Finite Group Databasebeta

← #13149570

Subgroups — #13149570

Order 1728.

1007 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 203 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
2#11721760C211yesmaximal, characteristic
3#64096⋊C3C212yesmaximal, complemented
4C4×#6196C212yesmaximal, complemented
5#11721097C211yesmaximal, characteristic
6#11721082C212yesmaximal, complemented
7#11721078C212yesmaximal, complemented
8#11721076C214yesmaximal
9#10594357—32nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#634⋊C3C2212yes—
12C2×#6199C2212yes—
13C2×#6196C2212yes—
14#88651C418yescomplemented
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#88542C2212yes—
17C4⋊#1084C2211yescharacteristic
18C4×#1084C2212yes—
19#3750⋊C3C2214yes—
20C4×(C3⋊Dic3)⋊C3C2214yes—
21C4×(C3⋊Dic3)⋊C3C2212yes—
22#88451C2212yes—
23#88449C2211yescharacteristic
24#88449C2211yescharacteristic
25#88448C2214yes—
26#88447C2214yes—
27C2×D3×Dic6—32no—
28#64096D312yescomplemented
29C3×#892—32no—
30C3×#892—61no—
31C3×#887—32no—
32C4×#627—34no—
33#63878—34no—
34#63877—34no—
35#63855—32no—
36#63844—32no—
37#63843—32no—
38#63837—34no—
39C22×#272C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084C2312yes—
43C2×#1084C2311yescharacteristic
44C4×#272C2×C414yes—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48#634⋊C3C2×C418yes—
49#6199C2×C418yes—
50#6196C2×C418yes—
51#5572—92no—
52#5516—91no—
53C2×#634C2×D312yes—
54C2×C3×C4×D3—32no—
55C2×C3×C4×D3—61no—
56C2×C3×Dic6—32no—
57C2×#627—34no—
58C2×D3×Dic3—34no—
59D3×Dic6—316no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D312yes—
62#3750C2×D314yes—
63C4×C3⋊Dic3C2×D312yes—
64C4×C3⋊Dic3C2×D314yes—
65C3×(C2×D3)⋊C4—34no—
66C3×(C2×D3)⋊C4—62no—
67C3×C4⋊Dic3—32no—
68C3×C4⋊Dic3—61no—
69C3×Dic3⋊C4—34no—
70C3×C4×Dic3—32no—
71C3×C4×Dic3—34no—
72C3×C4×Dic3—34no—
73C3×C4×Dic3—62no—
74#3724—34no—
75Dic3⋊Dic3—32no—
76Dic3⋊Dic3—34no—
77Dic3⋊Dic3—32no—
78(C2×D3)⋊Dic3—32no—
79(C2×D3)⋊Dic3—32no—
80Dic32—34no—
81Dic32—34no—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×#272—21no—
84C2×#272—21no—
85C2×#272C22×C412yes—
86C4×H(3,𝔽3)—22no—
87C4×H(3,𝔽3)C22×C412yes—
88#1084—22no—
89#1084C22×C412yes—
90(C3⋊Dic3)⋊C3—22no—
91(C3⋊Dic3)⋊C3—22no—
92(C3⋊Dic3)⋊C3C22×C418yes—
93C2×D3×Q8—92no—
94C22×Dic6—91no—
95C3×C4×Q8—92no—
96C3×#117—91no—
97Q8×Dic3—92no—
98#926—91no—
99#892—32no—
100#892—61no—
101#892—94no—
102#887—32no—
103#887—92no—
104#887—92no—
105#887—94no—
106#877—94no—
107#872—94no—
108#872—92no—
109C4×Dic6—92no—
110C4×Dic6—94no—
111C4×Dic6—92no—
112C4×Dic6—92no—
113C22×C3×D3—32no—
114C22×C3×D3—61no—
115C2×C32×C4—21no—
116C2×C32×C4—22no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D312yes—
119C2×C3⋊Dic3C22×D312yes—
120C2×C3⋊Dic3C22×D312yes—
121C2×C3⋊Dic3C22×D314yes—
122#634C4×D318yes—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—62no—
130C3×C4×D3—38no—
131C3×C4×D3—64no—
132C3×Dic6—38no—
133#627—316no—
134D3×Dic3—316no—
135#519—271noSylow
136C2×H(3,𝔽3)C23×C411yescharacteristic
137C2×H(3,𝔽3)2−1+411yescharacteristic
138C2×H(3,𝔽3)2−1+411yescharacteristic
139#272—22no—
140C2×C3×Q8—92no—
141C22×C3×C4—91no—
142C22×Dic3—92no—
143D3×Q8—916no—
144C2×C4×D3—32no—
145C2×C4×D3—61no—
146C2×C4×D3—94no—
147C2×Dic6—32no—
148C2×Dic6—92no—
149C2×Dic6—98no—
150C2×Dic6—92no—
151C2×Dic6—94no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—61no—
154C3×C4⋊C4—94no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4D3211yescharacteristic
157C3×C22⋊C4—92no—
158C3×C42—92no—
159C3×C42—94no—
160C3×C42—92no—
161#217—92no—
162(C2×D3)⋊C4—34no—
163(C2×D3)⋊C4—62no—
164(C2×D3)⋊C4—92no—
165(C2×D3)⋊C4—92no—
166C4⋊Dic3—32no—
167C4⋊Dic3—61no—
168C4⋊Dic3—91no—
169C4⋊Dic3—91no—
170C4⋊Dic3—94no—
171Dic3⋊C4—34no—
172Dic3⋊C4—94no—
173Dic3⋊C4—94no—
174Dic3⋊C4—92no—
175Dic3⋊C4—94no—
176Dic3⋊C4—92no—
177C4×Dic3—34no—
