13139031

Subgroups — #13139031

Order 1728.

783 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 95 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×#88655C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic, complemented
3#11721152C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#11720318C212yesmaximal, complemented
6D3×#821—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2—21no—
8C2×C4×#272C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6199—21no—
11C2×#6198—21no—
12#88655—22no—
13#88655C414yescomplemented
14C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×#6180C2211yescharacteristic, complemented
16C8×#272—22no—
17C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
18#88449—21no—
19C2×D3×D12—32no—
20C2×C3×D3×C8—32no—
21C2×C3×D3×C8—61no—
22D3×C4⋊Dic3—32no—
23#63648D312yescomplemented
24C3×#821—32no—
25#63564—32no—
26#63563—32no—
27C22×#272D411yescharacteristic
28C22×(C3⋊D3)⋊C3—22no—
29C2×((C3⋊D3)⋊C3)⋊C2—24no—
30C2×((C3⋊D3)⋊C3)⋊C2—42no—
31C2×((C3⋊D3)⋊C3)⋊C2—42no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084D411yescharacteristic
34C4×#272C2×C412yes—
35C4×#272D412yes—
36C2×(C3⋊Dic3)⋊C3—22no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
39#6199—42no—
40#6198—24no—
41C8×H(3,𝔽3)—21no—
42#6180—21no—
43C2×#821—91no—
44D3×#102—92no—
45C22×D32—62no—
46C2×C4⋊(C3⋊D3)C2×D312yes—
47C2×C3×D12—32no—
48C2×C3×C4×D3—32no—
49C2×C3×C4×D3—61no—
50C2×#626—62no—
51C2×D3×Dic3—62no—
52D3×D12—38no—
53D3×D12—64no—
54C2×C32×C8—21no—
55C2×C32×C8C2×D312yescomplemented
56C4⋊(C3⋊Dic3)C2×D312yescomplemented
57C3×C4⋊Dic3—32no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—61no—
60C3×D3×C8—64no—
61C3×D3×C8—64no—
62Dic3⋊Dic3—62no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272C22⋊C412yes—
65C2×#272D812yes—
66C2×#272SD1612yes—
67C2×(C3⋊D3)⋊C3—24no—
68C2×(C3⋊D3)⋊C3—42no—
69C2×(C3⋊D3)⋊C3—42no—
70((C3⋊D3)⋊C3)⋊C2—48no—
71C4×H(3,𝔽3)C22×C411yescharacteristic
72C4×H(3,𝔽3)C2×D411yescharacteristic
73#1084C22⋊C412yes—
74(C3⋊Dic3)⋊C3—42no—
75C2×D3×D4—92no—
76C22×D12—91no—
77C22×C3×C8—91no—
78C2×C4⋊Dic3—91no—
79C2×D3×C8—32no—
80C2×D3×C8—61no—
81D3×C4⋊C4—92no—
82C3×#102—92no—
83#832—92no—
84#821—32no—
85#821—92no—
86#821—92no—
87#809—92no—
88C22×C3⋊D3—22no—
89C22×C3×D3—32no—
90C22×C3×D3—61no—
91C22×C3×D3—62no—
92C2×D32—68no—
93C2×D32—124no—
94C2×D32—124no—
95C2×C32×C4—21no—
96C2×C32×C4C22×D312yes—
97C2×C3⋊Dic3—22no—
98C4⋊(C3⋊D3)—22no—
99C4⋊(C3⋊D3)C4×D314yescomplemented
100C2×C3×Dic3—32no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—62no—
103C3×D12—34no—
104C3×D12—62no—
105C3×C4×D3—34no—
106C3×C4×D3—34no—
107C3×C4×D3—62no—
108C3×C4×D3—62no—
109#626—68no—
110D3×Dic3—124no—
111C32×C8—22no—
112C32×C8—22no—
113C3×C3⋊C8—62no—
114C3×C3⋊C8—62no—
115C2×#102—271noSylow
116C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
117C2×H(3,𝔽3)C2×D811yescharacteristic
118C2×H(3,𝔽3)C2×SD1611yescharacteristic
119#272#10214yescomplemented
120(C3⋊D3)⋊C3—44no—
121C23×D3—182no—
122C23×D3—181no—
123C2×C3×D4—92no—
124C22×C3×C4—91no—
125C2×#67—182no—
126C22×Dic3—181no—
127D3×D4—98no—
128D3×D4—184no—
129C2×D12—32no—
130C2×D12—92no—
131C2×D12—94no—
132C2×D12—182no—
133C2×D12—181no—
134C2×C4×D3—32no—
135C2×C4×D3—61no—
136C2×C4×D3—182no—
137C2×C3×C8—32no—
138C2×C3×C8—61no—
139C2×C3×C8—91no—
140C2×C3×C8—182no—
141C2×C3×C8D3211yescharacteristic
142C3×C4⋊C4—92no—
143C4⋊Dic3—32no—
144C4⋊Dic3—92no—
145C4⋊Dic3—181no—
146Dic3⋊C4—182no—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—61no—
149D3×C8—64no—
150D3×C8—64no—
151C22×C32—21no—
152C22×C32D3×D412yes—
153C2×C3⋊D3—24no—
154C2×C3⋊D3—42no—
155C2×C3⋊D3—42no—
156C2×C3×D3—34no—
157C2×C3×D3—34no—
158C2×C3×D3—34no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—64no—
163C2×C3×D3—122no—
164C2×C3×D3—122no—
165D32—1216no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4C2×C4×D312yes—
