Finite Group Databasebeta

← #13147763

Subgroups — #13147763

Order 1728.

740 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 139 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#11721715C211yesmaximal, characteristic, complemented
2#64081⋊C3C211yesmaximal, characteristic
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721111C214yesmaximal
5#11721107C212yesmaximal
6#11721085C214yesmaximal, complemented
7#11721083C212yesmaximal, complemented
8#10593867—31nomaximal
9#10593862—31nomaximal
10C2×#6229C2212yes—
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2214yes—
14#3752⋊C3C2212yes—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16#3750⋊C3C2212yes—
17#3750⋊C3C2214yes—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19#88450C2214yes—
20#88449C2214yes—
21#88448C2214yes—
22#88447C2214yes—
23#64081D311yescharacteristic
24#64079D311yescharacteristic, complemented
25C3×#872—31no—
26C3×#870—31no—
27C3×#869—31no—
28C3×#869—31no—
29C3×#869—61no—
30#63871—32no—
31#63858—34no—
32#63856—32no—
33#63848—34no—
34#63847—34no—
35#63846—34no—
36#63843—32no—
37Dic3×Dic6—32no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2314yes—
42#6229—24no—
43C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2312yes—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46C4×(C3⋊D3)⋊C3—22no—
47#5731—91no—
48#5698—91no—
49#5518—91no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C2×C3×C4×D3—31no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—62no—
55C32×C42—21no—
56C32×C42C2×D311yescharacteristic
57C32×C42C2×D311yescharacteristic
58#3752C2×D312yes—
59C4⋊(C3⋊Dic3)C2×D312yes—
60#3750C2×D314yes—
61#3750C2×D312yes—
62C4×C3⋊Dic3C2×D311yescharacteristic
63C3×(C2×D3)⋊C4—32no—
64C3×C4⋊Dic3—32no—
65C3×C4⋊Dic3—34no—
66C3×C4⋊Dic3—34no—
67C3×C4⋊Dic3—64no—
68C3×Dic3⋊C4—34no—
69C3×Dic3⋊C4—32no—
70C3×C4×Dic3—31no—
71#3724—34no—
72Dic3⋊Dic3—34no—
73Dic3⋊Dic3—34no—
74Dic3⋊Dic3—34no—
75#3722—34no—
76(C2×D3)⋊Dic3—34no—
77Dic32—34no—
78Dic32—34no—
79C22×H(3,𝔽3)C2411yescharacteristic
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—22no—
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)#4112yes—
84#1084—24no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—22no—
87(C3⋊Dic3)⋊C3—24no—
88C3×C4⋊Q8—91no—
89C3×#125—91no—
90C3×#117—91no—
91#946—92no—
92Q8×Dic3—92no—
93#924—92no—
94#898—94no—
95#898—94no—
96#897—92no—
97#890—94no—
98#889—94no—
99#889—94no—
100#887—92no—
101#879—94no—
102#872—31no—
103#872—91no—
104#870—31no—
105#870—91no—
106#869—31no—
107#869—31no—
108#869—61no—
109C4×Dic6—92no—
110C22×C3⋊D3C22×D311yescharacteristic
111C22×C3×D3—31no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D312yes—
115C2×C32×C4C22×D312yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3C22×D312yes—
119C2×C3⋊Dic3C22×D314yes—
120C2×C3⋊Dic3C22×D311yescharacteristic
121C4×C3⋊D3—22no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—64no—
128C3×C4×D3—62no—
129C3×Dic6—64no—
130C3×Dic6—64no—
131C3×Dic6—124no—
132#541—271noSylow
133C2×H(3,𝔽3)C2×#4111yescharacteristic
134C2×H(3,𝔽3)2−1+412yes—
135(C3⋊D3)⋊C3—41no—
136C2×C3×Q8—92no—
137C22×C3×C4—91no—
138C2×C4×D3—31no—
139C2×C4×D3—91no—
140C2×Dic6—32no—
141C2×Dic6—32no—
142C2×Dic6—62no—
143C3×C4⋊C4—92no—
144C3×C4⋊C4—92no—
145C3×C4⋊C4—94no—
146C3×C4⋊C4—94no—
147C3×C22⋊C4—92no—
148C3×C42—31no—
149C3×C42—31no—
150C3×C42—61no—
151C3×C42—91no—
152C3×C42D3211yescharacteristic
153#217—94no—
154(C2×D3)⋊C4—32no—
155(C2×D3)⋊C4—92no—
156(C2×D3)⋊C4—94no—
157(C2×D3)⋊C4—94no—
158C4⋊Dic3—34no—
159C4⋊Dic3—34no—
160C4⋊Dic3—32no—
