Finite Group Databasebeta

← #13147852

Subgroups — #13147852

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#11721718C211yesmaximal, characteristic
3C4×#634⋊C3C212yesmaximal, complemented
4#11721089C211yesmaximal, characteristic
5#11721084C212yesmaximal, complemented
6#11721077C214yesmaximal, complemented
7#11721076C214yesmaximal
8#10593885—32nomaximal
9C2×C4×#272C2211yescharacteristic
10C2×#634⋊C3C2212yes—
11C2×#6199C2212yes—
12C2×#6196C2212yes—
13#88651C418yescomplemented
14C42×H(3,𝔽3)C2211yescharacteristic
15#88542C2212yes—
16#88540C2212yes—
17C4×#1084C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2212yes—
19#3750⋊C3C2214yes—
20C4×(C3⋊Dic3)⋊C3C2214yes—
21#88451C2212yes—
22#88448C2214yes—
23#88447C2214yes—
24#88447C2212yes—
25C2×D3×Dic6—32no—
26C4×#634D312yescomplemented
27C3×#872—32no—
28C3×#872—61no—
29C3×C4×Dic6—32no—
30#63878—34no—
31#63877—34no—
32#63844—32no—
33Dic3×Dic6—32no—
34#63841—32no—
35#63838—34no—
36#63837—34no—
37C22×#272C2311yescharacteristic
38C2×C4×H(3,𝔽3)C2312yes—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084C2312yes—
41C2×#1084C2311yescharacteristic
42C4×#272C2×C414yes—
43C2×(C3⋊Dic3)⋊C3C2314yes—
44C2×(C3⋊Dic3)⋊C3C2314yes—
45#634⋊C3C2×C418yes—
46#6199C2×C418yes—
47#6196C2×C418yes—
48#5572—92no—
49#5516—91no—
50C2×#634C2×D312yes—
51C2×C3×C4×D3—32no—
52C2×C3×C4×D3—61no—
53C2×C3×Dic6—32no—
54C2×#627—34no—
55C2×D3×Dic3—34no—
56D3×Dic6—316no—
57C32×C42—21no—
58C32×C42C2×D312yes—
59C4⋊(C3⋊Dic3)C2×D312yes—
60#3750C2×D314yes—
61C4×C3⋊Dic3C2×D314yes—
62C3×(C2×D3)⋊C4—32no—
63C3×(C2×D3)⋊C4—32no—
64C3×(C2×D3)⋊C4—62no—
65C3×C4⋊Dic3—32no—
66C3×Dic3⋊C4—32no—
67C3×Dic3⋊C4—32no—
68C3×Dic3⋊C4—34no—
69C3×Dic3⋊C4—62no—
70C3×C4×Dic3—34no—
71C3×C4×Dic3—32no—
72C3×C4×Dic3—61no—
73#3724—34no—
74Dic3⋊Dic3—34no—
75(C2×D3)⋊Dic3—34no—
76Dic32—34no—
77Dic32—34no—
78Dic32—34no—
79C22×H(3,𝔽3)C2411yescharacteristic
80C2×#272—22no—
81C2×#272C22×C412yes—
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)C22×C412yes—
84#1084—22no—
85#1084C22×C412yes—
86(C3⋊Dic3)⋊C3—24no—
87(C3⋊Dic3)⋊C3C22×C418yes—
88C2×D3×Q8—92no—
89C22×Dic6—91no—
90C3×C4×Q8—92no—
91C3×#117—91no—
92Q8×Dic3—92no—
93#926—91no—
94#892—94no—
95#892—92no—
96#887—94no—
97#887—92no—
98#887—94no—
99#877—94no—
100#872—32no—
101#872—61no—
102#872—94no—
103C4×Dic6—32no—
104C4×Dic6—92no—
105C4×Dic6—94no—
106C4×Dic6—92no—
107C22×C3×D3—32no—
108C22×C3×D3—61no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4C22×D312yes—
112C2×C32×C4C22×D312yes—
113C2×C32×C4C22×D312yes—
114C2×C3⋊Dic3C22×D314yes—
115C2×C3⋊Dic3C22×D314yes—
116#634C4×D318yes—
117C2×C3×Dic3—34no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—62no—
124C3×C4×D3—38no—
125C3×C4×D3—64no—
126C3×Dic6—38no—
127#627—316no—
128D3×Dic3—316no—
129#519—271noSylow
130C2×H(3,𝔽3)C23×C411yescharacteristic
131C2×H(3,𝔽3)2−1+412yes—
132#272—22no—
133C2×C3×Q8—92no—
134C22×C3×C4—91no—
135C22×Dic3—92no—
136D3×Q8—916no—
137C2×C4×D3—32no—
138C2×C4×D3—61no—
139C2×C4×D3—94no—
140C2×Dic6—32no—
141C2×Dic6—92no—
142C2×Dic6—98no—
143C2×Dic6—92no—
144C2×Dic6—94no—
145C3×C4⋊C4—94no—
146C3×C4⋊C4—92no—
147C3×C4⋊C4—92no—
148C3×C22⋊C4—92no—
149C3×C42—32no—
150C3×C42—61no—
151C3×C42—94no—
152C3×C42—91no—
153C3×C42D3211yescharacteristic
154#217—92no—
155(C2×D3)⋊C4—32no—
156(C2×D3)⋊C4—32no—
157(C2×D3)⋊C4—62no—
158(C2×D3)⋊C4—94no—
159C4⋊Dic3—32no—
160C4⋊Dic3—92no—
161C4⋊Dic3—94no—
162Dic3⋊C4—32no—
163Dic3⋊C4—32no—
164Dic3⋊C4—34no—
165Dic3⋊C4—62no—
