Finite Group Databasebeta

← #13140995

Subgroups — #13140995

Order 1728.

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

normal subgroups only

This group has 732 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
2#11721863C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C212yesmaximal, complemented
4#11720323C211yesmaximal, characteristic, complemented
5#11720308C212yesmaximal, complemented
6#10591651—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2—21no—
8C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
9C2×C4×#272C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×#6198—21no—
12#88655—24no—
13#88542—21no—
14C4⋊#1084C2211yescharacteristic, complemented
15#88540—21no—
16#3755⋊C3—24no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18C2×D3×D12—32no—
19C2×#3755D312yescomplemented
20C3×#946—32no—
21C3×#946—61no—
22C2×C3×#215—32no—
23#63564—32no—
24#63562—32no—
25(C2×D3)⋊(C3⋊C8)—32no—
26C22×#272D411yescharacteristic
27C22×(C3⋊D3)⋊C3—22no—
28C2×((C3⋊D3)⋊C3)⋊C2—24no—
29C2×((C3⋊D3)⋊C3)⋊C2—42no—
30C2×((C3⋊D3)⋊C3)⋊C2—42no—
31Q8×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—21no—
35C2×#1084D411yescharacteristic
36C4×#272—22no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4⋊((C3⋊D3)⋊C3)D414yes—
39#6198—24no—
40(C3⋊(C3⋊C8))⋊C3—22no—
41#5169—92no—
42#5046—91no—
43C22×D32—62no—
44C2×C32×Q8—21no—
45C2×C32×Q8C2×D312yescomplemented
46C2×C4⋊(C3⋊D3)C2×D312yes—
47C2×C3×D12—32no—
48C2×C3×C4×D3—32no—
49C2×C3×C4×D3—61no—
50C2×#626—62no—
51D3×D12—64no—
52D3×D12—64no—
53#3755—24no—
54C2×C3⋊(C3⋊C8)C2×D312yescomplemented
55C3×#215—64no—
56C3×(C2×D3)⋊C4—62no—
57C3×(C2×D3)⋊C4—62no—
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—61no—
60C3×Dic3⋊C4—62no—
61C3×Dic3⋊C4—62no—
62C2×C3×C3⋊C8—32no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—21no—
65C2×#272—21no—
66C2×#272SD1612yescomplemented
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)C2×D412yes—
73#1084—21no—
74#1084—41no—
75C2×D3×D4—92no—
76C22×D12—91no—
77C2×C3×SD16—92no—
78C3×C22⋊Q8—91no—
79#946—32no—
80#946—61no—
81C2×#215—32no—
82C2×#215—92no—
83#832—92no—
84#830—91no—
85#821—92no—
86(C2×D3)⋊C8—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—
95C32×Q8—22no—
96C32×Q8—22no—
97C32×Q8—42no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4C22×D312yes—
102C4⋊(C3⋊D3)—22no—
103C4⋊(C3⋊D3)#6714yes—
104C2×C3×Dic3—32no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C3×D12—34no—
109C3×D12—62no—
110C3×C4×D3—62no—
111C3×C4×D3—62no—
112C3×C4×D3—122no—
113#626—68no—
114C3⋊(C3⋊C8)—22no—
115C3×C3⋊C8—62no—
116#450—271noSylow
117C2×H(3,𝔽3)#12011yescharacteristic
118C2×H(3,𝔽3)C2×SD1611yescharacteristic
119C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
120#272—22no—
121(C3⋊D3)⋊C3—44no—
122C23×D3—182no—
123C23×D3—181no—
124C2×C3×Q8—32no—
125C2×C3×Q8—61no—
126C2×C3×Q8D3211yescharacteristic
127C2×C3×D4—92no—
128C22×C3×C4—91no—
129C2×#67—182no—
130D3×D4—184no—
131D3×D4—184no—
132C2×D12—32no—
133C2×D12—92no—
134C2×D12—184no—
135C2×D12—181no—
136C2×C4×D3—32no—
137C2×C4×D3—61no—
138C3×SD16—184no—
139C2×C3×C8—92no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—181no—
142C3×C22⋊C4—181no—
143#215—64no—
144#215—184no—
145(C2×D3)⋊C4—62no—
146(C2×D3)⋊C4—62no—
147C4⋊Dic3—32no—
148C4⋊Dic3—61no—
149Dic3⋊C4—62no—
150Dic3⋊C4—62no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—92no—
153C22×C32—21no—
154C22×C32D3×D412yes—
155C2×C3⋊D3—24no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3—42no—
158C2×C3×D3—34no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—64no—
163C2×C3×D3—121no—
164C2×C3×D3—121no—
165C2×C3×D3—122no—
166C2×C3×D3—122no—
