Finite Group Databasebeta

← #13149431

Subgroups — #13149431

Order 1728.

1050 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1050 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×((C3⋊D3)⋊C3)⋊Q8C212yesmaximal, complemented
2#11721755C211yesmaximal, characteristic, complemented
3#64106⋊C3C212yesmaximal, complemented
4#11721138C212yesmaximal
5C4×#6198C211yesmaximal, characteristic
6#11721134C214yesmaximal, complemented
7#11721117C212yesmaximal, complemented
8#11721099C211yesmaximal, characteristic, complemented
9#10594383—32nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2212yes—
12C2×#634⋊C3C2212yes—
13C2×#6198C2211yescharacteristic
14C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
15C2×#6196C2214yes—
16((C3⋊D3)⋊C3)⋊Q8—28no—
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18#88540C2212yes—
19C4×#1084C2211yescharacteristic
20C4×#1084C2212yes—
21#3752⋊C3C2214yes—
22C4⋊((C3⋊Dic3)⋊C3)C2214yes—
23#88450C2212yes—
24#88449C2212yes—
25#88449C2212yes—
26#88449C2211yescharacteristic
27C2×#3797—32no—
28C2×D3×Dic6—32no—
29#64106D312yescomplemented
30C3×#897—32no—
31C3×#887—32no—
32C3×#887—61no—
33#63918—34no—
34C4×#626—32no—
35#63895—34no—
36#63893—34no—
37#63880—34no—
38#63866—32no—
39C22×(C3⋊D3)⋊C3C2312yes—
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084C2311yescharacteristic
43C2×#1084C2312yes—
44C2×#1084C2311yescharacteristic
45#6229D414yescomplemented
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C4×(C3⋊D3)⋊C3—24no—
49#634⋊C3—24no—
50#6198Q814yescomplemented
51((C3⋊D3)⋊C3)⋊C4—28no—
52#6196—28no—
53#5632—92no—
54Q8×D12—91no—
55C2×C4×C3⋊D3C2×D312yes—
56C2×#634C2×D312yes—
57C2×C3×C4×D3—32no—
58C2×C3×Dic6—32no—
59C2×C3×Dic6—32no—
60C2×C3×Dic6—61no—
61C2×#627—34no—
62C2×#627—34no—
63C2×#626—32no—
64C2×#624—34no—
65C2×D3×Dic3—34no—
66#3797—68no—
67D3×Dic6—68no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4C2×D312yes—
70#3752C2×D314yes—
71C4⋊(C3⋊Dic3)C2×D314yes—
72C3×(C2×D3)⋊C4—34no—
73C3×C4⋊Dic3—34no—
74C3×Dic3⋊C4—34no—
75C3×Dic3⋊C4—62no—
76C3×C4×Dic3—32no—
77C3×C4×Dic3—34no—
78C3×C4×Dic3—62no—
79C3×C4×Dic3—61no—
80Dic3⋊Dic3—34no—
81Dic3⋊Dic3—34no—
82Dic3⋊Dic3—32no—
83#3722—32no—
84(C2×D3)⋊Dic3—32no—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—22no—
88C2×(C3⋊D3)⋊C3C2×Q814yescomplemented
89C4×H(3,𝔽3)—22no—
90C4×H(3,𝔽3)C2×D412yes—
91#1084—21no—
92#1084C2×D414yes—
93#1084C2×Q812yescomplemented
94(C3⋊Dic3)⋊C3—22no—
95(C3⋊Dic3)⋊C3—24no—
96C2×D3×Q8—92no—
97C2×D3×Q8—92no—
98C22×Dic6—92no—
99C3×C22⋊Q8—92no—
100C3×C4×Q8—91no—
101#944—92no—
102C4×#67—92no—
103#897—32no—
104#897—92no—
105#897—94no—
106#896—94no—
107#888—94no—
108#887—32no—
109#887—61no—