169C32×C4D3×D412yes—
170C3⋊Dic3—42no—
171C3×Dic3—34no—
172C3×Dic3—62no—
173C3×Dic3—122no—
174C22×D4—271no—
175C22×C8—91no—
176C2×C4⋊C4—271no—
177#102—92no—
178#102—272no—
179H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
180C23×C3—91no—
181C23×C3—182no—
182C22×D3—32no—
183C22×D3—61no—
184C22×D3—62no—
185C22×D3—182no—
186C22×D3—188no—
187C22×D3—184no—
188C22×D3—364no—
189C22×D3—364no—
190C22×D3—362no—
191C22×D3—362no—
192C3×D4—94no—
193C3×D4—182no—
194C2×C3×C4—32no—
195C2×C3×C4—61no—
196C2×C3×C4—91no—
197C2×C3×C4—92no—
198C2×C3×C4—92no—
199C2×C3×C4—182no—
200C2×C3×C4C2×D3211yescharacteristic
201#67—188no—
202C2×Dic3—32no—
203C2×Dic3—61no—
204C2×Dic3—62no—
205C2×Dic3—182no—
206C2×Dic3—362no—
207D12—34no—
208D12—62no—
209D12—94no—
210D12—182no—
211D12—184no—
212C4×D3—34no—
213C4×D3—34no—
214C4×D3—62no—
215C4×D3—62no—
216C4×D3—364no—
217C3×C8—21no—
218C3×C8—62no—
219C3×C8—62no—
220C3×C8—181no—
221C3⋊C8—62no—
222C3⋊C8—62no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32D3×C22⋊C412yes—
227C2×C32D3×D812yes—
228C2×C32D3×SD1612yes—
229C3⋊D3—44no—
230C3×D3—38no—
231C3×D3—64no—
232C3×D3—124no—
233C24—541no—
234C2×D4—92no—
235C2×D4—274no—
236C2×D4—542no—
237C2×D4—541no—
238C22×C4—91no—
239C22×C4—541no—
240C2×C8—91no—
241C2×C8—182no—
242C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
243C4⋊C4—92no—
244C4⋊C4—541no—
245C22×C3—32no—
246C22×C3—61no—
247C22×C3—92no—
248C22×C3—92no—
249C22×C3—92no—
250C22×C3—184no—
251C22×C3—362no—
252C22×C3—362no—
253C22×C3#380211yescharacteristic
254C2×D3—34no—
255C2×D3—34no—
256C2×D3—34no—
257C2×D3—62no—
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—64no—
261C2×D3—122no—
262C2×D3—122no—
263C2×D3—184no—
264C2×D3—362no—
265C2×D3—362no—
266C2×D3—3616no—
267C2×D3—368no—
268C3×C4—32no—
269C3×C4—32no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4—92no—
273C3×C4—362no—
274C3×C4C4×D3211yescharacteristic
275C3×C4#381111yescharacteristic
276Dic3—34no—
277Dic3—62no—
278Dic3—122no—
279Dic3—362no—
280C32—21no—
281C32D3×#10212yescomplemented
282C23—91no—
283C23—182no—
284C23—544no—
285C23—1082no—
286C23—1082no—
287D4—94no—
288D4—182no—
289D4—544no—
290C2×C4—91no—
291C2×C4—92no—
292C2×C4—92no—
293C2×C4—182no—
294C2×C4—1082no—
295C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C8—21no—
297C8—181no—
298C2×C3—32no—
299C2×C3—32no—
300C2×C3—32no—
301C2×C3—61no—
302C2×C3—61no—
303C2×C3—61no—
304C2×C3—94no—
305C2×C3—364no—
306C2×C3#6379611yescharacteristic
307C2×C3#6379711yescharacteristic
308C2×C3#6392111yescharacteristic
309D3—38no—
310D3—64no—
311D3—124no—
312D3—364no—
313C22—92no—
314C22—92no—
315C22—92no—
316C22—184no—
317C22—362no—
318C22—362no—
319C22—1088no—
320C22#8865511yescharacteristic
321C4—92no—
322C4—362no—
323C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
324C4#8867711yescharacteristic
325C3—32no—
326C3—61no—
327C3#1059094011yescharacteristic
328C2—94no—
329C2—364no—
330C2#1172083711yescharacteristic
331C2#1172083811yescharacteristic
332C2#1172116111yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#88655C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#102
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C8×#272C2semidirect
3. #11721152C2semidirect
4. #63648⋊C3C2semidirect
5. #11720318C2semidirect
13. #88655C4semidirect
14. C2×C8×H(3,𝔽3)C22semidirect
15. C2×#6180C22semidirect
17. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
23. #63648D3semidirect
38. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
55. C2×C32×C8C2×D3semidirect
56. C4⋊(C3⋊Dic3)C2×D3semidirect
99. C4⋊(C3⋊D3)C4×D3semidirect
119. #272#102semidirect
179. H(3,𝔽3)C2×#102semidirect
242. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
281. C32D3×#102semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C2×C8×#2722
#117211522
#63648⋊C32
#117203182
D3×#8213