167D32—1216no—
168C32×C4—22no—
169C32×C4—42no—
170C32×C4—42no—
171C32×C4D3×D412yes—
172C32×C4C2×#6712yes—
173C3×Dic3—62no—
174C3×Dic3—121no—
175C3×Dic3—122no—
176C3×Dic3—122no—
177C22×D4—271no—
178C2×SD16—92no—
179C22⋊Q8—91no—
180#102—272no—
181C22⋊C8—271no—
182H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
183C23×C3—91no—
184C23×C3—182no—
185C22×D3—32no—
186C22×D3—61no—
187C22×D3—62no—
188C22×D3—182no—
189C22×D3—184no—
190C22×D3—188no—
191C22×D3—364no—
192C22×D3—364no—
193C22×D3—362no—
194C22×D3—362no—
195C3×Q8—22no—
196C3×Q8—62no—
197C3×Q8—62no—
198C3×Q8—122no—
199C3×D4—94no—
200C3×D4—182no—
201C2×C3×C4—21no—
202C2×C3×C4—32no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—61no—
206C2×C3×C4—91no—
207C2×C3×C4—182no—
208C2×C3×C4—181no—
209C2×C3×C4C2×D3211yescharacteristic
210#67—188no—
211C2×Dic3—32no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215D12—34no—
216D12—62no—
217D12—94no—
218D12—182no—
219D12—184no—
220C4×D3—62no—
221C4×D3—62no—
222C4×D3—122no—
223C3×C8—182no—
224C3⋊C8—62no—
225C3⋊C8—182no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32D3×SD1612yes—
230C2×C32#91412yes—
231C2×C32#93712yes—
232C3⋊D3—44no—
233C3×D3—64no—
234C3×D3—122no—
235C3×D3—124no—
236C24—541no—
237C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
238C2×D4—92no—
239C2×D4—541no—
240C2×D4—544no—
241C22×C4—91no—
242SD16—184no—
243C2×C8—92no—
244C4⋊C4—91no—
245C4⋊C4—181no—
246C22⋊C4—181no—
247C22×C3—32no—
248C22×C3—61no—
249C22×C3—92no—
250C22×C3—184no—
251C22×C3—181no—
252C22×C3—181no—
253C22×C3—362no—
254C22×C3—362no—
255C22×C3#381111yescharacteristic
256C2×D3—34no—
257C2×D3—62no—
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—64no—
261C2×D3—121no—
262C2×D3—121no—
263C2×D3—122no—
264C2×D3—122no—
265C2×D3—184no—
266C2×D3—362no—
267C2×D3—362no—
268C2×D3—3616no—
269C2×D3—368no—
270C3×C4—32no—
271C3×C4—32no—
272C3×C4—41no—
273C3×C4—62no—
274C3×C4—122no—
275C3×C4—122no—
276C3×C4—181no—
277C3×C4—361no—
278C3×C4D3×#6712yes—
279Dic3—62no—
280Dic3—121no—
281Dic3—122no—
282Dic3—122no—
283C32—21no—
284C32#516912yescomplemented
285C23—91no—
286C23—182no—
287C23—544no—
288C23—1082no—
289C23—1082no—
290Q8—22no—
291D4—94no—
292D4—182no—
293D4—544no—
294C2×C4—21no—
295C2×C4—91no—
296C2×C4—182no—
297C2×C4—181no—
298C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C8—182no—
300C2×C3—32no—
301C2×C3—32no—
302C2×C3—32no—
303C2×C3—61no—
304C2×C3—61no—
305C2×C3—61no—
306C2×C3—182no—
307C2×C3—364no—
308C2×C3#6394011yescharacteristic
309C2×C3#6394111yescharacteristic
310C2×C3#6398311yescharacteristic
311D3—64no—
312D3—122no—
313D3—124no—
314D3—364no—
315C22—92no—
316C22—181no—
317C22—181no—
318C22—184no—
319C22—362no—
320C22—362no—
321C22—1088no—
322C22#8867711yescharacteristic
323C4—41no—
324C4—181no—
325C4—361no—
326C4#8867612yes—
327C3—32no—
328C3—61no—
329C3#1059165211yescharacteristic
330C2—182no—
331C2—364no—
332C2#1172122811yescharacteristic
333C2#1172122911yescharacteristic
334C2#1172132411yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×#88655C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#450
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11721863C2semidirect
3. C2×#3755⋊C3C2semidirect
4. #11720323C2semidirect
5. #11720308C2semidirect
8. C2×Q8×H(3,𝔽3)C22semidirect
14. C4⋊#1084C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
19. C2×#3755D3semidirect
45. C2×C32×Q8C2×D3semidirect
54. C2×C3⋊(C3⋊C8)C2×D3semidirect
66. C2×#272SD16semidirect
182. H(3,𝔽3)#450semidirect
237. C2×Q8((C3⋊D3)⋊C3)⋊C2semidirect
284. C32#5169semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
#117218632
C2×#3755⋊C32
#117203232
#117203082
#105916513