Finite Group Databasebeta

← #13149564

Subgroups — #13149564

Order 1728.

812 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 141 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#11721755C211yesmaximal, characteristic, complemented
2#64101⋊C3C212yesmaximal, complemented
3#11721145C212yesmaximal
4C4×#6198C211yesmaximal, characteristic
5#11721117C211yesmaximal, characteristic, complemented
6C4×#6196C212yesmaximal, complemented
7#11721109C214yesmaximal, complemented
8#11721107C212yesmaximal, complemented
9#10594565—32nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2212yes—
12C2×#6198C2211yescharacteristic
13C2×#6196C2212yes—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88540C2212yes—
16C4×#1084C2212yes—
17C4×#1084C2211yescharacteristic
18#3752⋊C3C2214yes—
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20C4×(C3⋊Dic3)⋊C3C2214yes—
21#88450C2214yes—
22#88450C2212yes—
23#88449C2212yes—
24#88449C2211yescharacteristic
25#88448C2214yes—
26#64101D312yescomplemented
27C3×#892—32no—
28C3×#887—32no—
29C3×#887—61no—
30#63918—32no—
31C4×#627—34no—
32C4×#626—32no—
33#63871—32no—
34#63868—34no—
35#63862—34no—
36#63856—32no—
37#63852—34no—
38C22×(C3⋊D3)⋊C3C2312yes—
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2312yes—
42C2×#1084C2311yescharacteristic
43C2×#1084C2311yescharacteristic
44#6229—22no—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C4×(C3⋊D3)⋊C3—24no—
48#6198—22no—
49#6196—24no—
50#5584—91no—
51#5533—92no—
52C2×C4×C3⋊D3C2×D312yes—
53C2×C3×C4×D3—32no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—61no—
56C2×#627—34no—
57C2×#626—32no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D312yes—
60#3752C2×D314yes—
61C4⋊(C3⋊Dic3)C2×D312yes—
62C4×C3⋊Dic3C2×D314yes—
63C3×(C2×D3)⋊C4—34no—
64C3×C4⋊Dic3—32no—
65C3×Dic3⋊C4—34no—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—34no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—34no—
70C3×C4×Dic3—62no—
71C3×C4×Dic3—61no—
72#3724—34no—
73Dic3⋊Dic3—34no—
74Dic3⋊Dic3—34no—
75Dic3⋊Dic3—32no—
76#3722—34no—
77#3722—32no—
78(C2×D3)⋊Dic3—34no—
79(C2×D3)⋊Dic3—32no—
80C22×H(3,𝔽3)C2411yescharacteristic
81C2×(C3⋊D3)⋊C3—22no—
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—22no—
84C4×H(3,𝔽3)—22no—
85C4×H(3,𝔽3)#4112yes—
86#1084—22no—
87#1084—21no—
88#1084#4112yes—
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3—22no—
91C3×C4×Q8—91no—
92C3×#117—92no—
93#946—92no—
94C4×#67—92no—
95#924—92no—
96#898—94no—
97#897—92no—
98#892—32no—
99#892—92no—
100#888—92no—
101#887—32no—
102#887—61no—
103#885—94no—
104#879—94no—
105#876—94no—
106#875—92no—
107C4×D12—91no—
108#869—91no—
109C4×Dic6—92no—
110C4×Dic6—94no—
111C22×C3⋊D3C22×D312yes—
112C22×C3×D3—32no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D312yes—
117C2×C3⋊Dic3C22×D312yes—
118C2×C3⋊Dic3C22×D314yes—
119C4×C3⋊D3—24no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—61no—
128C3×C4×D3—64no—
129C3×Dic6—64no—
130C3×Dic6—64no—
131#627—68no—
132#626—64no—
133#556—271noSylow
134C2×H(3,𝔽3)C2×#4111yescharacteristic
135C2×H(3,𝔽3)C2×#4111yescharacteristic
136C2×H(3,𝔽3)2−1+411yescharacteristic
137(C3⋊D3)⋊C3—42no—
138C2×C3×Q8—91no—
139C22×C3×C4—92no—
140C2×#67—92no—
141C2×D12—91no—
142C2×C4×D3—32no—
143C2×C4×D3—92no—
144C2×Dic6—32no—
145C2×Dic6—61no—
146C2×Dic6—92no—
147C2×Dic6—94no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—92no—
151C3×C4⋊C4—92no—
152C3×C4⋊C4D3211yescharacteristic
153C3×C22⋊C4—94no—
154C3×C42—94no—
155C3×C42—92no—
156C3×C42—91no—
157#217—92no—
158#217—94no—
159(C2×D3)⋊C4—34no—
160(C2×D3)⋊C4—94no—
161(C2×D3)⋊C4—92no—
162(C2×D3)⋊C4—94no—
163(C2×D3)⋊C4—92no—
164(C2×D3)⋊C4—94no—
165C4⋊Dic3—32no—
166C4⋊Dic3—92no—
167C4⋊Dic3—91no—
168C4⋊Dic3—92no—
169C4⋊Dic3—94no—
170Dic3⋊C4—34no—
171Dic3⋊C4—62no—
172Dic3⋊C4—94no—
