Finite Group Databasebeta

← #13148454

Subgroups — #13148454

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
1C22×#6196C211yesmaximal, characteristic
2#11721733C211yesmaximal, characteristic
3#64086⋊C3C212yesmaximal
4#11721298C211yesmaximal, characteristic
5#11721296C212yesmaximal
6C4×#6196C212yesmaximal, complemented
7#11721078C212yesmaximal, complemented
8#11721077C214yesmaximal, complemented
9#10594245—32nomaximal
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6196C418yescomplemented
13C2×#6196C2212yes—
14C2×#6196C2211yescharacteristic
15C2×#6196C2211yescharacteristic
16C22⋊C4×H(3,𝔽3)C2211yescharacteristic
17#88552C2211yescharacteristic
18#88540C2212yes—
19C4×#1084C2212yes—
20#3757⋊C3C2212yes—
21#3750⋊C3C2214yes—
22C4×(C3⋊Dic3)⋊C3C2214yes—
23#88449C2212yes—
24#88448C2214yes—
25#88447C2214yes—
26#88447C2212yes—
27C22×#627—32no—
28#64086D312yes—
29C3×#877—32no—
30C3×#877—32no—
31C3×#877—61no—
32#63957—32no—
33#63955—32no—
34#63955—32no—
35C4×#627—34no—
36#63838—34no—
37#63837—34no—
38#63837—34no—
39C23×H(3,𝔽3)C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2312yes—
41C2×#1084C2×C414yes—
42C2×#1084C2312yes—
43C2×#1084C2311yescharacteristic
44C2×#1084C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3C2×C418yes—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48C2×(C3⋊Dic3)⋊C3C2312yes—
49#6196C2×C418yes—
50#6196C2×C414yes—
51#6196C2×C414yes—
52#5516—91no—
53#5516—92no—
54C22×C3⋊Dic3C2×D312yes—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—61no—
58C2×#627—316no—
59C2×#627—32no—
60C2×#627—32no—
61C2×#627—32no—
62C2×#627—32no—
63C32×C22⋊C4—21no—
64C32×C22⋊C4C2×D312yes—
65#3757C2×D312yes—
66#3750C2×D314yes—
67C4×C3⋊Dic3C2×D314yes—
68C3×#217—32no—
69C3×#217—32no—
70C3×#217—61no—
71C3×Dic3⋊C4—34no—
72C3×Dic3⋊C4—34no—
73C3×Dic3⋊C4—62no—
74C3×C4×Dic3—34no—
75C3×C4×Dic3—34no—
76C3×C4×Dic3—62no—
77#3724—34no—
78Dic3⋊Dic3—34no—
79Dic3⋊Dic3—34no—
80Dic32—34no—
81Dic32—34no—
82Dic32—34no—
83C22×H(3,𝔽3)—22no—
84C22×H(3,𝔽3)C22×C412yes—
85C22×H(3,𝔽3)C2411yescharacteristic
86C4×H(3,𝔽3)—22no—
87#1084—22no—
88#1084C22×C412yes—
89#1084C22×C412yes—
90(C3⋊Dic3)⋊C3—24no—
91(C3⋊Dic3)⋊C3C22×C414yes—
92(C3⋊Dic3)⋊C3C22×C414yes—
93C22×Dic6—91no—
94C22×Dic6—92no—
95C3×#117—92no—
96C3×#117—91no—
97#926—91no—
98#926—92no—
99#887—94no—
100#887—94no—
101#887—94no—
102#877—32no—
103#877—32no—
104#877—61no—
105#877—92no—
106#877—92no—
107#877—92no—
108#877—92no—
109C4×Dic6—92no—
110C4×Dic6—94no—
111C4×Dic6—92no—
112C4×Dic6—94no—
113C23×C32—21no—
114C23×C32C22×D312yes—
115C2×C32×C4—22no—
116C2×C32×C4C22×D314yes—
117C2×C3⋊Dic3C4×D318yes—
118C2×C3⋊Dic3C22×D314yes—
119C2×C3⋊Dic3C22×D312yes—
120C2×C3⋊Dic3C22×D312yes—
121C2×C3×Dic3—38no—
122C2×C3×Dic3—38no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133#627—38no—
134#627—38no—
135#627—38no—
136#627—38no—
137#519—271noSylow
138C2×H(3,𝔽3)—22no—
139C2×H(3,𝔽3)C23×C411yescharacteristic
140C2×H(3,𝔽3)2−1+412yes—
141C22×C3×C4—92no—
142C22×C3×C4—91no—
143C22×Dic3—32no—
144C22×Dic3—32no—
145C22×Dic3—61no—
146C22×Dic3—92no—
147C2×Dic6—91no—
148C2×Dic6—91no—
149C2×Dic6—92no—
150C2×Dic6—98no—
151C2×Dic6—92no—
152C2×Dic6—92no—
153C2×Dic6—92no—
154C2×Dic6—92no—
155C2×Dic6—916no—
156C3×C4⋊C4—94no—
157C3×C4⋊C4—92no—
158C3×C22⋊C4—32no—
159C3×C22⋊C4—61no—
160C3×C22⋊C4—92no—
161C3×C22⋊C4—91no—
162C3×C22⋊C4D3211yescharacteristic
163C3×C42—94no—
164C3×C42—92no—
165#217—32no—
166#217—32no—
167#217—61no—
168#217—92no—
169C4⋊Dic3—92no—
170C4⋊Dic3—94no—
171Dic3⋊C4—34no—
172Dic3⋊C4—34no—
173Dic3⋊C4—62no—
174Dic3⋊C4—94no—
175Dic3⋊C4—94no—
176Dic3⋊C4—94no—
177Dic3⋊C4—94no—
178C4×Dic3—34no—