161C4⋊Dic3—64no—
162C4⋊Dic3—92no—
163C4⋊Dic3—94no—
164C4⋊Dic3—94no—
165Dic3⋊C4—34no—
166Dic3⋊C4—32no—
167Dic3⋊C4—94no—
168Dic3⋊C4—92no—
169Dic3⋊C4—94no—
170Dic3⋊C4—94no—
171Dic3⋊C4—94no—
172Dic3⋊C4—94no—
173C4×Dic3—31no—
174C4×Dic3—91no—
175C4×Dic3—94no—
176C4×Dic3—94no—
177C4×Dic3—94no—
178C22×C32—21no—
179C22×C32C23×D311yescharacteristic
180C22×C32C23×D311yescharacteristic
181C2×C3⋊D3—21no—
182C2×C3⋊D3—22no—
183C2×C3×D3—61no—
184C2×C3×D3—62no—
185C32×C4—24no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4#23512yes—
190C32×C4#23912yes—
191C3⋊Dic3—21no—
192C3⋊Dic3—24no—
193C3⋊Dic3—22no—
194C3×Dic3—64no—
195C3×Dic3—61no—
196C3×Dic3—64no—
197C3×Dic3—62no—
198C3×Dic3—64no—
199C3×Dic3—124no—
200C4⋊Q8—91no—
201#126—274no—
202#125—91no—
203#125—274no—
204C22⋊Q8—272no—
205C4×Q8—272no—
206#117—91no—
207H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
208C23×C3—91no—
209C22×D3—31no—
210C22×D3—91no—
211C3×Q8—184no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—62no—
217C2×C3×C4—61no—
218C2×C3×C4—92no—
219C2×C3×C4—94no—
220C2×C3×C4—91no—
221C2×C3×C4—94no—
222C2×C3×C4—182no—
223C2×C3×C4C2×D3212yes—
224C2×C3×C4C2×D3211yescharacteristic
225C2×Dic3—34no—
226C2×Dic3—34no—
227C2×Dic3—32no—
228C2×Dic3—34no—
229C2×Dic3—31no—
230C2×Dic3—64no—
231C2×Dic3—92no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C4×D3—62no—
235C4×D3—182no—
236Dic6—64no—
237Dic6—64no—
238Dic6—124no—
239C2×C32—22no—
240C2×C32—21no—
241C2×C32C2×#23511yescharacteristic
242C2×C32C2×#23911yescharacteristic
243C2×C32#101012yes—
244C2×C32#101012yes—
245C3⋊D3—41no—
246C3×D3—121no—
247C2×Q8—92no—
248C22×C4—91no—
249C4⋊C4—92no—
250C4⋊C4—92no—
251C4⋊C4—94no—
252C4⋊C4—94no—
253C4⋊C4—274no—
254C4⋊C4—274no—
255C22⋊C4—92no—
256C22⋊C4—274no—
257C42—91no—
258C42—274no—
259C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—61no—
263C22×C3—181no—
264C22×C3—182no—
265C22×C3C22×D3211yescharacteristic
266C2×D3—62no—
267C2×D3—61no—
268C2×D3—181no—
269C2×D3—182no—
270C3×C4—22no—
271C3×C4—32no—
272C3×C4—32no—
273C3×C4—62no—
274C3×C4—62no—
275C3×C4—64no—
276C3×C4—62no—
277C3×C4—181no—
278C3×C4—182no—
279C3×C4—184no—
280C3×C4—184no—
281C3×C4#379912yes—
282Dic3—64no—
283Dic3—64no—
284Dic3—62no—
285Dic3—64no—
286Dic3—61no—
287Dic3—124no—
288Dic3—181no—
289Dic3—184no—
290Dic3—182no—
291C32—21no—
292C32#551811yescharacteristic, complemented
293C32#569811yescharacteristic, complemented
294C23—91no—
295Q8—184no—
296C2×C4—92no—
297C2×C4—94no—
298C2×C4—94no—
299C2×C4—91no—
300C2×C4—182no—
301C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
302C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
303C2×C3—32no—
304C2×C3—32no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—62no—
308C2×C3—61no—
309C2×C3—361no—
310C2×C3#6429812yes—
311C2×C3C2×#379911yescharacteristic
312D3—121no—
313D3—361no—
314C22—182no—
315C22—181no—
316C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
317C4—22no—
318C4—182no—
319C4—184no—
320C4—181no—
321C4—184no—
322C4#8865212yes—
323C3—31no—
324C3—31no—
325C3—61no—
326C3#1059388111yescharacteristic
327C2—361no—
328C2#1172248312yes—
329C2C2×#8865211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721715C2semidirect
3. #64079⋊C3C2semidirect
6. #11721085C2semidirect
7. #11721083C2semidirect
24. #64079D3semidirect
207. H(3,𝔽3)#541semidirect
292. C32#5518semidirect
293. C32#5698semidirect

Maximal subgroups

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

subgroupindex
#117217152
#64081⋊C32
#64079⋊C32
#117211112
#117211072
#117210852
#117210832
#105938673
#105938623