Finite Group Databasebeta

← #13138045

Subgroups — #13138045

Order 1728.

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

normal subgroups only

This group has 799 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
2C4⋊C4×#272C211yesmaximal, characteristic, complemented
3C2×#88434C211yesmaximal, characteristic, complemented
4#63636⋊C3C212yesmaximal, complemented
5#11720308C212yesmaximal, complemented
6D3×#809—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2—21no—
8C2×C4×#272—21no—
9C2×C4×#272C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×#6198—21no—
12#88655—22no—
13#88655C414yescomplemented
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
15C4⋊#1084C2211yescharacteristic, complemented
16#88540—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18#88434—22no—
19C2×D3×D12—32no—
20C3×D3×C4⋊C4—32no—
21C3×D3×C4⋊C4—61no—
22C2×D3×C3⋊C8—32no—
23#63636D312yescomplemented
24C3×#809—32no—
25#63564—32no—
26#63562—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)—21no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084D411yescharacteristic
36C4×#272—42no—
37C4×#272C2×C412yes—
38C4×#272D412yes—
39C4⋊((C3⋊D3)⋊C3)—22no—
40C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
41#6198—24no—
42(C3⋊(C3⋊C8))⋊C3—22no—
43C2×#809—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×C3×C4×D3—62no—
51C2×C3×C4×D3—62no—
52C2×#626—62no—
53D3×D12—38no—
54D3×D12—64no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4C2×D312yescomplemented
57C2×C3⋊(C3⋊C8)C2×D312yescomplemented
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—61no—
60C3×Dic3⋊C4—62no—
61C3×Dic3⋊C4—62no—
62C2×C3×C3⋊C8—32no—
63D3×C3⋊C8—64no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272C22⋊C412yes—
66C2×#272D812yes—
67C2×#272SD1612yes—
68C2×(C3⋊D3)⋊C3—24no—
69C2×(C3⋊D3)⋊C3—42no—
70C2×(C3⋊D3)⋊C3—42no—
71((C3⋊D3)⋊C3)⋊C2—48no—
72C4×H(3,𝔽3)—41no—
73C4×H(3,𝔽3)C22×C411yescharacteristic
74C4×H(3,𝔽3)C2×D411yescharacteristic
75#1084—41no—
76#1084C22⋊C412yes—
77C2×D3×D4—92no—
78C22×D12—91no—
79C2×C3×C4⋊C4—91no—
80C22×C3⋊C8—91no—
81C2×D3×C8—92no—
82D3×C4⋊C4—32no—
83D3×C4⋊C4—61no—
84C3×#102—92no—
85#832—92no—
86#821—92no—
87#809—32no—
88#809—92no—
89#809—92no—
90C22×C3⋊D3—22no—
91C22×C3×D3—32no—
92C22×C3×D3—61no—
93C22×C3×D3—62no—
94C2×D32—68no—
95C2×D32—124no—
96C2×D32—124no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4C22×D312yes—
101C4⋊(C3⋊D3)—22no—
102C4⋊(C3⋊D3)C4×D314yescomplemented
103C2×C3×Dic3—32no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C3×D12—34no—
108C3×D12—62no—
109C3×C4×D3—34no—
110C3×C4×D3—34no—
111C3×C4×D3—62no—
112C3×C4×D3—62no—
113C3×C4×D3—124no—
114C3×C4×D3—124no—
115#626—68no—
116C3⋊(C3⋊C8)—22no—
117C3×C3⋊C8—62no—
118C2×#102—271noSylow
119C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
120C2×H(3,𝔽3)C2×D811yescharacteristic
121C2×H(3,𝔽3)C2×SD1611yescharacteristic
122#272#10214yescomplemented
123(C3⋊D3)⋊C3—44no—
124C23×D3—182no—
125C23×D3—181no—
126C2×C3×D4—92no—
127C22×C3×C4—91no—
128C22×C3×C4—181no—
129C2×#67—182no—
130D3×D4—98no—
131D3×D4—184no—
132C2×D12—32no—
133C2×D12—92no—
134C2×D12—94no—
135C2×D12—182no—
136C2×D12—181no—
137C2×C4×D3—32no—
138C2×C4×D3—61no—
139C2×C4×D3—62no—
140C2×C4×D3—62no—
141C2×C3×C8—92no—
142C3×C4⋊C4—32no—
143C3×C4⋊C4—61no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—181no—
146C3×C4⋊C4D3211yescharacteristic
147C4⋊Dic3—32no—
148C4⋊Dic3—61no—
149Dic3⋊C4—62no—
150Dic3⋊C4—62no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—92no—
153C2×C3⋊C8—182no—
154D3×C8—184no—
155C22×C32—21no—
156C22×C32D3×D412yes—
157C2×C3⋊D3—24no—
158C2×C3⋊D3—42no—
159C2×C3⋊D3—42no—
160C2×C3×D3—34no—
161C2×C3×D3—34no—
162C2×C3×D3—34no—
163C2×C3×D3—62no—
164C2×C3×D3—62no—
165C2×C3×D3—62no—
166C2×C3×D3—64no—
167C2×C3×D3—122no—
168C2×C3×D3—122no—
169D32—1216no—
170C32×C4—21no—
171C32×C4—21no—
172C32×C4—42no—
173C32×C4—42no—
