Finite Group Databasebeta

← #13138046

Subgroups — #13138046

Order 1728.

741 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 741 conjugacy classes of subgroups — too many to draw. Only its 87 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
2#11721760C211yesmaximal, characteristic, complemented
3C2×#88435C211yesmaximal, characteristic, complemented
4#63636⋊C3C212yesmaximal, complemented
5#11720308C212yesmaximal, complemented
6#10590592—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2—21no—
8C2×C4×#272C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6198—21no—
11#88655—22no—
12#88655C414yescomplemented
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
14#88542—21no—
15C4⋊#1084C2211yescharacteristic, complemented
16C4×#1084—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18#88435—22no—
19C2×D3×D12—32no—
20C3×#892—32no—
21C3×#892—61no—
22C2×#3711—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×#272C2×C412yes—
37C4×#272D412yes—
38C4⋊((C3⋊D3)⋊C3)—22no—
39C4⋊((C3⋊D3)⋊C3)C2×C414yes—
40#6198—24no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42#5001—91no—
43#4770—92no—
44C22×D32—62no—
45C2×C4⋊(C3⋊D3)C2×D312yes—
46C2×C3×D12—32no—
47C2×C3×C4×D3—32no—
48C2×C3×C4×D3—61no—
49C2×#626—62no—
50D3×D12—38no—
51D3×D12—64no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4C2×D312yescomplemented
54C2×C3⋊(C3⋊C8)C2×D312yescomplemented
55C3×(C2×D3)⋊C4—62no—
56C3×(C2×D3)⋊C4—62no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—61no—
59C3×C4×Dic3—62no—
60C3×C4×Dic3—62no—
61C2×C3×C3⋊C8—32no—
62#3711—64no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—21no—
65C2×#272—21no—
66C2×#272C22⋊C412yes—
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)—41no—
72C4×H(3,𝔽3)C22×C411yescharacteristic
73C4×H(3,𝔽3)C2×D411yescharacteristic
74#1084—41no—
75#1084C22⋊C412yes—
76C2×D3×D4—92no—
77C22×D12—91no—
78C3×#117—91no—
79C2×#208—91no—
80C2×ΓL(1,𝔽25)—92no—
81#892—32no—
82#892—61no—
83C3×#102—92no—
84#832—92no—
85#821—92no—
86#809—32no—
87#809—92no—
88#809—92no—
89C22×C3⋊D3—22no—
90C22×C3×D3—32no—
91C22×C3×D3—61no—
92C22×C3×D3—62no—
93C2×D32—68no—
94C2×D32—124no—
95C2×D32—124no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4C22×D312yes—
100C4⋊(C3⋊D3)—22no—
101C4⋊(C3⋊D3)C4×D314yes—
102C2×C3×Dic3—32no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C3×D12—34no—
107C3×D12—62no—
108C3×C4×D3—34no—
109C3×C4×D3—34no—
110C3×C4×D3—62no—
111C3×C4×D3—62no—
112#626—68no—
113C3⋊(C3⋊C8)—22no—
114C3×C3⋊C8—62no—
115#418—271noSylow
116C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
117C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
118C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
119#272—22no—
120(C3⋊D3)⋊C3—44no—
121C23×D3—182no—
122C23×D3—181no—
123C2×C3×D4—92no—
124C22×C3×C4—91no—
125C2×#67—182no—
126D3×D4—98no—
127D3×D4—184no—
128C2×D12—32no—
129C2×D12—92no—
130C2×D12—94no—
131C2×D12—182no—
132C2×D12—181no—
133C2×C4×D3—32no—
134C2×C4×D3—61no—
135C2×C3×C8—92no—
136C3×C4⋊C4—32no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—91no—
139C3×C4⋊C4D3211yescharacteristic
140C3×C22⋊C4—181no—
141C3×C42—181no—
142(C2×D3)⋊C4—62no—
143(C2×D3)⋊C4—62no—
144C4⋊Dic3—32no—
145C4⋊Dic3—61no—
146C4×Dic3—62no—
147C4×Dic3—62no—
148#208—182no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—92no—
151ΓL(1,𝔽25)—184no—
152C22×C32—21no—
153C22×C32D3×D412yes—
154C2×C3⋊D3—24no—
155C2×C3⋊D3—42no—
156C2×C3⋊D3—42no—
157C2×C3×D3—34no—
158C2×C3×D3—62no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—64no—
162C2×C3×D3—121no—
163C2×C3×D3—121no—
164C2×C3×D3—122no—
165C2×C3×D3—122no—
166D32—1216no—
167C32×C4—21no—
168C32×C4—21no—
169C32×C4—42no—
170C32×C4—42no—
171C32×C4C2×C4×D312yes—
