Finite Group Databasebeta

← #13149402

Subgroups — #13149402

Order 1728.

836 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 141 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
1#11721761C211yesmaximal, characteristic
2#64096⋊C3C212yesmaximal, complemented
3C4×#6196C211yesmaximal, characteristic, complemented
4#11721114C212yesmaximal
5#11721099C211yesmaximal, characteristic, complemented
6#11721097C212yesmaximal, complemented
7#11721096C212yesmaximal, complemented
8#11721092C214yesmaximal, complemented
9#10594335—32nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272C2212yes—
12C2×#634⋊C3C2212yes—
13C2×#6196C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88542C2212yes—
16C4×#1084C2211yescharacteristic
17#3750⋊C3C2214yes—
18C4×(C3⋊Dic3)⋊C3C2214yes—
19C4×(C3⋊Dic3)⋊C3C2212yes—
20#88451C2214yes—
21#88451C2214yes—
22#88449C2212yes—
23#88449C2212yes—
24#88449C2211yescharacteristic
25#88448C2212yes—
26#64096D312yescomplemented
27C3×#893—32no—
28C3×#893—61no—
29C3×#887—32no—
30C4×#627—32no—
31#63866—32no—
32#63863—32no—
33#63862—34no—
34#63860—32no—
35#63859—34no—
36#63855—34no—
37#63851—34no—
38C22×#272C2312yes—
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2311yescharacteristic
42C2×#1084C2311yescharacteristic
43C4⋊#272—22no—
44C4×#272—24no—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48#634⋊C3—24no—
49#6196—22no—
50#5568—92no—
51#5542—91no—
52C2×#634C2×D312yes—
53C2×C3×D12—32no—
54C2×C3×D12—61no—
55C2×C3×C4×D3—34no—
56C2×C3×C4×D3—62no—
57C2×C3×Dic6—32no—
58C2×#627—32no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D312yes—
61#3750C2×D314yes—
62C4×C3⋊Dic3C2×D314yes—
63C4×C3⋊Dic3C2×D312yes—
64C3×(C2×D3)⋊C4—34no—
65C3×(C2×D3)⋊C4—62no—
66C3×Dic3⋊C4—34no—
67C3×C4×Dic3—34no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—61no—
71#3724—32no—
72Dic3⋊Dic3—34no—
73Dic3⋊Dic3—32no—
74Dic3⋊Dic3—34no—
75Dic3⋊Dic3—32no—
76(C2×D3)⋊Dic3—34no—
77(C2×D3)⋊Dic3—34no—
78(C2×D3)⋊Dic3—34no—
79(C2×D3)⋊Dic3—34no—
80C22×H(3,𝔽3)C2411yescharacteristic
81C2×#272—22no—
82C2×#272—22no—
83C2×#272—22no—
84C4×H(3,𝔽3)—22no—
85C4×H(3,𝔽3)#4112yes—
86#1084—21no—
87#1084#4112yes—
88(C3⋊Dic3)⋊C3—22no—
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3—22no—
91C3×C4×Q8—92no—
92C3×C4×D4—91no—
93#947—92no—
94#944—92no—
95#936—92no—
96#926—92no—
97#924—92no—
98#898—94no—
99#896—94no—
100#893—32no—
101#893—61no—
102#887—32no—
103#887—92no—
104#879—94no—
105#876—94no—
106#875—92no—
107#872—94no—
108#869—91no—
109C4×Dic6—91no—
110C4×Dic6—92no—
111C22×C3×D3—34no—
112C22×C3×D3—62no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D312yes—
117C2×C3⋊Dic3C22×D312yes—
118C2×C3⋊Dic3C22×D314yes—
119C2×C3⋊Dic3C22×D312yes—
120#634—24no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—64no—
129C3×D12—64no—
130C3×C4×D3—68no—
131C3×C4×D3—124no—
132C3×Dic6—64no—
133#627—64no—
134#556—271noSylow
135C2×H(3,𝔽3)C2×#4111yescharacteristic
136C2×H(3,𝔽3)C2×#4111yescharacteristic
137C2×H(3,𝔽3)2−1+411yescharacteristic
138#272—42no—
139C2×C3×Q8—92no—
140C2×C3×D4—91no—
141C22×C3×C4—92no—
142C2×D12—32no—
143C2×D12—61no—
144C2×C4×D3—34no—
145C2×C4×D3—62no—
146C2×Dic6—32no—
147C2×Dic6—92no—
148C2×Dic6—91no—
149C2×Dic6—92no—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—61no—
152C3×C4⋊C4—94no—
153C3×C4⋊C4D3211yescharacteristic
154C3×C22⋊C4—92no—
155C3×C42—92no—
156C3×C42—94no—
157C3×C42—91no—
158#217—94no—
159#217—94no—
160(C2×D3)⋊C4—34no—
161(C2×D3)⋊C4—62no—
162(C2×D3)⋊C4—94no—
163(C2×D3)⋊C4—94no—
164(C2×D3)⋊C4—94no—
165(C2×D3)⋊C4—94no—
166C4⋊Dic3—91no—
167C4⋊Dic3—92no—
168C4⋊Dic3—92no—
169C4⋊Dic3—92no—
170Dic3⋊C4—34no—
171Dic3⋊C4—94no—
172Dic3⋊C4—92no—
173Dic3⋊C4—92no—
174Dic3⋊C4—92no—
175Dic3⋊C4—94no—
176Dic3⋊C4—94no—
177C4×Dic3—32no—