179C4×Dic3—34no—
180C4×Dic3—62no—
181C4×Dic3—94no—
182C4×Dic3—94no—
183C4×Dic3—92no—
184C4×Dic3—94no—
185C4×Dic3—94no—
186C4×Dic3—94no—
187C22×C32—24no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—42no—
191C22×C32C2×C4×D314yes—
192C22×C32C23×D312yes—
193C32×C4—24no—
194C32×C4—42no—
195C3⋊Dic3—24no—
196C3⋊Dic3C2×C4×D314yes—
197C3⋊Dic3C2×C4×D314yes—
198C3×Dic3—34no—
199C3×Dic3—34no—
200C3×Dic3—34no—
201C3×Dic3—34no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—64no—
205C3×Dic3—64no—
206C3×Dic3—122no—
207C22×Q8—271no—
208C4×Q8—272no—
209C4×Q8—272no—
210C4×Q8—274no—
211#117—91no—
212#117—92no—
213#117—272no—
214#117—271no—
215H(3,𝔽3)#51911yescharacteristic, complemented, Sylow
216C23×C3—32no—
217C23×C3—61no—
218C23×C3C2×D3211yescharacteristic
219C2×C3×C4—34no—
220C2×C3×C4—62no—
221C2×C3×C4—92no—
222C2×C3×C4—92no—
223C2×C3×C4—98no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—94no—
227C2×C3×C4—94no—
228C2×C3×C4—92no—
229C2×C3×C4C2×D3212yes—
230C2×Dic3—32no—
231C2×Dic3—32no—
232C2×Dic3—38no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—38no—
236C2×Dic3—34no—
237C2×Dic3—34no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—64no—
241C2×Dic3—62no—
242C2×Dic3—98no—
243C2×Dic3—92no—
244C2×Dic3—92no—
245C2×Dic3—94no—
246Dic6—94no—
247Dic6—94no—
248Dic6—98no—
249Dic6—98no—
250Dic6—98no—
251Dic6—98no—
252Dic6—98no—
253C2×C32—24no—
254C2×C32—21no—
255C2×C32—22no—
256C2×C32—42no—
257C2×C32C22×C4×D312yes—
258C2×C32#101014yes—
259C2×Q8—271no—
260C2×Q8—271no—
261C2×Q8—272no—
262C2×Q8—278no—
263C22×C4—91no—
264C22×C4—92no—
265C4⋊C4—92no—
266C4⋊C4—94no—
267C4⋊C4—272no—
268C4⋊C4—274no—
269C22⋊C4—91no—
270C22⋊C4—92no—
271C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C42—92no—
273C42—94no—
274C42—272no—
275C42—274no—
276C22×C3—22no—
277C22×C3—32no—
278C22×C3—34no—
279C22×C3—64no—
280C22×C3—61no—
281C22×C3—62no—
282C22×C3—122no—
283C22×C3C4×D3212yes—
284C22×C3C22×D3211yescharacteristic
285C3×C4—22no—
286C3×C4—64no—
287C3×C4—94no—
288C3×C4—94no—
289C3×C4—92no—
290C3×C4—92no—
291C3×C4—122no—
292C3×C4—184no—
293C3×C4—182no—
294Dic3—34no—
295Dic3—34no—
296Dic3—34no—
297Dic3—34no—
298Dic3—62no—
299Dic3—62no—
300Dic3—64no—
301Dic3—64no—
302Dic3—94no—
303Dic3—94no—
304Dic3—122no—
305Dic3—184no—
306C32—21no—
307C32#551612yescomplemented
308C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
309Q8—274no—
310Q8—274no—
311Q8—278no—
312C2×C4—91no—
313C2×C4—92no—
314C2×C4—91no—
315C2×C4—92no—
316C2×C4—94no—
317C2×C4—98no—
318C2×C4—92no—
319C2×C4—94no—
320C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
321C2×C3—22no—
322C2×C3—34no—
323C2×C3—32no—
324C2×C3—64no—
325C2×C3—62no—
326C2×C3—61no—
327C2×C3—122no—
328C2×C3C2×C4×D3211yescharacteristic
329C2×C3#6430912yes—
330C22—22no—
331C22C4×((C3⋊D3)⋊C3)⋊C212yes—
332C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C4—22no—
334C4—92no—
335C4—92no—
336C4—94no—
337C4—94no—
338C4—182no—
339C4—184no—
340C3—32no—
341C3—61no—
342C3#1059424511yescharacteristic
343C2—22no—
344C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
345C2#1172250212yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
cosocleC22×H(3,𝔽3)C24
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#519
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
6. C4×#6196C2semidirect
7. #11721078C2semidirect
8. #11721077C2semidirect
12. C2×#6196C4semidirect
215. H(3,𝔽3)#519semidirect
307. C32#5516semidirect

Maximal subgroups

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

subgroupindex
C22×#61962
#117217332
#64086⋊C32
#117212982
#117212962
C4×#61962
#117210782
#117210772
#105942453