Finite Group Databasebeta

← #13138043

Subgroups — #13138043

Order 1728.

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

normal subgroups only

This group has 647 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×#88651C211yesmaximal, characteristic, complemented
2#11721763C211yesmaximal, characteristic
3C2×#88437C212yesmaximal, complemented
4#63637⋊C3C212yesmaximal, complemented
5#11720323C211yesmaximal, characteristic, complemented
6#10590603—32nomaximal
7C2×C4⋊#272—21no—
8C2×C4⋊#272C2211yescharacteristic
9C2×C4×#272C2211yescharacteristic
10C2×#634⋊C3C2212yes—
11C2×#6199—21no—
12C2×#6196—21no—
13#88651—24no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88542—21no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#88437—24no—
18C2×D3×Dic6—32no—
19C3×#895—32no—
20C3×#895—61no—
21C2×#3716—32no—
22C2×#3715—32no—
23#63637D312yescomplemented
24C3×#810—32no—
25(C2×D3)⋊(C3⋊C8)—32no—
26C22×#272—21no—
27C22×#272D411yescharacteristic
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084D411yescharacteristic
31C4⋊#272—22no—
32C4⋊#272—42no—
33C4×#272—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3D414yescomplemented
37#6199—24no—
38#6196—24no—
39(C3⋊(C3⋊C8))⋊C3—22no—
40#5040—91no—
41#4806—92no—
42C2×#634C2×D312yes—
43C2×C3×D12—32no—
44C2×C3×D12—62no—
45C2×C3×D12—61no—
46C2×C3×D12—62no—
47C2×C3×C4×D3—32no—
48C2×C3×C4×D3—61no—
49C2×C3×Dic6—32no—
50C2×#627—62no—
51C2×D3×Dic3—62no—
52D3×Dic6—64no—
53D3×Dic6—64no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4C2×D312yes—
56C2×C3⋊(C3⋊C8)C2×D312yes—
57C3×(C2×D3)⋊C4—62no—
58C3×(C2×D3)⋊C4—62no—
59C2×C3×C3⋊C8—32no—
60#3716—64no—
61#3715—64no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×#272—21no—
64C2×#272—21no—
65C2×#272—41no—
66C2×#272—41no—
67C2×#272—41no—
68C2×#272SD1612yescomplemented
69C4×H(3,𝔽3)—41no—
70C4×H(3,𝔽3)C2×D412yes—
71#1084—21no—
72(C3⋊Dic3)⋊C3—24no—
73C2×D3×Q8—92no—
74C22×Dic6—91no—
75C3×#121—91no—
76C2×#215—92no—
77C2×#214—92no—
78C2×#204—92no—
79#895—32no—
80#895—61no—
81C3×#103—92no—
82#830—91no—
83(C2×D3)⋊C8—92no—
84#810—32no—
85#810—92no—
86C22×C3×D3—32no—
87C22×C3×D3—61no—
88C22×C3×D3—62no—
89C22×C3×D3—62no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—21no—
93C2×C32×C4C22×D312yes—
94C2×C3⋊Dic3—22no—
95#634—22no—
96#634D1214yescomplemented
97C2×C3×Dic3—32no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—62no—
100C3×D12—62no—
101C3×D12—62no—
102C3×D12—122no—
103C3×D12—124no—
104C3×D12—124no—
105C3×C4×D3—62no—
106C3×C4×D3—62no—
107C3×C4×D3—122no—
108C3×Dic6—34no—
109C3×Dic6—62no—
110#627—68no—
111D3×Dic3—68no—
112C3⋊(C3⋊C8)—22no—
113C3×C3⋊C8—62no—
114#448—271noSylow
115C2×H(3,𝔽3)#12011yescharacteristic
116C2×H(3,𝔽3)C2×SD1611yescharacteristic
117C2×H(3,𝔽3)#13711yescharacteristic
118#272—22no—
119#272—81no—
120C2×C3×Q8—92no—
121C2×C3×D4—91no—
122C2×C3×D4—181no—
123C22×C3×C4—91no—
124C22×Dic3—181no—
125D3×Q8—184no—
126D3×Q8—184no—
127C2×D12—32no—
128C2×D12—61no—
129C2×D12—62no—
130C2×D12—62no—
131C2×C4×D3—32no—
132C2×C4×D3—61no—
133C2×C4×D3—182no—
134C2×Dic6—32no—
135C2×Dic6—92no—
136C2×Dic6—184no—
137C2×Dic6—182no—
138C2×Dic6—181no—
139C2×C3×C8—92no—
140C3×C4⋊C4—32no—
141C3×C4⋊C4—61no—
142C3×C4⋊C4D3211yescharacteristic
143C3×C22⋊C4—181no—
144#215—184no—
145#214—184no—
146(C2×D3)⋊C4—62no—
147(C2×D3)⋊C4—62no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—92no—
150#204—184no—
151C22×C32—21no—
152C22×C32D3×D412yes—
153C2×C3×D3—34no—
154C2×C3×D3—62no—
155C2×C3×D3—62no—
156C2×C3×D3—62no—
157C2×C3×D3—121no—