178C4×Dic3—32no—
179C4×Dic3—34no—
180C4×Dic3—61no—
181C4×Dic3—92no—
182C4×Dic3—94no—
183C22×C32—21no—
184C22×C32C23×D312yes—
185C2×C3×D3—64no—
186C2×C3×D3—64no—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C2×C3×D3—122no—
190C2×C3×D3—122no—
191C32×C4—24no—
192C32×C4—22no—
193C32×C4—42no—
194C32×C4#23514yes—
195C3⋊Dic3—24no—
196C3⋊Dic3—22no—
197C3⋊Dic3—22no—
198C3×Dic3—34no—
199C3×Dic3—62no—
200C3×Dic3—64no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—121no—
205C4⋊Q8—271no—
206#126—274no—
207#124—272no—
208C22⋊Q8—272no—
209C4×Q8—92no—
210C4×Q8—271no—
211C4×D4—91no—
212#117—272no—
213H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
214C23×C3—92no—
215C22×D3—34no—
216C22×D3—62no—
217C3×Q8—184no—
218C3×D4—182no—
219C2×C3×C4—32no—
220C2×C3×C4—34no—
221C2×C3×C4—61no—
222C2×C3×C4—62no—
223C2×C3×C4—91no—
224C2×C3×C4—92no—
225C2×C3×C4—94no—
226C2×C3×C4—92no—
227C2×C3×C4—91no—
228C2×C3×C4—184no—
229C2×C3×C4C2×D3212yes—
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—32no—
232C2×Dic3—32no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—34no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—92no—
239C2×Dic3—92no—
240C2×Dic3—94no—
241D12—64no—
242D12—64no—
243C4×D3—68no—
244C4×D3—124no—
245Dic6—64no—
246Dic6—184no—
247Dic6—184no—
248Dic6—182no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32C2×#23512yes—
253C2×C32#100712yes—
254C2×C32D3×#4112yes—
255C3×D3—124no—
256C3×D3—124no—
257C2×Q8—92no—
258C2×Q8—271no—
259C2×D4—91no—
260C22×C4—92no—
261C4⋊C4—94no—
262C4⋊C4—271no—
263C4⋊C4—272no—
264C4⋊C4—272no—
265C4⋊C4—272no—
266C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
267C22⋊C4—92no—
268C22⋊C4—274no—
269C22⋊C4—274no—
270C42—94no—
271C42—91no—
272C42—92no—
273C22×C3—32no—
274C22×C3—61no—
275C22×C3—182no—
276C22×C3—182no—
277C22×C3—182no—
278C22×C3C22×D3211yescharacteristic
279C2×D3—64no—
280C2×D3—64no—
281C2×D3—64no—
282C2×D3—64no—
283C2×D3—122no—
284C2×D3—122no—
285C3×C4—22no—
286C3×C4—34no—
287C3×C4—64no—
288C3×C4—62no—
289C3×C4—92no—
290C3×C4—122no—
291C3×C4—182no—
292C3×C4—184no—
293C3×C4—182no—
294C3×C4—181no—
295C3×C4#379812yes—
296Dic3—34no—
297Dic3—62no—
298Dic3—62no—
299Dic3—64no—
300Dic3—62no—
301Dic3—62no—
302Dic3—121no—
303Dic3—182no—
304Dic3—184no—
305Dic3—182no—
306C32—21no—
307C32#556812yescomplemented
308C23—92no—
309Q8—184no—
310Q8—542no—
311D4—182no—
312C2×C4—91no—
313C2×C4—92no—
314C2×C4—91no—
315C2×C4—92no—
316C2×C4—94no—
317C2×C4—184no—
318C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
319C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C2×C3—32no—
321C2×C3—32no—
322C2×C3—32no—
323C2×C3—61no—
324C2×C3—61no—
325C2×C3—61no—
326C2×C3—362no—
327C2×C3C2×#379811yescharacteristic
328C2×C3#6431211yescharacteristic
329C2×C3#6431611yescharacteristic
330D3—124no—
331D3—124no—
332C22—182no—
333C22—182no—
334C22—182no—
335C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
336C4—22no—
337C4—92no—
338C4—182no—
339C4—181no—
340C4—182no—
341C4—184no—
342C4#8865012yes—
343C3—32no—
344C3—61no—
345C3#1059435011yescharacteristic
346C2—362no—
347C2C2×#8865011yescharacteristic
348C2#1172250511yescharacteristic
349C2#1172251711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC2×C4⋊#272C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#556
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64096⋊C3C2semidirect
3. C4×#6196C2semidirect
5. #11721099C2semidirect
6. #11721097C2semidirect
7. #11721096C2semidirect
8. #11721092C2semidirect
26. #64096D3semidirect
213. H(3,𝔽3)#556semidirect
307. C32#5568semidirect

Maximal subgroups

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

subgroupindex
#117217612
#64096⋊C32
C4×#61962
#117211142
#117210992
#117210972
#117210962
#117210922
#105943353