Finite Group Databasebeta

← #13149393

Subgroups — #13149393

Order 1728.

1074 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 161 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×#88651C212yesmaximal, complemented
2#11721761C211yesmaximal, characteristic
3#64097⋊C3C212yesmaximal, complemented
4#11721149C212yesmaximal
5#11721148C214yesmaximal, complemented
6#11721117C211yesmaximal, characteristic, complemented
7C4×#6196C211yesmaximal, characteristic, complemented
8#11721096C212yesmaximal, complemented
9#10594376—32nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272C2212yes—
12C2×#634⋊C3C2214yes—
13C2×#6199C2214yes—
14C2×#6196C2212yes—
15C2×#6196C2211yescharacteristic
16#88651—28no—
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18#88542C2212yes—
19C4×#1084C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21#3750⋊C3C2214yes—
22C4×(C3⋊Dic3)⋊C3C2212yes—
23#88451C2214yes—
24#88449C2211yescharacteristic
25#88449C2212yes—
26#88448C2212yes—
27C2×D3×Dic6—34no—
28#64097D312yescomplemented
29C3×#893—32no—
30C3×#893—61no—
31C3×#888—32no—
32#63918—32no—
33C4×#627—32no—
34#63898—32no—
35#63896—34no—
36Dic3×D12—32no—
37#63887—34no—
38#63851—34no—
39C22×#272C2312yes—
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084C2311yescharacteristic
43C2×#1084C2311yescharacteristic
44C4⋊#272Q814yescomplemented
45C4×#272—24no—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48C2×(C3⋊Dic3)⋊C3C2312yes—
49#634⋊C3—28no—
50#6199—28no—
51#6196—24no—
52#6196D414yescomplemented
53#5727—92no—
54D4×Dic6—91no—
55C2×#634C2×D314yes—
56C2×C3×D12—32no—
57C2×C3×D12—61no—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—62no—
60C2×C3×Dic6—34no—
61C2×#627—32no—
62C2×#627—34no—
63C2×D3×Dic3—34no—
64C2×D3×Dic3—34no—
65D3×Dic6—616no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D312yes—
68C4⋊(C3⋊Dic3)C2×D312yes—
69#3750C2×D314yes—
70C4×C3⋊Dic3C2×D312yes—
71C3×(C2×D3)⋊C4—34no—
72C3×(C2×D3)⋊C4—62no—
73C3×C4⋊Dic3—32no—
74C3×Dic3⋊C4—34no—
75C3×C4×Dic3—32no—
76C3×C4×Dic3—32no—
77C3×C4×Dic3—61no—
78#3724—32no—
79Dic3⋊Dic3—32no—
80Dic3⋊Dic3—32no—
81Dic3⋊Dic3—34no—
82(C2×D3)⋊Dic3—34no—
83(C2×D3)⋊Dic3—34no—
84C22×H(3,𝔽3)C2411yescharacteristic
85C2×#272—22no—
86C2×#272—22no—
87C2×#272C2×Q814yescomplemented
88C4×H(3,𝔽3)—22no—
89C4×H(3,𝔽3)C2×Q812yescomplemented
90#1084—21no—
91#1084C2×D412yes—
92(C3⋊Dic3)⋊C3—24no—
93(C3⋊Dic3)⋊C3—22no—
94(C3⋊Dic3)⋊C3C2×D414yes—
95C2×D3×Q8—94no—
96C22×Dic6—92no—
97C3×C4⋊Q8—92no—
98C3×C4×D4—91no—
99#946—94no—
100D4×Dic3—92no—
101#924—92no—
102#896—94no—
103#896—94no—
104#893—32no—
105#893—61no—
106#893—92no—
107#888—32no—
108#888—92no—
109#888—92no—
110#878—94no—
111C4×D12—92no—
112#869—91no—
113C4×Dic6—91no—