174C32×C4C2×C4×D312yes—
175C32×C4D3×D412yes—
176C3×Dic3—34no—
177C3×Dic3—62no—
178C3×Dic3—122no—
179C3×Dic3—122no—
180C22×D4—271no—
181C22×C8—271no—
182C2×C4⋊C4—91no—
183#102—92no—
184#102—272no—
185H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
186C23×C3—91no—
187C23×C3—182no—
188C22×D3—32no—
189C22×D3—61no—
190C22×D3—62no—
191C22×D3—182no—
192C22×D3—188no—
193C22×D3—184no—
194C22×D3—364no—
195C22×D3—364no—
196C22×D3—362no—
197C22×D3—362no—
198C3×D4—94no—
199C3×D4—182no—
200C2×C3×C4—21no—
201C2×C3×C4—32no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—61no—
205C2×C3×C4—91no—
206C2×C3×C4—92no—
207C2×C3×C4—92no—
208C2×C3×C4—181no—
209C2×C3×C4—362no—
210C2×C3×C4C2×D3211yescharacteristic
211#67—188no—
212C2×Dic3—32no—
213C2×Dic3—61no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216D12—34no—
217D12—62no—
218D12—94no—
219D12—182no—
220D12—184no—
221C4×D3—34no—
222C4×D3—34no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—124no—
226C4×D3—124no—
227C3×C8—182no—
228C3⋊C8—62no—
229C3⋊C8—182no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32D3×C22⋊C412yes—
234C2×C32D3×D812yes—
235C2×C32D3×SD1612yes—
236C3⋊D3—44no—
237C3×D3—38no—
238C3×D3—64no—
239C3×D3—124no—
240C24—541no—
241C2×D4—92no—
242C2×D4—274no—
243C2×D4—542no—
244C2×D4—541no—
245C22×C4—91no—
246C22×C4—181no—
247C2×C8—92no—
248C2×C8—542no—
249C4⋊C4—91no—
250C4⋊C4—181no—
251C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
252C22×C3—32no—
253C22×C3—61no—
254C22×C3—92no—
255C22×C3—92no—
256C22×C3—92no—
257C22×C3—184no—
258C22×C3—362no—
259C22×C3—362no—
260C22×C3#381111yescharacteristic
261C2×D3—34no—
262C2×D3—34no—
263C2×D3—34no—
264C2×D3—62no—
265C2×D3—62no—
266C2×D3—62no—
267C2×D3—64no—
268C2×D3—122no—
269C2×D3—122no—
270C2×D3—184no—
271C2×D3—362no—
272C2×D3—362no—
273C2×D3—3616no—
274C2×D3—368no—
275C3×C4—32no—
276C3×C4—32no—
277C3×C4—41no—
278C3×C4—61no—
279C3×C4—61no—
280C3×C4—92no—
281C3×C4—122no—
282C3×C4—122no—
283C3×C4—361no—
284C3×C4C4×D3211yescharacteristic
285C3×C4#380211yescharacteristic
286Dic3—34no—
287Dic3—62no—
288Dic3—122no—
289Dic3—122no—
290C32—21no—
291C32D3×#10212yescomplemented
292C23—91no—
293C23—182no—
294C23—544no—
295C23—1082no—
296C23—1082no—
297D4—94no—
298D4—182no—
299D4—544no—
300C2×C4—21no—
301C2×C4—91no—
302C2×C4—92no—
303C2×C4—92no—
304C2×C4—181no—
305C2×C4—362no—
306C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
307C8—182no—
308C2×C3—32no—
309C2×C3—32no—
310C2×C3—32no—
311C2×C3—61no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—94no—
315C2×C3—364no—
316C2×C3#6392111yescharacteristic
317C2×C3#6392611yescharacteristic
318C2×C3#6394011yescharacteristic
319D3—38no—
320D3—64no—
321D3—124no—
322D3—364no—
323C22—92no—
324C22—92no—
325C22—92no—
326C22—184no—
327C22—362no—
328C22—362no—
329C22—1088no—
330C22#8867711yescharacteristic
331C4—41no—
332C4—92no—
333C4—361no—
334C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C4#8865511yescharacteristic
336C3—32no—
337C3—61no—
338C3#1059059311yescharacteristic
339C2—94no—
340C2—364no—
341C2#1172116111yescharacteristic
342C2#1172119011yescharacteristic
343C2#1172122811yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×#88655C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×#272C2semidirect
3. C2×#88434C2semidirect
4. #63636⋊C3C2semidirect
5. #11720308C2semidirect
13. #88655C4semidirect
14. C4⋊C4×H(3,𝔽3)C22semidirect
15. C4⋊#1084C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
23. #63636D3semidirect
40. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
56. C32×C4⋊C4C2×D3semidirect
57. C2×C3⋊(C3⋊C8)C2×D3semidirect
102. C4⋊(C3⋊D3)C4×D3semidirect
122. #272#102semidirect
185. H(3,𝔽3)C2×#102semidirect
251. C4⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
291. C32D3×#102semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C4⋊C4×#2722
C2×#884342
#63636⋊C32
#117203082
D3×#8093