Finite Group Databasebeta

← #13138047

Subgroups — #13138047

Order 1728.

787 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 77 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, complemented
2#11721763C211yesmaximal, characteristic, complemented
3C2×#88436C212yesmaximal, complemented
4#63636⋊C3C212yesmaximal, complemented
5#11720323C211yesmaximal, characteristic, complemented
6#10590595—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2—21no—
8C2×C4⋊#272—21no—
9C2×C4⋊#272C2211yescharacteristic, complemented
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
12C2×#6198—21no—
13#88655—24no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
15#88542—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
17#88436—24no—
18C2×D3×D12—32no—
19C3×#895—32no—
20C3×#895—61no—
21C2×#3714—32no—
22C2×#3713—32no—
23#63636D312yescomplemented
24C3×#809—32no—
25(C2×D3)⋊(C3⋊C8)—32no—
26C22×#272—21no—
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×#1084D411yescharacteristic
35C4⋊#272—22no—
36C4⋊#272—42no—
37C4×#272—22no—
38C4⋊((C3⋊D3)⋊C3)—22no—
39C4⋊((C3⋊D3)⋊C3)D414yescomplemented
40#6198—24no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42#5036—91no—
43#4773—92no—
44C22×D32—62no—
45C2×C4⋊(C3⋊D3)C2×D312yes—
46C2×C3×D12—32no—
47C2×C3×D12—32no—
48C2×C3×D12—62no—
49C2×C3×D12—61no—
50C2×C3×D12—62no—
51C2×C3×C4×D3—32no—
52C2×C3×C4×D3—61no—
53C2×#626—62no—
54D3×D12—64no—
55D3×D12—64no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D312yescomplemented
58C2×C3⋊(C3⋊C8)C2×D312yescomplemented
59C3×(C2×D3)⋊C4—62no—
60C3×(C2×D3)⋊C4—62no—
61C2×C3×C3⋊C8—32no—
62#3714—64no—
63#3713—64no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—21no—
66C2×#272—21no—
67C2×#272—41no—
68C2×#272—41no—
69C2×#272—41no—
70C2×#272D812yescomplemented
71C2×(C3⋊D3)⋊C3—24no—
72C2×(C3⋊D3)⋊C3—42no—
73C2×(C3⋊D3)⋊C3—42no—
74((C3⋊D3)⋊C3)⋊C2—48no—
75C4×H(3,𝔽3)—41no—
76C4×H(3,𝔽3)C2×D412yes—
77#1084—21no—
78C2×D3×D4—92no—
79C22×D12—91no—
80C3×#121—91no—
81C2×#213—92no—
82C2×#213—92no—
83C2×D24—92no—
84#895—32no—
85#895—61no—
86C3×#102—92no—
87#830—91no—
88(C2×D3)⋊C8—92no—
89#809—32no—
90#809—92no—
91C22×C3⋊D3—22no—
92C22×C3×D3—32no—
93C22×C3×D3—61no—
94C22×C3×D3—62no—
95C22×C3×D3—62no—
96C22×C3×D3—62no—
97C2×D32—68no—
98C2×D32—124no—
99C2×D32—124no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4C22×D312yes—
104C4⋊(C3⋊D3)—22no—
105C4⋊(C3⋊D3)D1214yescomplemented
106C2×C3×Dic3—32no—
107C2×C3×Dic3—61no—
108C3×D12—34no—
109C3×D12—62no—
110C3×D12—62no—
111C3×D12—62no—
112C3×D12—122no—
113C3×D12—124no—
114C3×D12—124no—
115C3×C4×D3—62no—
116C3×C4×D3—62no—
117C3×C4×D3—122no—
118#626—68no—
119C3⋊(C3⋊C8)—22no—
120C3×C3⋊C8—62no—
121#447—271noSylow
122C2×H(3,𝔽3)#12011yescharacteristic
123C2×H(3,𝔽3)C2×D811yescharacteristic
124C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
125#272—22no—
126#272—81no—
127(C3⋊D3)⋊C3—44no—
128C23×D3—182no—
129C23×D3—181no—
130C2×C3×D4—92no—
131C2×C3×D4—91no—
132C2×C3×D4—181no—
133C22×C3×C4—91no—
134C2×#67—182no—
135D3×D4—184no—
136D3×D4—184no—
137C2×D12—32no—
138C2×D12—32no—
139C2×D12—61no—
140C2×D12—62no—
141C2×D12—62no—
142C2×D12—92no—
143C2×D12—184no—
144C2×D12—181no—
145C2×C4×D3—32no—
146C2×C4×D3—61no—
147C2×C3×C8—92no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4D3211yescharacteristic
151C3×C22⋊C4—181no—
152#213—184no—
153#213—184no—
154(C2×D3)⋊C4—62no—
155(C2×D3)⋊C4—62no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—92no—
158D24—184no—
159C22×C32—21no—
160C22×C32D3×D412yes—
161C2×C3⋊D3—24no—
162C2×C3⋊D3—42no—
163C2×C3⋊D3—42no—
164C2×C3×D3—34no—
165C2×C3×D3—62no—
166C2×C3×D3—62no—
167C2×C3×D3—62no—
168C2×C3×D3—64no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C2×C3×D3—122no—
172C2×C3×D3—122no—
173C2×C3×D3—122no—
174C2×C3×D3—122no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C2×C3×D3—122no—
179D32—1216no—
180C32×C4—22no—
181C32×C4—42no—