178C4×Dic3—34no—
179C4×Dic3—32no—
180C4×Dic3—62no—
181C4×Dic3—94no—
182C4×Dic3—92no—
183C4×Dic3—94no—
184C4×Dic3—94no—
185C4×Dic3—94no—
186C22×C32—21no—
187C22×C32C23×D312yes—
188C2×C3×D3—34no—
189C2×C3×D3—62no—
190C2×C3×D3—62no—
191C2×C3×D3—62no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C32×C4—24no—
195C32×C4—22no—
196C32×C4—42no—
197C32×C4C2×C4×D314yes—
198C3⋊Dic3—22no—
199C3⋊Dic3—22no—
200C3⋊Dic3C2×C4×D318yes—
201C3×Dic3—38no—
202C3×Dic3—34no—
203C3×Dic3—62no—
204C3×Dic3—62no—
205C3×Dic3—64no—
206C3×Dic3—62no—
207C3×Dic3—122no—
208C22×Q8—271no—
209C4×Q8—92no—
210C4×Q8—272no—
211C4×Q8—272no—
212C4×Q8—272no—
213#117—91no—
214#117—271no—
215#117—274no—
216H(3,𝔽3)#51911yescharacteristic, complemented, Sylow
217C23×C3—91no—
218C22×D3—32no—
219C22×D3—61no—
220C3×Q8—98no—
221C2×C3×C4—32no—
222C2×C3×C4—34no—
223C2×C3×C4—61no—
224C2×C3×C4—62no—
225C2×C3×C4—94no—
226C2×C3×C4—91no—
227C2×C3×C4—94no—
228C2×C3×C4—92no—
229C2×C3×C4—92no—
230C2×C3×C4—92no—
231C2×C3×C4C2×D3212yes—
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—32no—
234C2×Dic3—34no—
235C2×Dic3—34no—
236C2×Dic3—32no—
237C2×Dic3—32no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C2×Dic3—94no—
241C2×Dic3—92no—
242C2×Dic3—92no—
243C2×Dic3—98no—
244C4×D3—38no—
245C4×D3—64no—
246C4×D3—916no—
247Dic6—38no—
248Dic6—98no—
249Dic6—98no—
250Dic6—916no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32C22×C4×D312yes—
255C2×C32#100712yes—
256C2×C32#101012yes—
257C3×D3—64no—
258C3×D3—122no—
259C2×Q8—92no—
260C2×Q8—272no—
261C2×Q8—278no—
262C22×C4—91no—
263C22×C4—272no—
264C4⋊C4—91no—
265C4⋊C4—94no—
266C4⋊C4—271no—
267C4⋊C4—271no—
268C4⋊C4—274no—
269C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C22⋊C4—92no—
271C22⋊C4—272no—
272C42—92no—
273C42—94no—
274C42—92no—
275C42—274no—
276C22×C3—32no—
277C22×C3—61no—
278C22×C3—92no—
279C22×C3—181no—
280C22×C3—181no—
281C22×C3C22×D3211yescharacteristic
282C2×D3—34no—
283C2×D3—62no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—121no—
287C2×D3—121no—
288C3×C4—22no—
289C3×C4—34no—
290C3×C4—64no—
291C3×C4—62no—
292C3×C4—98no—
293C3×C4—92no—
294C3×C4—122no—
295C3×C4—182no—
296C3×C4—182no—
297C3×C4—182no—
298C3×C4C4×D3212yes—
299Dic3—34no—
300Dic3—38no—
301Dic3—62no—
302Dic3—64no—
303Dic3—62no—
304Dic3—62no—
305Dic3—98no—
306Dic3—122no—
307Dic3—182no—
308Dic3—182no—
309C32—21no—
310C32#557212yescomplemented
311C23—91no—
312Q8—98no—
313Q8—278no—
314C2×C4—91no—
315C2×C4—94no—
316C2×C4—94no—
317C2×C4—92no—
318C2×C4—92no—
319C2×C4—92no—
320C2×C4—278no—
321C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
322C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
323C2×C3—32no—
324C2×C3—32no—
325C2×C3—32no—
326C2×C3—61no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3—182no—
330C2×C3C2×C4×D3211yescharacteristic
331C2×C3#6430911yescharacteristic
332C2×C3#6431611yescharacteristic
333D3—64no—
334D3—122no—
335C22—92no—
336C22—181no—
337C22—181no—
338C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
339C4—22no—
340C4—92no—
341C4—98no—
342C4—182no—
343C4—182no—
344C4—182no—
345C4C4×((C3⋊D3)⋊C3)⋊C212yes—
346C3—32no—
347C3—61no—
348C3#1059434711yescharacteristic
349C2—182no—
350C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
351C2#1172250211yescharacteristic
352C2#1172251711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×#272C23
cosocleC22×H(3,𝔽3)C24
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#519
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #64096⋊C3C2semidirect
4. C4×#6196C2semidirect
6. #11721082C2semidirect
7. #11721078C2semidirect
14. #88651C4semidirect
28. #64096D3semidirect
216. H(3,𝔽3)#519semidirect
310. C32#5572semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
#117217602
#64096⋊C32
C4×#61962
#117210972
#117210822
#117210782
#117210762
#105943573