114C4×Dic6—92no—
115C22×C3×D3—34no—
116C22×C3×D3—62no—
117C2×C32×C4—21no—
118C2×C32×C4—22no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D312yes—
121C2×C3⋊Dic3C22×D312yes—
122C2×C3⋊Dic3C22×D314yes—
123C2×C3⋊Dic3C22×D312yes—
124#634—28no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C3×D12—38no—
133C3×D12—64no—
134C3×C4×D3—68no—
135C3×C4×D3—124no—
136C3×Dic6—68no—
137#627—38no—
138#627—68no—
139D3×Dic3—68no—
140D3×Dic3—68no—
141D4×Q8—271noSylow
142C2×H(3,𝔽3)C22×D411yescharacteristic
143C2×H(3,𝔽3)C22×Q811yescharacteristic
144C2×H(3,𝔽3)2−1+411yescharacteristic
145#272—24no—
146C2×C3×Q8—94no—
147C2×C3×D4—91no—
148C22×C3×C4—92no—
149C22×Dic3—94no—
150D3×Q8—1816no—
151C2×D12—32no—
152C2×D12—61no—
153C2×C4×D3—34no—
154C2×C4×D3—62no—
155C2×C4×D3—94no—
156C2×C4×D3—94no—
157C2×Dic6—34no—
158C2×Dic6—94no—
159C2×Dic6—91no—
160C2×Dic6—92no—
161C2×Dic6—92no—
162C2×Dic6—94no—
163C2×Dic6—188no—
164C3×C4⋊C4—32no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4—92no—
167C3×C4⋊C4—94no—
168C3×C4⋊C4D3211yescharacteristic
169C3×C22⋊C4—92no—
170C3×C42—92no—
171C3×C42—91no—
172#217—94no—
173(C2×D3)⋊C4—34no—
174(C2×D3)⋊C4—62no—
175(C2×D3)⋊C4—94no—
176(C2×D3)⋊C4—94no—
177C4⋊Dic3—32no—
178C4⋊Dic3—92no—
179C4⋊Dic3—91no—
180C4⋊Dic3—92no—
181C4⋊Dic3—92no—
182Dic3⋊C4—34no—
183Dic3⋊C4—94no—
184Dic3⋊C4—92no—
185Dic3⋊C4—92no—
186Dic3⋊C4—92no—
187Dic3⋊C4—94no—
188C4×Dic3—32no—
189C4×Dic3—32no—
190C4×Dic3—61no—
191C4×Dic3—92no—
192C22×C32—21no—
193C22×C32C23×D312yes—
194C2×C3×D3—38no—
195C2×C3×D3—64no—
196C2×C3×D3—64no—
197C2×C3×D3—64no—
198C2×C3×D3—122no—
199C2×C3×D3—122no—
200C32×C4—24no—
201C32×C4—22no—
202C32×C4—42no—
203C32×C4D3×Q814yescomplemented
204C3⋊Dic3—24no—
205C3⋊Dic3—22no—
206C3⋊Dic3D3×D414yes—
207C3×Dic3—34no—
208C3×Dic3—34no—
209C3×Dic3—62no—
210C3×Dic3—64no—
211C3×Dic3—62no—
212C3×Dic3—62no—
213C3×Dic3—121no—
214C22×Q8—272no—
215C4⋊Q8—92no—
216C4⋊Q8—271no—
217C22⋊Q8—274no—
218C22⋊Q8—272no—
219C4×Q8—271no—
220C4×D4—91no—
221C4×D4—272no—
222H(3,𝔽3)D4×Q811yescharacteristic, complemented, Sylow
223C23×C3—92no—
224C22×D3—34no—
225C22×D3—62no—
226C3×Q8—188no—
227C3×D4—94no—
228C2×C3×C4—32no—
229C2×C3×C4—34no—
230C2×C3×C4—61no—
231C2×C3×C4—62no—
232C2×C3×C4—92no—
233C2×C3×C4—91no—
234C2×C3×C4—92no—
235C2×C3×C4—94no—
236C2×C3×C4—91no—
237C2×C3×C4—184no—
238C2×C3×C4C2×D3212yes—
239C2×C3×C4C2×D3211yescharacteristic
240C2×Dic3—32no—
241C2×Dic3—32no—
242C2×Dic3—32no—
243C2×Dic3—32no—
244C2×Dic3—34no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—92no—
248C2×Dic3—92no—
249C2×Dic3—94no—
250C2×Dic3—188no—
251D12—38no—
252D12—64no—
253C4×D3—68no—
254C4×D3—124no—
255C4×D3—188no—
256C4×D3—188no—
257Dic6—68no—
258Dic6—94no—
