Finite Group Databasebeta

← #13149576

Subgroups — #13149576

Order 1728.

1107 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1107 conjugacy classes of subgroups — too many to draw. Only its 179 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
2C4⋊C4×#272C211yesmaximal, characteristic
3#64097⋊C3C212yesmaximal, complemented
4#11721152C211yesmaximal, characteristic
5#11721147C212yesmaximal
6C4×#6199C211yesmaximal, characteristic
7#11721116C212yesmaximal, complemented
8#11721079C214yesmaximal, complemented
9D3×#888—32nomaximal
10C2×C4×#272C2212yes—
11C2×C4×#272C2211yescharacteristic
12C2×#634⋊C3C2214yes—
13C2×#6199C2212yes—
14C2×#6199C2211yescharacteristic
15C2×#6199C2211yescharacteristic
16C2×#6196C2214yes—
17#88651—28no—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540C2212yes—
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22#3750⋊C3C2214yes—
23C4×(C3⋊Dic3)⋊C3C2212yes—
24#88449C2212yes—
25#88448C2214yes—
26#88447C2212yes—
27C2×D3×Dic6—34no—
28#64097D312yescomplemented
29C3×D3×C4⋊C4—32no—
30C3×D3×C4⋊C4—61no—
31C3×#888—32no—
32#63919—32no—
33#63918—32no—
34D3×C4⋊Dic3—32no—
35D3×Dic3⋊C4—34no—
36C4×D3×Dic3—32no—
37#63840—34no—
38#63839—34no—
39C22×#272C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084C2312yes—
43C2×#1084C2311yescharacteristic
44C4×#272—24no—
45C4×#272Q814yescomplemented
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48C2×(C3⋊Dic3)⋊C3C2312yes—
49#634⋊C3—28no—
50#6199—24no—
51#6199D414yes—
52#6199Q814yescomplemented
53#6196—28no—
54D3×C4⋊Q8—92no—
55C2×#888—91no—
56C2×#634C2×D314yes—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—34no—
59C2×C3×C4×D3—61no—
60C2×C3×C4×D3—62no—
61C2×C3×Dic6—34no—
62C2×#627—34no—
63C2×#627—34no—
64C2×D3×Dic3—32no—
65C2×D3×Dic3—32no—
66C2×D3×Dic3—34no—
67D3×Dic6—616no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4C2×D312yes—
70C4⋊(C3⋊Dic3)C2×D312yes—
71#3750C2×D314yes—
72C4×C3⋊Dic3C2×D312yes—
73C3×C4⋊Dic3—32no—
74C3×C4⋊Dic3—32no—
75C3×C4⋊Dic3—61no—
76C3×Dic3⋊C4—34no—
77C3×Dic3⋊C4—34no—
78C3×Dic3⋊C4—62no—
79C3×C4×Dic3—32no—
80#3724—34no—
81Dic3⋊Dic3—34no—
82Dic3⋊Dic3—34no—
83Dic32—32no—
84Dic32—32no—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×#272C2×D412yes—
87C2×#272C2×Q812yes—
88C2×#272C2×Q812yes—
89C4×H(3,𝔽3)—22no—
90C4×H(3,𝔽3)C2×Q812yescomplemented
91#1084—22no—
92#1084C2×Q812yescomplemented
93(C3⋊Dic3)⋊C3—24no—
94(C3⋊Dic3)⋊C3C2×D414yes—
95(C3⋊Dic3)⋊C3C2×Q814yescomplemented
96C2×D3×Q8—94no—
97C22×Dic6—92no—
98C3×C4⋊Q8—92no—
99C2×C3×C4⋊C4—91no—
100#944—94no—
101C2×C4⋊Dic3—91no—
102C2×Dic3⋊C4—92no—
103C2×C4×Dic3—91no—
104D3×C4⋊C4—32no—
105D3×C4⋊C4—61no—
106D3×C4⋊C4—94no—
107D3×C4⋊C4—92no—
108#888—32no—
109#888—92no—
110#888—92no—
111#888—94no—
112#888—94no—
113#888—92no—
114C42×D3—92no—
