Finite Group Databasebeta

← #13147833

Subgroups — #13147833

Order 1728.

1193 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1193 conjugacy classes of subgroups — too many to draw. Only its 207 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×#88650C211yesmaximal, characteristic
2C42×#272C211yesmaximal, characteristic, complemented
3#64081⋊C3C212yesmaximal, complemented
4#11721138C212yesmaximal, complemented
5#11721124C214yesmaximal, complemented
6C4×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
7#11721097C211yesmaximal, characteristic, complemented
8#11721078C212yesmaximal, complemented
9#10593965—32nomaximal
10C2×C4×#272C2212yes—
11C2×C4×#272C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2212yes—
13C2×#6200C2212yes—
14C2×#6198C2211yescharacteristic
15C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
16C2×#6196C2211yescharacteristic
17#88650C418yescomplemented
18C42×H(3,𝔽3)C2211yescharacteristic
19C4×#1084C2212yes—
20C4×#1084C2211yescharacteristic
21#3752⋊C3C2214yes—
22#3750⋊C3C2214yes—
23C4×(C3⋊Dic3)⋊C3C2212yes—
24#88451C2212yes—
25#88449C2212yes—
26#88447C2214yes—
27C2×#3798—32no—
28#64081D312yescomplemented
29C3×#872—32no—
30C3×C42×D3—32no—
31C3×C42×D3—61no—
32#63904—34no—
33C4×#624—32no—
34#63880—34no—
35#63879—34no—
36C4×D3×Dic3—32no—
37#63855—32no—
38#63837—34no—
39C22×#272C2311yescharacteristic
40C22×(C3⋊D3)⋊C3C2312yes—
41C2×C4×H(3,𝔽3)C2312yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084C2311yescharacteristic
44C2×#1084C2312yes—
45C4×#272—24no—
46C4×#272C2×C414yescomplemented
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C2×(C3⋊Dic3)⋊C3C2312yes—
49C4×(C3⋊D3)⋊C3C2×C418yescomplemented
50#6200C2×C418yescomplemented
51#6198C2×C414yescomplemented
52((C3⋊D3)⋊C3)⋊C4—28no—
53#6196C2×C414yescomplemented
54#5522—92no—
55C4×#235—91no—
56C2×C4×C3⋊D3C2×D312yes—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—32no—
59C2×C3×C4×D3—34no—
60C2×C3×C4×D3—61no—
61C2×C3×C4×D3—62no—
62C2×#627—32no—
63C2×#626—32no—
64C2×#626—32no—
65C2×#625—32no—
66C2×#624—34no—
67C2×D3×Dic3—34no—
68#3798—316no—
69C32×C42—21no—
70C32×C42C2×D312yes—
71#3752C2×D314yes—
72#3750C2×D314yes—
73C4×C3⋊Dic3C2×D312yes—
74C3×(C2×D3)⋊C4—34no—
75C3×Dic3⋊C4—34no—
76C3×C4×Dic3—34no—
77C3×C4×Dic3—32no—
78C3×C4×Dic3—32no—
79C3×C4×Dic3—62no—
80C3×C4×Dic3—61no—
81Dic3⋊Dic3—34no—
82(C2×D3)⋊Dic3—34no—
83Dic32—34no—
84Dic32—34no—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×#272—22no—
87C2×#272C22×C412yes—
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3C22×C414yes—
90C4×H(3,𝔽3)—22no—
91C4×H(3,𝔽3)C22×C412yes—
92#1084C22×C412yes—
93#1084#4114yescomplemented
94(C3⋊Dic3)⋊C3—24no—
95(C3⋊Dic3)⋊C3C22×C414yes—
96C2×#235—91no—
97C2×#235—92no—
98C3×#117—92no—
99C2×C3×C42—91no—
100C4×#67—94no—
101#926—91no—
102C2×C4×Dic3—92no—
