Finite Group Databasebeta

← #13149607

Subgroups — #13149607

Order 1728.

797 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 147 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#11721760C211yesmaximal, characteristic
2#64097⋊C3C212yesmaximal, complemented
3#11721152C211yesmaximal, characteristic
4#11721148C214yesmaximal
5#11721099C212yesmaximal, complemented
6#11721098C212yesmaximal, complemented
7#11721097C211yesmaximal, characteristic
8#11721093C212yesmaximal, complemented
9#10594502—32nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#634⋊C3C2214yes—
12C2×#6199C2212yes—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14#88542C2212yes—
15C4⋊#1084C2211yescharacteristic
16C4×#1084C2212yes—
17C4⋊((C3⋊Dic3)⋊C3)C2212yes—
18#3750⋊C3C2214yes—
19C4×(C3⋊Dic3)⋊C3C2212yes—
20#88451C2212yes—
21#88449C2214yes—
22#88449C2211yescharacteristic
23#88449C2211yescharacteristic
24#88449C2212yes—
25#88448C2214yes—
26#64097D312yescomplemented
27C3×#892—32no—
28C3×#892—61no—
29C3×#888—32no—
30#63896—34no—
31D3×C4⋊Dic3—32no—
32#63887—34no—
33#63872—32no—
34#63869—32no—
35#63867—34no—
36#63866—34no—
37#63855—32no—
38C22×#272C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2312yes—
42C2×#1084C2311yescharacteristic
43C4×#272Q814yescomplemented
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47#634⋊C3—44no—
48#6199—42no—
49#5729—92no—
50#5519—91no—
51C2×#634C2×D314yes—
52C2×C3×C4×D3—32no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—34no—
55C2×D3×Dic3—34no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D312yes—
58C4⋊(C3⋊Dic3)C2×D312yes—
59#3750C2×D314yes—
60C4×C3⋊Dic3C2×D312yes—
61C3×(C2×D3)⋊C4—34no—
62C3×(C2×D3)⋊C4—62no—
63C3×C4⋊Dic3—32no—
64C3×C4⋊Dic3—32no—
65C3×C4⋊Dic3—61no—
66C3×Dic3⋊C4—34no—
67C3×C4×Dic3—32no—
68C3×C4×Dic3—34no—
69C3×C4×Dic3—62no—
70#3724—34no—
71Dic3⋊Dic3—32no—
72Dic3⋊Dic3—32no—
73Dic3⋊Dic3—34no—
74Dic3⋊Dic3—34no—
75Dic3⋊Dic3—34no—
76Dic3⋊Dic3—34no—
77(C2×D3)⋊Dic3—32no—
78(C2×D3)⋊Dic3—32no—
79C22×H(3,𝔽3)C2411yescharacteristic
80C2×#272—21no—
81C2×#272—21no—
82C2×#272C2×Q812yes—
83C4×H(3,𝔽3)—22no—
84C4×H(3,𝔽3)C2×Q812yes—
85#1084—22no—
86#1084C2×Q812yes—
87(C3⋊Dic3)⋊C3—24no—
88(C3⋊Dic3)⋊C3—22no—
89(C3⋊Dic3)⋊C3—22no—
90C3×C4⋊Q8—92no—
91C3×#117—91no—
92#946—94no—
93#944—94no—
94#926—91no—
95C2×C4⋊Dic3—91no—
96#896—94no—
97#892—32no—
98#892—61no—
99D3×C4⋊C4—92no—
100#890—92no—
101#890—92no—
102#889—94no—
103#888—32no—
104#888—92no—
105#878—94no—
106#872—92no—
107#870—92no—
108#870—92no—
109#869—92no—
110C22×C3×D3—32no—
111C22×C3×D3—61no—
112C2×C32×C4—21no—
113C2×C32×C4—22no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D312yes—
116C2×C3⋊Dic3C22×D312yes—
117C2×C3⋊Dic3C22×D312yes—
118C2×C3⋊Dic3C22×D314yes—
119#634—44no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—62no—
127C3×C4×D3—38no—
128C3×C4×D3—64no—
129C3×Dic6—124no—
130D3×Dic3—124no—
131#544—271noSylow
132C2×H(3,𝔽3)C22×Q811yescharacteristic
133C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
134C2×H(3,𝔽3)2−1+411yescharacteristic
135#272—22no—
136C2×C3×Q8—94no—
137C22×C3×C4—91no—
138C22×Dic3—92no—
139C2×C4×D3—32no—
140C2×C4×D3—61no—
141C2×C4×D3—94no—
142C2×Dic6—34no—
143C2×Dic6—94no—
144C3×C4⋊C4—32no—
145C3×C4⋊C4—61no—
146C3×C4⋊C4—94no—
147C3×C4⋊C4—92no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4D3211yescharacteristic
150C3×C22⋊C4—92no—
151C3×C42—92no—
152C3×C42—92no—
153#217—92no—
154(C2×D3)⋊C4—34no—
155(C2×D3)⋊C4—62no—
156(C2×D3)⋊C4—92no—
157(C2×D3)⋊C4—92no—
158C4⋊Dic3—32no—
159C4⋊Dic3—32no—
160C4⋊Dic3—61no—
161C4⋊Dic3—92no—
162C4⋊Dic3—91no—
163C4⋊Dic3—92no—
164C4⋊Dic3—94no—
165C4⋊Dic3—91no—
166C4⋊Dic3—94no—
167Dic3⋊C4—34no—
168Dic3⋊C4—94no—
169Dic3⋊C4—94no—
170Dic3⋊C4—94no—