259Dic6—98no—
260Dic6—188no—
261Dic6—188no—
262Dic6—184no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32C2×D3×D412yes—
267C2×C32C2×D3×Q812yes—
268C2×C32#100712yes—
269C3×D3—68no—
270C3×D3—124no—
271C2×Q8—94no—
272C2×Q8—271no—
273C2×Q8—272no—
274C2×Q8—548no—
275C2×D4—91no—
276C22×C4—92no—
277C22×C4—274no—
278C4⋊C4—92no—
279C4⋊C4—94no—
280C4⋊C4—271no—
281C4⋊C4—272no—
282C4⋊C4—272no—
283C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
284C22⋊C4—92no—
285C22⋊C4—274no—
286C42—91no—
287C42—92no—
288C22×C3—32no—
289C22×C3—61no—
290C22×C3—94no—
291C22×C3—182no—
292C22×C3—182no—
293C22×C3C22×D3211yescharacteristic
294C2×D3—38no—
295C2×D3—64no—
296C2×D3—64no—
297C2×D3—64no—
298C2×D3—122no—
299C2×D3—122no—
300C3×C4—22no—
301C3×C4—34no—
302C3×C4—64no—
303C3×C4—62no—
304C3×C4—94no—
305C3×C4—92no—
306C3×C4—122no—
307C3×C4—184no—
308C3×C4—182no—
309C3×C4—181no—
310C3×C4#379712yes—
311Dic3—34no—
312Dic3—34no—
313Dic3—62no—
314Dic3—62no—
315Dic3—64no—
316Dic3—62no—
317Dic3—94no—
318Dic3—121no—
319Dic3—184no—
320Dic3—182no—
321C32—21no—
322C32#572712yescomplemented
323C23—92no—
324Q8—188no—
325Q8—274no—
326Q8—544no—
327D4—94no—
328C2×C4—91no—
329C2×C4—92no—
330C2×C4—91no—
331C2×C4—92no—
332C2×C4—94no—
333C2×C4—184no—
334C2×C4—548no—
335C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
336C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C2×C3—32no—
338C2×C3—32no—
339C2×C3—32no—
340C2×C3—61no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—184no—
344C2×C3C2×#379711yescharacteristic
345C2×C3D32×D411yescharacteristic
346C2×C3#6431611yescharacteristic
347D3—68no—
348D3—124no—
349C22—94no—
350C22—182no—
351C22—182no—
352C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
353C4—22no—
354C4—92no—
355C4—94no—
356C4—184no—
357C4—181no—
358C4—182no—
359C4#8865112yescomplemented
360C3—32no—
361C3—61no—
362C3#1059437911yescharacteristic
363C2—184no—
364C2C2×#8865111yescharacteristic
365C2D4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C2#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 radicalC2×C4⋊#272C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)D4×Q8
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
1. C2×#88651C2semidirect
3. #64097⋊C3C2semidirect
5. #11721148C2semidirect
6. #11721117C2semidirect
7. C4×#6196C2semidirect
8. #11721096C2semidirect
28. #64097D3semidirect
44. C4⋊#272Q8semidirect
52. #6196D4semidirect
87. C2×#272C2×Q8semidirect
89. C4×H(3,𝔽3)C2×Q8semidirect
203. C32×C4D3×Q8semidirect
222. H(3,𝔽3)D4×Q8semidirect
322. C32#5727semidirect
359. C4#88651semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
#117217612
#64097⋊C32
#117211492
#117211482
#117211172
C4×#61962
#117210962
#105943763