182C32×C4—42no—
183C32×C4C2×D1212yes—
184C32×C4D3×D412yes—
185C3×Dic3—62no—
186C3×Dic3—121no—
187C22×D4—271no—
188C2×D8—272no—
189#121—91no—
190#102—92no—
191C22⋊C8—271no—
192H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
193C23×C3—91no—
194C23×C3—182no—
195C23×C3—181no—
196C22×D3—32no—
197C22×D3—61no—
198C22×D3—62no—
199C22×D3—62no—
200C22×D3—62no—
201C22×D3—182no—
202C22×D3—184no—
203C22×D3—188no—
204C22×D3—364no—
205C22×D3—364no—
206C22×D3—362no—
207C22×D3—362no—
208C3×D4—94no—
209C3×D4—182no—
210C3×D4—182no—
211C3×D4—362no—
212C2×C3×C4—21no—
213C2×C3×C4—32no—
214C2×C3×C4—62no—
215C2×C3×C4—62no—
216C2×C3×C4—61no—
217C2×C3×C4—91no—
218C2×C3×C4—182no—
219C2×C3×C4C2×D3211yescharacteristic
220#67—188no—
221C2×Dic3—32no—
222C2×Dic3—61no—
223D12—34no—
224D12—62no—
225D12—62no—
226D12—62no—
227D12—94no—
228D12—122no—
229D12—124no—
230D12—124no—
231D12—182no—
232D12—184no—
233C4×D3—62no—
234C4×D3—62no—
235C4×D3—122no—
236C3×C8—182no—
237C3⋊C8—62no—
238C3⋊C8—182no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32#88212yes—
243C2×C32D3×D812yes—
244C2×C32#91412yes—
245C3⋊D3—44no—
246C3×D3—64no—
247C3×D3—122no—
248C3×D3—124no—
249C3×D3—242no—
250C3×D3—242no—
251C24—541no—
252C2×D4—92no—
253C2×D4—91no—
254C2×D4—181no—
255C2×D4—541no—
256C2×D4—544no—
257C22×C4—91no—
258D8—544no—
259C2×C8—92no—
260C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
261C22⋊C4—181no—
262C22×C3—32no—
263C22×C3—61no—
264C22×C3—92no—
265C22×C3—184no—
266C22×C3—181no—
267C22×C3—181no—
268C22×C3—362no—
269C22×C3—362no—
270C22×C3—361no—
271C22×C3—361no—
272C22×C3—361no—
273C22×C3#381111yescharacteristic
274C2×D3—34no—
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—64no—
279C2×D3—121no—
280C2×D3—121no—
281C2×D3—122no—
282C2×D3—122no—
283C2×D3—122no—
284C2×D3—122no—
285C2×D3—122no—
286C2×D3—122no—
287C2×D3—122no—
288C2×D3—122no—
289C2×D3—184no—
290C2×D3—362no—
291C2×D3—362no—
292C2×D3—3616no—
293C2×D3—368no—
294C3×C4—32no—
295C3×C4—32no—
296C3×C4—41no—
297C3×C4—62no—
298C3×C4—122no—
299C3×C4—122no—
300C3×C4—181no—
301C3×C4D3×D1212yes—
302Dic3—62no—
303Dic3—121no—
304C32—21no—
305C32#477312yescomplemented
306C23—91no—
307C23—182no—
308C23—181no—
309C23—544no—
310C23—1082no—
311C23—1082no—
312D4—94no—
313D4—182no—
314D4—182no—
315D4—362no—
316D4—544no—
317C2×C4—21no—
318C2×C4—91no—
319C2×C4—182no—
320C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
321C8—182no—
322C2×C3—32no—
323C2×C3—32no—
324C2×C3—32no—
325C2×C3—61no—
326C2×C3—61no—
327C2×C3—61no—
328C2×C3—182no—
329C2×C3—364no—
330C2×C3—721no—
331C2×C3#6392311yescharacteristic
332C2×C3#6392611yescharacteristic
333C2×C3#6394111yescharacteristic
334D3—64no—
335D3—122no—
336D3—124no—
337D3—242no—
338D3—242no—
339D3—364no—
340C22—92no—
341C22—181no—
342C22—181no—
343C22—184no—
344C22—362no—
345C22—362no—
346C22—361no—
347C22—361no—
348C22—361no—
349C22—1088no—
350C22#8867711yescharacteristic
351C4—41no—
352C4—181no—
353C4#8865412yes—
354C3—32no—
355C3—61no—
356C3#1059059811yescharacteristic
357C2—182no—
358C2—364no—
359C2—721no—
360C2#1172116311yescharacteristic
361C2#1172119011yescharacteristic
362C2#1172122911yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#447
LCSG ▷ C2×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
1. C2×#88655C2semidirect
2. #11721763C2semidirect
3. C2×#88436C2semidirect
4. #63636⋊C3C2semidirect
5. #11720323C2semidirect
9. C2×C4⋊#272C22semidirect
14. C4⋊C4×H(3,𝔽3)C22semidirect
16. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
23. #63636D3semidirect
39. C4⋊((C3⋊D3)⋊C3)D4semidirect
57. C32×C4⋊C4C2×D3semidirect
58. C2×C3⋊(C3⋊C8)C2×D3semidirect
70. C2×#272D8semidirect
105. C4⋊(C3⋊D3)D12semidirect
192. H(3,𝔽3)#447semidirect
260. C4⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
305. C32#4773semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
#117217632
C2×#884362
#63636⋊C32
#117203232
#105905953