158C2×C3×D3—121no—
159C2×C3×D3—122no—
160C2×C3×D3—122no—
161C2×C3×D3—122no—
162C2×C3×D3—122no—
163C2×C3×D3—122no—
164C2×C3×D3—122no—
165C32×C4—22no—
166C32×C4—42no—
167C32×C4—42no—
168C32×C4C2×D1212yes—
169C32×C4D3×D412yes—
170C3⋊Dic3—24no—
171C3×Dic3—64no—
172C3×Dic3—62no—
173C3×Dic3—121no—
174C22×Q8—271no—
175C2×SD16—272no—
176#121—91no—
177#103—92no—
178C22⋊C8—271no—
179H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
180C23×C3—91no—
181C23×C3—181no—
182C22×D3—32no—
183C22×D3—61no—
184C22×D3—62no—
185C22×D3—62no—
186C3×Q8—94no—
187C3×Q8—182no—
188C3×D4—182no—
189C3×D4—362no—
190C2×C3×C4—21no—
191C2×C3×C4—32no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—61no—
195C2×C3×C4—91no—
196C2×C3×C4—182no—
197C2×C3×C4—182no—
198C2×C3×C4C2×D3211yescharacteristic
199C2×Dic3—32no—
200C2×Dic3—61no—
201C2×Dic3—62no—
202C2×Dic3—182no—
203C2×Dic3—184no—
204D12—62no—
205D12—62no—
206D12—122no—
207D12—124no—
208D12—124no—
209C4×D3—62no—
210C4×D3—62no—
211C4×D3—122no—
212C4×D3—188no—
213Dic6—34no—
214Dic6—62no—
215Dic6—94no—
216Dic6—182no—
217Dic6—184no—
218Dic6—188no—
219C3×C8—182no—
220C3⋊C8—62no—
221C3⋊C8—182no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32#88212yes—
226C2×C32D3×SD1612yes—
227C2×C32#91812yes—
228C3×D3—64no—
229C3×D3—122no—
230C3×D3—242no—
231C3×D3—242no—
232C2×Q8—92no—
233C2×Q8—541no—
234C2×Q8—544no—
235C2×D4—91no—
236C2×D4—181no—
237C22×C4—91no—
238C22×C4—541no—
239SD16—544no—
240C2×C8—92no—
241C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
242C22⋊C4—181no—
243C22×C3—32no—
244C22×C3—61no—
245C22×C3—92no—
246C22×C3—181no—
247C22×C3—181no—
248C22×C3—361no—
249C22×C3—361no—
250C22×C3—361no—
251C22×C3#381111yescharacteristic
252C2×D3—34no—
253C2×D3—62no—
254C2×D3—62no—
255C2×D3—62no—
256C2×D3—121no—
257C2×D3—121no—
258C2×D3—122no—
259C2×D3—122no—
260C2×D3—122no—
261C2×D3—122no—
262C2×D3—122no—
263C2×D3—122no—
264C3×C4—32no—
265C3×C4—32no—
266C3×C4—41no—
267C3×C4—62no—
268C3×C4—122no—
269C3×C4—122no—
270C3×C4—184no—
271C3×C4—181no—
272C3×C4D3×D1212yes—
273Dic3—62no—
274Dic3—64no—
275Dic3—121no—
276Dic3—184no—
277C32—21no—
278C32#480612yescomplemented
279C23—91no—
280C23—181no—
281Q8—94no—
282Q8—182no—
283Q8—544no—
284D4—182no—
285D4—362no—
286C2×C4—21no—
287C2×C4—91no—
288C2×C4—182no—
289C2×C4—182no—
290C2×C4—544no—
291C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C8—182no—
293C2×C3—32no—
294C2×C3—32no—
295C2×C3—32no—
296C2×C3—61no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—182no—
300C2×C3—721no—
301C2×C3#6392311yescharacteristic
302C2×C3#6393011yescharacteristic
303C2×C3#6394511yescharacteristic
304D3—64no—
305D3—122no—
306D3—242no—
307D3—242no—
308C22—92no—
309C22—181no—
310C22—181no—
311C22—361no—
312C22—361no—
313C22—361no—
314C22#8867711yescharacteristic
315C4—41no—
316C4—181no—
317C4—184no—
318C4#8865412yes—
319C3—32no—
320C3—61no—
321C3#1059060611yescharacteristic
322C2—182no—
323C2—721no—
324C2#1172116311yescharacteristic
325C2#1172118811yescharacteristic
326C2#1172122711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721763C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#448
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×#88651C2semidirect
3. C2×#88437C2semidirect
4. #63637⋊C3C2semidirect
5. #11720323C2semidirect
23. #63637D3semidirect
36. #634⋊C3D4semidirect
68. C2×#272SD16semidirect
96. #634D12semidirect
179. H(3,𝔽3)#448semidirect
278. C32#4806semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
#117217632
C2×#884372
#63637⋊C32
#117203232
#105906033