103#893—94no—
104#887—94no—
105#67⋊C4—94no—
106#67⋊C4—94no—
107C4×D12—92no—
108#872—32no—
109#872—92no—
110#872—92no—
111C42×D3—32no—
112C42×D3—61no—
113C42×D3—92no—
114C42×D3—92no—
115C4×Dic6—92no—
116C22×C3⋊D3C22×D312yes—
117C22×C3×D3—32no—
118C22×C3×D3—32no—
119C22×C3×D3—61no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D312yes—
124C2×C3⋊Dic3C22×D314yes—
125C2×C3⋊Dic3C22×D312yes—
126C4×C3⋊D3C4×D318yescomplemented
127C2×C3×Dic3—32no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—61no—
133C3×C4×D3—38no—
134C3×C4×D3—38no—
135C3×C4×D3—64no—
136C3×C4×D3—68no—
137C3×C4×D3—68no—
138#627—38no—
139#626—38no—
140#626—38no—
141#625—38no—
142#624—68no—
143D3×Dic3—68no—
144C4×#41—271noSylow
145C2×H(3,𝔽3)C23×C411yescharacteristic
146C2×H(3,𝔽3)C2×#4112yes—
147#272—22no—
148(C3⋊D3)⋊C3—24no—
149C22×C3×C4—92no—
150C22×C3×C4—91no—
151C22×C3×C4—92no—
152C2×#67—92no—
153C2×#67—92no—
154C2×#67—92no—
155C22×Dic3—94no—
156#235—98no—
157#235—916no—
158C2×D12—91no—
159C2×D12—92no—
160C2×C4×D3—32no—
161C2×C4×D3—32no—
162C2×C4×D3—34no—
163C2×C4×D3—61no—
164C2×C4×D3—62no—
165C2×C4×D3—92no—
166C2×C4×D3—94no—
167C2×C4×D3—94no—
168C2×Dic6—91no—
169C2×Dic6—92no—
170C3×C4⋊C4—94no—
171C3×C22⋊C4—94no—
172C3×C42—32no—
173C3×C42—61no—
174C3×C42—92no—
175C3×C42—92no—
176C3×C42—91no—
177C3×C42D3211yescharacteristic
178#217—92no—
179(C2×D3)⋊C4—34no—
180(C2×D3)⋊C4—94no—
181(C2×D3)⋊C4—94no—
182C4⋊Dic3—92no—
183Dic3⋊C4—34no—
184Dic3⋊C4—94no—
185Dic3⋊C4—94no—
186C4×Dic3—32no—
187C4×Dic3—34no—
188C4×Dic3—32no—
189C4×Dic3—62no—
190C4×Dic3—61no—
191C4×Dic3—92no—
192C4×Dic3—94no—
193C4×Dic3—94no—
194C4×Dic3—94no—
195C22×C32—21no—
196C22×C32C23×D312yes—
197C2×C3⋊D3—24no—
198C2×C3⋊D3C2×C4×D314yes—
199C2×C3×D3—34no—
200C2×C3×D3—34no—
201C2×C3×D3—62no—
202C2×C3×D3—64no—
203C2×C3×D3—64no—
204C2×C3×D3—64no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4—22no—
208C32×C4C2×C4×D314yes—
209C3⋊Dic3—24no—
210C3⋊Dic3C2×C4×D314yes—
211C3×Dic3—34no—
212C3×Dic3—34no—
213C3×Dic3—38no—
214C3×Dic3—62no—
215C3×Dic3—64no—
216C3×Dic3—64no—
217C2×#41—271no—
218C4×Q8—272no—
219C4×D4—272no—
220C4×D4—274no—
221#117—92no—
222#117—271no—
223C2×C42—91no—
224C2×C42—272no—
225H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
226C23×C3—92no—
227C23×C3—91no—
228C22×D3—32no—
229C22×D3—32no—
230C22×D3—61no—
231C22×D3—92no—
232C2×C3×C4—34no—
233C2×C3×C4—32no—
234C2×C3×C4—62no—
235C2×C3×C4—61no—
236C2×C3×C4—98no—
237C2×C3×C4—92no—
238C2×C3×C4—92no—
239C2×C3×C4—91no—
240C2×C3×C4—94no—
241C2×C3×C4—94no—
242C2×C3×C4—184no—
243C2×C3×C4C2×D3212yes—
244C2×C3×C4C2×D3211yescharacteristic
245#67—98no—
246#67—98no—
247#67—98no—
248C2×Dic3—34no—
249C2×Dic3—32no—
250C2×Dic3—32no—
251C2×Dic3—34no—
252C2×Dic3—62no—
253C2×Dic3—61no—
254C2×Dic3—92no—
255C2×Dic3—94no—
256C2×Dic3—188no—
257D12—94no—