172C32×C4D3×D412yes—
173C3×Dic3—34no—
174C3×Dic3—62no—
175C3×Dic3—122no—
176C3×Dic3—122no—
177C22×D4—271no—
178C2×M16—271no—
179#117—91no—
180#102—92no—
181#102—272no—
182H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
183C23×C3—91no—
184C23×C3—182no—
185C22×D3—32no—
186C22×D3—61no—
187C22×D3—62no—
188C22×D3—182no—
189C22×D3—188no—
190C22×D3—184no—
191C22×D3—364no—
192C22×D3—364no—
193C22×D3—362no—
194C22×D3—362no—
195C3×D4—94no—
196C3×D4—182no—
197C2×C3×C4—21no—
198C2×C3×C4—32no—
199C2×C3×C4—62no—
200C2×C3×C4—62no—
201C2×C3×C4—61no—
202C2×C3×C4—91no—
203C2×C3×C4—92no—
204C2×C3×C4—92no—
205C2×C3×C4—181no—
206C2×C3×C4C2×D3211yescharacteristic
207#67—188no—
208C2×Dic3—32no—
209C2×Dic3—61no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212D12—34no—
213D12—62no—
214D12—94no—
215D12—182no—
216D12—184no—
217C4×D3—34no—
218C4×D3—34no—
219C4×D3—62no—
220C4×D3—62no—
221C3×C8—182no—
222C3⋊C8—62no—
223C3⋊C8—182no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32D3×C22⋊C412yes—
228C2×C32#91112yes—
229C2×C32#91412yes—
230C3⋊D3—44no—
231C3×D3—64no—
232C3×D3—122no—
233C3×D3—124no—
234C24—541no—
235C2×D4—92no—
236C2×D4—274no—
237C2×D4—542no—
238C2×D4—541no—
239C22×C4—91no—
240M16—542no—
241C2×C8—92no—
242C4⋊C4—91no—
243C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
244C22⋊C4—181no—
245C42—181no—
246C22×C3—32no—
247C22×C3—61no—
248C22×C3—92no—
249C22×C3—184no—
250C22×C3—181no—
251C22×C3—181no—
252C22×C3—362no—
253C22×C3—362no—
254C22×C3#381111yescharacteristic
255C2×D3—34no—
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—62no—
259C2×D3—64no—
260C2×D3—121no—
261C2×D3—121no—
262C2×D3—122no—
263C2×D3—122no—
264C2×D3—184no—
265C2×D3—362no—
266C2×D3—362no—
267C2×D3—3616no—
268C2×D3—368no—
269C3×C4—32no—
270C3×C4—32no—
271C3×C4—41no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—92no—
275C3×C4—122no—
276C3×C4—122no—
277C3×C4—361no—
278C3×C4C4×D3211yescharacteristic
279C3×C4#380211yescharacteristic
280Dic3—34no—
281Dic3—62no—
282Dic3—122no—
283Dic3—122no—
284C32—21no—
285C32#477012yescomplemented
286C23—91no—
287C23—182no—
288C23—544no—
289C23—1082no—
290C23—1082no—
291D4—94no—
292D4—182no—
293D4—544no—
294C2×C4—21no—
295C2×C4—91no—
296C2×C4—92no—
297C2×C4—92no—
298C2×C4—181no—
299C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C8—182no—
301C2×C3—32no—
302C2×C3—32no—
303C2×C3—32no—
304C2×C3—61no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3—182no—
308C2×C3—364no—
309C2×C3#6392111yescharacteristic
310C2×C3#6392711yescharacteristic
311C2×C3#6394111yescharacteristic
312D3—64no—
313D3—122no—
314D3—124no—
315D3—364no—
316C22—92no—
317C22—181no—
318C22—181no—
319C22—184no—
320C22—362no—
321C22—362no—
322C22—1088no—
323C22#8867711yescharacteristic
324C4—41no—
325C4—92no—
326C4—361no—
327C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
328C4#8865511yescharacteristic
329C3—32no—
330C3—61no—
331C3#1059059411yescharacteristic
332C2—182no—
333C2—364no—
334C2#1172116111yescharacteristic
335C2#1172119111yescharacteristic
336C2#1172122911yescharacteristic
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)#418
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. #11721760C2semidirect
3. C2×#88435C2semidirect
4. #63636⋊C3C2semidirect
5. #11720308C2semidirect
12. #88655C4semidirect
13. C4⋊C4×H(3,𝔽3)C22semidirect
15. C4⋊#1084C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
23. #63636D3semidirect
53. C32×C4⋊C4C2×D3semidirect
54. C2×C3⋊(C3⋊C8)C2×D3semidirect
182. H(3,𝔽3)#418semidirect
243. C4⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
285. C32#4770semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
#117217602
C2×#884352
#63636⋊C32
#117203082
#105905923