110#67⋊C4—94no—
111#878—94no—
112C4×D12—92no—
113C4×D12—91no—
114#869—92no—
115#869—91no—
116C22×C3⋊D3C22×D312yes—
117C22×C3×D3—32no—
118C2×C32×C4—21no—
119C2×C32×C4—22no—
120C2×C32×C4C22×D314yes—
121C2×C32×C4C22×D312yes—
122C2×C3⋊Dic3C22×D312yes—
123C2×C3⋊Dic3C22×D314yes—
124C4×C3⋊D3—24no—
125#634—24no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—32no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—61no—
134C3×C4×D3—64no—
135C3×Dic6—38no—
136C3×Dic6—64no—
137C3×Dic6—64no—
138#627—68no—
139#627—68no—
140#626—38no—
141#624—68no—
142D3×Dic3—68no—
143D4×Q8—271noSylow
144C2×H(3,𝔽3)C22×D411yescharacteristic
145C2×H(3,𝔽3)C22×Q811yescharacteristic
146C2×H(3,𝔽3)2−1+411yescharacteristic
147(C3⋊D3)⋊C3—24no—
148C2×C3×Q8—92no—
149C2×C3×Q8—91no—
150C22×C3×C4—92no—
151C2×#67—92no—
152C22×Dic3—94no—
153D3×Q8—188no—
154D3×Q8—188no—
155C2×D12—91no—
156C2×C4×D3—32no—
157C2×C4×D3—92no—
158C2×C4×D3—94no—
159C2×C4×D3—94no—
160C2×Dic6—32no—
161C2×Dic6—32no—
162C2×Dic6—61no—
163C2×Dic6—92no—
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⋊C4D3211yescharacteristic
173C3×C22⋊C4—94no—
174C3×C42—92no—
175C3×C42—91no—
176#217—92no—
177(C2×D3)⋊C4—34no—
178(C2×D3)⋊C4—94no—
179(C2×D3)⋊C4—92no—
180(C2×D3)⋊C4—92no—
181C4⋊Dic3—34no—
182C4⋊Dic3—94no—
183C4⋊Dic3—91no—
184C4⋊Dic3—92no—
185C4⋊Dic3—92no—
186Dic3⋊C4—34no—
187Dic3⋊C4—62no—
188Dic3⋊C4—94no—
189Dic3⋊C4—94no—
190Dic3⋊C4—92no—
191C4×Dic3—34no—
192C4×Dic3—32no—
193C4×Dic3—62no—
194C4×Dic3—61no—
195C22×C32—21no—
196C22×C32C23×D312yes—
197C2×C3⋊D3—22no—
198C2×C3⋊D3—22no—
199C2×C3⋊D3D3×Q814yescomplemented
200C2×C3×D3—34no—
201C2×C3×D3—62no—
202C2×C3×D3—62no—
203C32×C4—24no—
204C32×C4—22no—
205C32×C4—42no—
206C32×C4D3×D414yes—
207C3⋊Dic3—24no—
208C3⋊Dic3—22no—
209C3×Dic3—38no—
210C3×Dic3—34no—
211C3×Dic3—64no—
212C3×Dic3—62no—
213C3×Dic3—64no—
214C3×Dic3—62no—
215C3×Dic3—62no—
216C3×Dic3—121no—
217C22×Q8—272no—
218C4⋊Q8—272no—
219C4⋊Q8—271no—
220C22⋊Q8—92no—
221C22⋊Q8—274no—
222C4×Q8—91no—
223C4×D4—272no—
224C4×D4—271no—
225H(3,𝔽3)D4×Q811yescharacteristic, complemented, Sylow
226C23×C3—92no—
227C22×D3—32no—
228C22×D3—92no—
229C3×Q8—94no—
230C3×Q8—184no—
231C2×C3×C4—32no—
232C2×C3×C4—34no—
233C2×C3×C4—61no—
234C2×C3×C4—62no—
235C2×C3×C4—91no—
236C2×C3×C4—92no—
237C2×C3×C4—94no—
238C2×C3×C4—92no—
239C2×C3×C4—91no—
240C2×C3×C4—184no—
241C2×C3×C4C2×D3212yes—
242C2×C3×C4C2×D3211yescharacteristic
243#67—98no—
244C2×Dic3—32no—
245C2×Dic3—34no—
246C2×Dic3—32no—
247C2×Dic3—32no—
248C2×Dic3—34no—
249C2×Dic3—61no—
250C2×Dic3—62no—
251C2×Dic3—61no—
252C2×Dic3—94no—
253C2×Dic3—92no—
254C2×Dic3—188no—
255D12—94no—
256C4×D3—64no—
257C4×D3—184no—
258C4×D3—188no—