258D12—98no—
259C4×D3—38no—
260C4×D3—38no—
261C4×D3—64no—
262C4×D3—68no—
263C4×D3—68no—
264C4×D3—98no—
265C4×D3—188no—
266C4×D3—188no—
267Dic6—94no—
268Dic6—98no—
269C2×C32—21no—
270C2×C32—22no—
271C2×C32C22×C4×D312yes—
272C2×C32D3×#4114yes—
273C3⋊D3—24no—
274C3×D3—64no—
275C3×D3—64no—
276C3×D3—64no—
277#41—278no—
278C2×Q8—271no—
279C2×D4—271no—
280C2×D4—272no—
281C22×C4—91no—
282C22×C4—92no—
283C22×C4—92no—
284C22×C4—274no—
285C4⋊C4—94no—
286C4⋊C4—272no—
287C22⋊C4—94no—
288C22⋊C4—272no—
289C42—92no—
290C42—91no—
291C42—92no—
292C42—274no—
293C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
294C22×C3—32no—
295C22×C3—61no—
296C22×C3—94no—
297C22×C3—92no—
298C22×C3—184no—
299C22×C3—182no—
300C22×C3C22×D3211yescharacteristic
301C2×D3—34no—
302C2×D3—34no—
303C2×D3—62no—
304C2×D3—64no—
305C2×D3—64no—
306C2×D3—64no—
307C2×D3—94no—
308C2×D3—184no—
309C3×C4—22no—
310C3×C4—34no—
311C3×C4—64no—
312C3×C4—64no—
313C3×C4—62no—
314C3×C4—94no—
315C3×C4—92no—
316C3×C4—94no—
317C3×C4—184no—
318C3×C4C4×D3212yes—
319Dic3—34no—
320Dic3—38no—
321Dic3—34no—
322Dic3—62no—
323Dic3—64no—
324Dic3—64no—
325Dic3—94no—
326Dic3—184no—
327C32—21no—
328C32#552212yescomplemented
329C23—92no—
330C23—91no—
331Q8—274no—
332D4—274no—
333D4—278no—
334C2×C4—92no—
335C2×C4—91no—
336C2×C4—92no—
337C2×C4—94no—
338C2×C4—98no—
339C2×C4—94no—
340C2×C4—184no—
341C2×C4—548no—
342C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
343C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
344C2×C3—32no—
345C2×C3—34no—
346C2×C3—61no—
347C2×C3—62no—
348C2×C3—184no—
349C2×C3—182no—
350C2×C3C2×C4×D3211yescharacteristic
351C2×C3#6430612yes—
352D3—64no—
353D3—64no—
354D3—64no—
355D3—184no—
356C22—92no—
357C22—94no—
358C22—182no—
359C22—184no—
360C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
361C4—22no—
362C4—92no—
363C4—94no—
364C4—94no—
365C4—184no—
366C4C4×((C3⋊D3)⋊C3)⋊C212yes—
367C3—32no—
368C3—61no—
369C3#1059396611yescharacteristic
370C2—182no—
371C2—184no—
372C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
373C2#1172249112yes—
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×#88650C2
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C42×#272C2semidirect
3. #64081⋊C3C2semidirect
4. #11721138C2semidirect
5. #11721124C2semidirect
6. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
7. #11721097C2semidirect
8. #11721078C2semidirect
17. #88650C4semidirect
28. #64081D3semidirect
46. C4×#272C2×C4semidirect
49. C4×(C3⋊D3)⋊C3C2×C4semidirect
50. #6200C2×C4semidirect
51. #6198C2×C4semidirect
53. #6196C2×C4semidirect
93. #1084#41semidirect
126. C4×C3⋊D3C4×D3semidirect
225. H(3,𝔽3)C4×#41semidirect
328. C32#5522semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
C42×#2722
#64081⋊C32
#117211382
#117211242
C4×((C3⋊D3)⋊C3)⋊C42
#117210972
#117210782
#105939653