115#869—92no—
116C22×C3×D3—32no—
117C22×C3×D3—61no—
118C2×C32×C4—21no—
119C2×C32×C4—22no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D312yes—
122C2×C3⋊Dic3C22×D314yes—
123C2×C3⋊Dic3C22×D312yes—
124C2×C3⋊Dic3C22×D312yes—
125#634—28no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133C3×C4×D3—38no—
134C3×C4×D3—64no—
135C3×C4×D3—68no—
136C3×C4×D3—124no—
137C3×Dic6—68no—
138#627—68no—
139#627—68no—
140D3×Dic3—38no—
141D3×Dic3—38no—
142D3×Dic3—68no—
143C2×C4⋊Q8—271noSylow
144C2×H(3,𝔽3)C22×D411yescharacteristic
145C2×H(3,𝔽3)C22×Q811yescharacteristic
146C2×H(3,𝔽3)C22×Q811yescharacteristic
147#272C4⋊Q814yescomplemented
148C2×C3×Q8—94no—
149C22×C3×C4—91no—
150C22×C3×C4—92no—
151C22×Dic3—91no—
152C22×Dic3—91no—
153C22×Dic3—92no—
154D3×Q8—1816no—
155C2×C4×D3—32no—
156C2×C4×D3—34no—
157C2×C4×D3—61no—
158C2×C4×D3—62no—
159C2×C4×D3—92no—
160C2×C4×D3—92no—
161C2×C4×D3—94no—
162C2×Dic6—34no—
163C2×Dic6—94no—
164C2×Dic6—94no—
165C2×Dic6—94no—
166C2×Dic6—94no—
167C2×Dic6—188no—
168C3×C4⋊C4—32no—
169C3×C4⋊C4—61no—
170C3×C4⋊C4—94no—
171C3×C4⋊C4—92no—
172C3×C4⋊C4—92no—
173C3×C4⋊C4—91no—
174C3×C4⋊C4D3211yescharacteristic
175C3×C42—92no—
176C4⋊Dic3—32no—
177C4⋊Dic3—32no—
178C4⋊Dic3—61no—
179C4⋊Dic3—92no—
180C4⋊Dic3—92no—
181C4⋊Dic3—94no—
182Dic3⋊C4—34no—
183Dic3⋊C4—34no—
184Dic3⋊C4—62no—
185Dic3⋊C4—94no—
186Dic3⋊C4—94no—
187Dic3⋊C4—94no—
188Dic3⋊C4—94no—
189C4×Dic3—32no—
190C4×Dic3—92no—
191C4×Dic3—92no—
192C4×Dic3—92no—
193C4×Dic3—92no—
194C22×C32—21no—
195C22×C32C23×D312yes—
196C2×C3×D3—34no—
197C2×C3×D3—34no—
198C2×C3×D3—34no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C2×C3×D3—62no—
202C32×C4—24no—
203C32×C4—22no—
204C32×C4—42no—
205C32×C4D3×Q814yescomplemented
206C3⋊Dic3—24no—
207C3⋊Dic3D3×D414yes—
208C3⋊Dic3D3×Q814yescomplemented
209C3×Dic3—34no—
210C3×Dic3—34no—
211C3×Dic3—34no—
212C3×Dic3—62no—
213C3×Dic3—64no—
214C3×Dic3—64no—
215C3×Dic3—122no—
216C22×Q8—272no—
217C4⋊Q8—92no—
218C4⋊Q8—272no—
219C4⋊Q8—274no—
220C2×C4⋊C4—91no—
221C2×C4⋊C4—272no—
222C2×C4⋊C4—271no—
223C2×C42—271no—
224H(3,𝔽3)C2×C4⋊Q811yescharacteristic, complemented, Sylow
225C23×C3—91no—
226C22×D3—32no—
227C22×D3—61no—
228C3×Q8—188no—
229C2×C3×C4—32no—
230C2×C3×C4—34no—
231C2×C3×C4—61no—
232C2×C3×C4—62no—
233C2×C3×C4—92no—
234C2×C3×C4—92no—
235C2×C3×C4—91no—
236C2×C3×C4—94no—
237C2×C3×C4—94no—
238C2×C3×C4—92no—
239C2×C3×C4—184no—
240C2×C3×C4C2×D3212yes—
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—32no—
243C2×Dic3—32no—
244C2×Dic3—32no—
245C2×Dic3—34no—
246C2×Dic3—34no—
247C2×Dic3—61no—
248C2×Dic3—62no—
249C2×Dic3—92no—
250C2×Dic3—92no—
251C2×Dic3—94no—
252C2×Dic3—94no—
253C2×Dic3—94no—
254C2×Dic3—184no—
255C4×D3—38no—
256C4×D3—64no—
257C4×D3—68no—
258C4×D3—98no—
259C4×D3—98no—
260C4×D3—124no—