166Dic3⋊C4—94no—
167Dic3⋊C4—94no—
168Dic3⋊C4—94no—
169C4×Dic3—32no—
170C4×Dic3—34no—
171C4×Dic3—61no—
172C4×Dic3—94no—
173C4×Dic3—94no—
174C4×Dic3—92no—
175C4×Dic3—94no—
176C4×Dic3—94no—
177C4×Dic3—94no—
178C22×C32—21no—
179C22×C32C23×D312yes—
180C2×C3×D3—34no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—122no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—42no—
189C32×C4C2×C4×D314yes—
190C3⋊Dic3—24no—
191C3⋊Dic3C2×C4×D318yes—
192C3×Dic3—38no—
193C3×Dic3—34no—
194C3×Dic3—62no—
195C3×Dic3—64no—
196C3×Dic3—62no—
197C3×Dic3—62no—
198C3×Dic3—122no—
199C22×Q8—271no—
200C4×Q8—92no—
201C4×Q8—274no—
202C4×Q8—272no—
203#117—91no—
204#117—274no—
205#117—271no—
206H(3,𝔽3)#51911yescharacteristic, complemented, Sylow
207C23×C3—91no—
208C22×D3—32no—
209C22×D3—61no—
210C3×Q8—98no—
211C2×C3×C4—32no—
212C2×C3×C4—32no—
213C2×C3×C4—32no—
214C2×C3×C4—61no—
215C2×C3×C4—62no—
216C2×C3×C4—94no—
217C2×C3×C4—91no—
218C2×C3×C4—94no—
219C2×C3×C4—94no—
220C2×C3×C4—92no—
221C2×C3×C4C2×D3212yes—
222C2×C3×C4C2×D3211yescharacteristic
223C2×Dic3—32no—
224C2×Dic3—34no—
225C2×Dic3—32no—
226C2×Dic3—32no—
227C2×Dic3—34no—
228C2×Dic3—61no—
229C2×Dic3—62no—
230C2×Dic3—94no—
231C2×Dic3—94no—
232C2×Dic3—98no—
233C4×D3—38no—
234C4×D3—64no—
235C4×D3—916no—
236Dic6—38no—
237Dic6—98no—
238Dic6—98no—
239Dic6—916no—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32C22×C4×D312yes—
243C2×C32#100712yes—
244C2×C32#101012yes—
245C3×D3—64no—
246C3×D3—122no—
247C2×Q8—92no—
248C2×Q8—272no—
249C2×Q8—278no—
250C22×C4—91no—
251C22×C4—272no—
252C4⋊C4—92no—
253C4⋊C4—94no—
254C4⋊C4—92no—
255C4⋊C4—274no—
256C22⋊C4—92no—
257C22⋊C4—272no—
258C42—91no—
259C42—94no—
260C42—274no—
261C42—272no—
262C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C22×C3—32no—
264C22×C3—61no—
265C22×C3—92no—
266C22×C3—182no—
267C22×C3C22×D3211yescharacteristic
268C2×D3—34no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—62no—
272C2×D3—122no—
273C3×C4—22no—
274C3×C4—34no—
275C3×C4—62no—
276C3×C4—62no—
277C3×C4—62no—
278C3×C4—98no—
279C3×C4—92no—
280C3×C4—122no—
281C3×C4—184no—
282C3×C4—182no—
283C3×C4C4×D3212yes—
284Dic3—34no—
285Dic3—38no—
286Dic3—62no—
287Dic3—62no—
288Dic3—62no—
289Dic3—64no—
290Dic3—98no—
291Dic3—122no—
292Dic3—184no—
293C32—21no—
294C32#557212yescomplemented
295C23—91no—
296Q8—98no—
297Q8—278no—
298C2×C4—94no—
299C2×C4—91no—
300C2×C4—94no—
301C2×C4—92no—
302C2×C4—94no—
303C2×C4—278no—
304C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
305C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C2×C3—32no—
307C2×C3—32no—
308C2×C3—32no—
309C2×C3—62no—
310C2×C3—61no—
311C2×C3—182no—
312C2×C3#6429712yes—
313C2×C3C2×C4×D3211yescharacteristic
314D3—64no—
315D3—122no—
316C22—92no—
317C22—182no—
318C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
319C4—22no—
320C4—92no—
321C4—98no—
322C4—182no—
323C4—184no—
324C4C4×((C3⋊D3)⋊C3)⋊C212yes—
325C3—32no—
326C3—61no—
327C3#1059384811yescharacteristic
328C2—182no—
329C2#1172248212yes—
330C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×#272C23
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#519
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
3. C4×#634⋊C3C2semidirect
5. #11721084C2semidirect
6. #11721077C2semidirect
13. #88651C4semidirect
26. C4×#634D3semidirect
206. H(3,𝔽3)#519semidirect
294. C32#5572semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
#117217182
C4×#634⋊C32
#117210892
#117210842
#117210772
#117210762
#105938853