173Dic3⋊C4—92no—
174Dic3⋊C4—94no—
175Dic3⋊C4—94no—
176C4×Dic3—34no—
177C4×Dic3—32no—
178C4×Dic3—34no—
179C4×Dic3—61no—
180C4×Dic3—62no—
181C4×Dic3—94no—
182C22×C32—21no—
183C22×C32C23×D312yes—
184C2×C3⋊D3—22no—
185C2×C3⋊D3—22no—
186C2×C3⋊D3—22no—
187C2×C3×D3—62no—
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C32×C4—24no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4#23514yes—
194C3⋊Dic3—22no—
195C3⋊Dic3—24no—
196C3×Dic3—34no—
197C3×Dic3—62no—
198C3×Dic3—64no—
199C3×Dic3—62no—
200C3×Dic3—64no—
201C3×Dic3—64no—
202C3×Dic3—62no—
203C3×Dic3—121no—
204C4⋊Q8—271no—
205#126—274no—
206#124—272no—
207C22⋊Q8—272no—
208C4×Q8—91no—
209C4×Q8—272no—
210C4×D4—271no—
211#117—92no—
212H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
213C23×C3—92no—
214C22×D3—32no—
215C22×D3—92no—
216C3×Q8—182no—
217C2×C3×C4—32no—
218C2×C3×C4—34no—
219C2×C3×C4—61no—
220C2×C3×C4—62no—
221C2×C3×C4—91no—
222C2×C3×C4—92no—
223C2×C3×C4—94no—
224C2×C3×C4—92no—
225C2×C3×C4—91no—
226C2×C3×C4—184no—
227C2×C3×C4C2×D3212yes—
228C2×C3×C4C2×D3211yescharacteristic
229#67—184no—
230C2×Dic3—32no—
231C2×Dic3—32no—
232C2×Dic3—34no—
233C2×Dic3—32no—
234C2×Dic3—34no—
235C2×Dic3—61no—
236C2×Dic3—62no—
237C2×Dic3—61no—
238C2×Dic3—92no—
239C2×Dic3—94no—
240D12—182no—
241C4×D3—64no—
242C4×D3—184no—
243Dic6—64no—
244Dic6—64no—
245Dic6—188no—
246Dic6—184no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32C2×#23512yes—
251C2×C32D3×#4112yes—
252C2×C32#101012yes—
253C3⋊D3—42no—
254C3×D3—122no—
255C2×Q8—91no—
256C2×Q8—272no—
257C2×D4—271no—
258C22×C4—92no—
259C4⋊C4—92no—
260C4⋊C4—92no—
261C4⋊C4—271no—
262C4⋊C4—274no—
263C4⋊C4—272no—
264C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
265C22⋊C4—94no—
266C22⋊C4—272no—
267C22⋊C4—274no—
268C42—94no—
269C42—91no—
270C42—92no—
271C22×C3—32no—
272C22×C3—61no—
273C22×C3—182no—
274C22×C3—182no—
275C22×C3—182no—
276C22×C3C22×D3211yescharacteristic
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—182no—
281C2×D3—182no—
282C2×D3—182no—
283C3×C4—22no—
284C3×C4—34no—
285C3×C4—64no—
286C3×C4—62no—
287C3×C4—92no—
288C3×C4—122no—
289C3×C4—184no—
290C3×C4—182no—
291C3×C4—182no—
292C3×C4—181no—
293C3×C4#379812yes—
294Dic3—34no—
295Dic3—62no—
296Dic3—64no—
297Dic3—62no—
298Dic3—64no—
299Dic3—62no—
300Dic3—64no—
301Dic3—121no—
302Dic3—184no—
303Dic3—182no—
304C32—21no—
305C32#553312yescomplemented
306C23—92no—
307Q8—182no—
308Q8—544no—
309D4—542no—
310C2×C4—91no—
311C2×C4—92no—
312C2×C4—94no—
313C2×C4—91no—
314C2×C4—92no—
315C2×C4—184no—
316C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
317C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
318C2×C3—32no—
319C2×C3—32no—
320C2×C3—32no—
321C2×C3—61no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—362no—
325C2×C3C2×#379811yescharacteristic
326C2×C3#6430911yescharacteristic
327C2×C3#6431911yescharacteristic
328D3—122no—
329D3—362no—
330C22—182no—
331C22—182no—
332C22—182no—
333C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
334C4—22no—
335C4—92no—
336C4—181no—
337C4—184no—
338C4—182no—
339C4—182no—
340C4#8865012yes—
341C3—32no—
342C3—61no—
343C3#1059458111yescharacteristic
344C2—362no—
345C2C2×#8865011yescharacteristic
346C2#1172250211yescharacteristic
347C2#1172252011yescharacteristic
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)#556
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. #11721755C2semidirect
2. #64101⋊C3C2semidirect
5. #11721117C2semidirect
6. C4×#6196C2semidirect
7. #11721109C2semidirect
8. #11721107C2semidirect
26. #64101D3semidirect
212. H(3,𝔽3)#556semidirect
305. C32#5533semidirect

Maximal subgroups

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

subgroupindex
#117217552
#64101⋊C32
#117211452
C4×#61982
#117211172
C4×#61962
#117211092
#117211072
#105945653