259C4×D3—188no—
260Dic6—38no—
261Dic6—64no—
262Dic6—64no—
263Dic6—184no—
264Dic6—188no—
265Dic6—188no—
266Dic6—188no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32C2×D3×D412yes—
271C2×C32C2×D3×Q812yes—
272C2×C32#101012yes—
273C3⋊D3—24no—
274C3×D3—64no—
275C2×Q8—91no—
276C2×Q8—92no—
277C2×Q8—274no—
278C2×Q8—548no—
279C2×D4—271no—
280C22×C4—92no—
281C22×C4—274no—
282C4⋊C4—92no—
283C4⋊C4—94no—
284C4⋊C4—272no—
285C4⋊C4—271no—
286C4⋊C4—272no—
287C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
288C22⋊C4—94no—
289C22⋊C4—272no—
290C42—92no—
291C42—91no—
292C22×C3—32no—
293C22×C3—61no—
294C22×C3—94no—
295C22×C3—182no—
296C22×C3—182no—
297C22×C3C22×D3211yescharacteristic
298C2×D3—34no—
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—94no—
302C2×D3—182no—
303C2×D3—182no—
304C3×C4—22no—
305C3×C4—34no—
306C3×C4—64no—
307C3×C4—62no—
308C3×C4—94no—
309C3×C4—92no—
310C3×C4—122no—
311C3×C4—182no—
312C3×C4—184no—
313C3×C4—181no—
314C3×C4#380212yes—
315Dic3—38no—
316Dic3—34no—
317Dic3—64no—
318Dic3—62no—
319Dic3—62no—
320Dic3—64no—
321Dic3—62no—
322Dic3—121no—
323Dic3—182no—
324Dic3—184no—
325C32—21no—
326C32#563212yescomplemented
327C23—92no—
328Q8—94no—
329Q8—184no—
330Q8—548no—
331D4—274no—
332C2×C4—91no—
333C2×C4—92no—
334C2×C4—91no—
335C2×C4—94no—
336C2×C4—92no—
337C2×C4—184no—
338C2×C4—548no—
339C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
340C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
341C2×C3—32no—
342C2×C3—32no—
343C2×C3—32no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—184no—
348C2×C3C2×#380211yescharacteristic
349C2×C3#6430911yescharacteristic
350C2×C3D32×Q811yescharacteristic
351D3—64no—
352D3—184no—
353C22—94no—
354C22—182no—
355C22—182no—
356C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
357C4—22no—
358C4—92no—
359C4—94no—
360C4—181no—
361C4—182no—
362C4—184no—
363C4#8865512yes—
364C3—32no—
365C3—61no—
366C3#1059452611yescharacteristic
367C2—184no—
368C2C2×#8865511yescharacteristic
369C2#1172250211yescharacteristic
370C2Q8×((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 radicalC2×#6198C22
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×((C3⋊D3)⋊C3)⋊Q8C2semidirect
2. #11721755C2semidirect
3. #64106⋊C3C2semidirect
6. #11721134C2semidirect
7. #11721117C2semidirect
8. #11721099C2semidirect
29. #64106D3semidirect
45. #6229D4semidirect
50. #6198Q8semidirect
88. C2×(C3⋊D3)⋊C3C2×Q8semidirect
93. #1084C2×Q8semidirect
199. C2×C3⋊D3D3×Q8semidirect
225. H(3,𝔽3)D4×Q8semidirect
326. C32#5632semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊Q82
#117217552
#64106⋊C32
#117211382
C4×#61982
#117211342
#117211172
#117210992
#105943833