261C4×D3—188no—
262Dic6—68no—
263Dic6—188no—
264Dic6—188no—
265Dic6—188no—
266Dic6—188no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32C2×D3×D412yes—
271C2×C32C2×D3×Q812yes—
272C2×C32C2×D3×Q812yes—
273C3×D3—38no—
274C3×D3—64no—
275C2×Q8—94no—
276C2×Q8—274no—
277C2×Q8—548no—
278C22×C4—91no—
279C22×C4—92no—
280C22×C4—271no—
281C22×C4—271no—
282C22×C4—272no—
283C4⋊C4—92no—
284C4⋊C4—91no—
285C4⋊C4—94no—
286C4⋊C4—92no—
287C4⋊C4—274no—
288C4⋊C4—272no—
289C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C42—92no—
291C42—272no—
292C22×C3—32no—
293C22×C3—61no—
294C22×C3—92no—
295C22×C3—92no—
296C22×C3—92no—
297C22×C3C22×D3211yescharacteristic
298C2×D3—34no—
299C2×D3—34no—
300C2×D3—34no—
301C2×D3—62no—
302C2×D3—62no—
303C2×D3—62no—
304C3×C4—22no—
305C3×C4—34no—
306C3×C4—64no—
307C3×C4—62no—
308C3×C4—94no—
309C3×C4—94no—
310C3×C4—92no—
311C3×C4—122no—
312C3×C4—184no—
313C3×C4—182no—
314C3×C4#379712yes—
315Dic3—34no—
316Dic3—34no—
317Dic3—34no—
318Dic3—62no—
319Dic3—64no—
320Dic3—64no—
321Dic3—94no—
322Dic3—94no—
323Dic3—122no—
324Dic3—184no—
325C32—21no—
326C32D3×C4⋊Q812yescomplemented
327C23—91no—
328Q8—188no—
329Q8—548no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—91no—
333C2×C4—94no—
334C2×C4—92no—
335C2×C4—94no—
336C2×C4—184no—
337C2×C4—274no—
338C2×C4—274no—
339C2×C4—544no—
340C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
341C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C2×C3—32no—
343C2×C3—32no—
344C2×C3—32no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—61no—
348C2×C3—94no—
349C2×C3C2×#379711yescharacteristic
350C2×C3D32×D411yescharacteristic
351C2×C3D32×Q811yescharacteristic
352D3—38no—
353D3—64no—
354C22—92no—
355C22—92no—
356C22—92no—
357C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
358C4—22no—
359C4—94no—
360C4—94no—
361C4—92no—
362C4—182no—
363C4—184no—
364C4#8865112yescomplemented
365C3—32no—
366C3—61no—
367C3#1059437411yescharacteristic
368C2—94no—
369C2C2×#8865111yescharacteristic
370C2D4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
371C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
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)C2×C4⋊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
3. #64097⋊C3C2semidirect
7. #11721116C2semidirect
8. #11721079C2semidirect
28. #64097D3semidirect
45. C4×#272Q8semidirect
52. #6199Q8semidirect
90. C4×H(3,𝔽3)C2×Q8semidirect
92. #1084C2×Q8semidirect
95. (C3⋊Dic3)⋊C3C2×Q8semidirect
147. #272C4⋊Q8semidirect
205. C32×C4D3×Q8semidirect
208. C3⋊Dic3D3×Q8semidirect
224. H(3,𝔽3)C2×C4⋊Q8semidirect
326. C32D3×C4⋊Q8semidirect
364. C4#88651semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
C4⋊C4×#2722
#64097⋊C32
#117211522
#117211472
C4×#61992
#117211162
#117210792
D3×#8883