171Dic3⋊C4—94no—
172Dic3⋊C4—92no—
173Dic3⋊C4—92no—
174Dic3⋊C4—94no—
175Dic3⋊C4—94no—
176C4×Dic3—34no—
177C4×Dic3—32no—
178C4×Dic3—62no—
179C4×Dic3—92no—
180C22×C32—21no—
181C22×C32C23×D312yes—
182C2×C3×D3—34no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C32×C4—24no—
189C32×C4—22no—
190C32×C4—42no—
191C32×C4D3×Q814yes—
192C3⋊Dic3—24no—
193C3⋊Dic3—22no—
194C3⋊Dic3—22no—
195C3×Dic3—34no—
196C3×Dic3—62no—
197C3×Dic3—64no—
198C3×Dic3—62no—
199C3×Dic3—62no—
200C3×Dic3—64no—
201C3×Dic3—122no—
202C4⋊Q8—92no—
203C4⋊Q8—272no—
204#125—272no—
205#125—272no—
206C22⋊Q8—274no—
207#117—91no—
208#117—271no—
209C2×C4⋊C4—271no—
210H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
211C23×C3—91no—
212C22×D3—32no—
213C22×D3—61no—
214C3×Q8—364no—
215C2×C3×C4—32no—
216C2×C3×C4—34no—
217C2×C3×C4—61no—
218C2×C3×C4—62no—
219C2×C3×C4—91no—
220C2×C3×C4—94no—
221C2×C3×C4—92no—
222C2×C3×C4—92no—
223C2×C3×C4—94no—
224C2×C3×C4—92no—
225C2×C3×C4C2×D3212yes—
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—32no—
228C2×Dic3—34no—
229C2×Dic3—32no—
230C2×Dic3—32no—
231C2×Dic3—34no—
232C2×Dic3—61no—
233C2×Dic3—62no—
234C2×Dic3—92no—
235C2×Dic3—92no—
236C2×Dic3—94no—
237C2×Dic3—362no—
238C4×D3—38no—
239C4×D3—64no—
240C4×D3—364no—
241Dic6—124no—
242Dic6—364no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32#100412yes—
247C2×C32C2×D3×Q812yes—
248C2×C32#100712yes—
249C3×D3—64no—
250C3×D3—122no—
251C2×Q8—94no—
252C22×C4—91no—
253C22×C4—272no—
254C4⋊C4—92no—
255C4⋊C4—91no—
256C4⋊C4—94no—
257C4⋊C4—272no—
258C4⋊C4—271no—
259C4⋊C4—271no—
260C4⋊C4—274no—
261C4⋊C4—274no—
262C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C22⋊C4—92no—
264C22⋊C4—272no—
265C42—92no—
266C42—92no—
267C22×C3—32no—
268C22×C3—61no—
269C22×C3—92no—
270C22×C3—181no—
271C22×C3—181no—
272C22×C3C22×D3211yescharacteristic
273C2×D3—34no—
274C2×D3—62no—
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—121no—
278C2×D3—121no—
279C3×C4—22no—
280C3×C4—34no—
281C3×C4—64no—
282C3×C4—62no—
283C3×C4—92no—
284C3×C4—122no—
285C3×C4—184no—
286C3×C4—182no—
287C3×C4—182no—
288C3×C4—182no—
289C3×C4#379712yes—
290Dic3—34no—
291Dic3—62no—
292Dic3—64no—
293Dic3—62no—
294Dic3—64no—
295Dic3—62no—
296Dic3—122no—
297Dic3—182no—
298Dic3—184no—
299Dic3—182no—
300C32—21no—
301C32#572912yescomplemented
302C23—91no—
303Q8—364no—
304C2×C4—91no—
305C2×C4—94no—
306C2×C4—92no—
307C2×C4—92no—
308C2×C4—92no—
309C2×C4—94no—
310C2×C4—1082no—
311C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
312C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
313C2×C3—32no—
314C2×C3—32no—
315C2×C3—32no—
316C2×C3—61no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3—182no—
320C2×C3C2×#379711yescharacteristic
321C2×C3#6431311yescharacteristic
322C2×C3#6431611yescharacteristic
323D3—64no—
324D3—122no—
325C22—92no—
326C22—181no—
327C22—181no—
328C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C4—22no—
330C4—92no—
331C4—184no—
332C4—182no—
333C4—182no—
334C4—182no—
335C4#8865112yes—
336C3—32no—
337C3—61no—
338C3#1059435111yescharacteristic
339C2—182no—
340C2C2×#8865111yescharacteristic
341C2#1172250611yescharacteristic
342C2#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 radicalC22×#272C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
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. #64097⋊C3C2semidirect
5. #11721099C2semidirect
6. #11721098C2semidirect
8. #11721093C2semidirect
26. #64097D3semidirect
43. C4×#272Q8semidirect
210. H(3,𝔽3)#544semidirect
301. C32#5729semidirect

Maximal subgroups

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

subgroupindex
#117217602
#64097⋊C32
#117211522
#117211482
#117210992
#117210982
#117210972
#117210932
#105945023