Finite Group Databasebeta

← #13149398

Subgroups — #13149398

Order 1728.

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

normal subgroups only

This group has 812 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
1#11721755C211yesmaximal, characteristic
2#64097⋊C3C212yesmaximal
3#11721117C211yesmaximal, characteristic
4C4×#6196C211yesmaximal, characteristic
5#11721099C212yesmaximal
6#11721083C212yesmaximal
7#11721079C214yesmaximal
8#11721078C212yesmaximal
9#10594312—32nomaximal
10C2×#6229C2211yescharacteristic
11C2×#634⋊C3C2214yes—
12C2×#6196C2212yes—
13C2×#6196C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88540C2212yes—
16C4×#1084C2212yes—
17C4×#1084C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2212yes—
19#3750⋊C3C2214yes—
20C4×(C3⋊Dic3)⋊C3C2212yes—
21#88449C2212yes—
22#88449C2214yes—
23#88449C2211yescharacteristic
24#88448C2212yes—
25#88447C2214yes—
26#64097D312yes—
27C3×#888—32no—
28C3×#887—32no—
29C3×#887—61no—
30#63918—32no—
31C4×#627—32no—
32#63866—34no—
33#63843—32no—
34Dic3×Dic6—32no—
35#63840—34no—
36#63839—34no—
37#63837—34no—
38C2×C4×H(3,𝔽3)C2312yes—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084C2311yescharacteristic
41C2×#1084C2311yescharacteristic
42C2×#1084C2312yes—
43#6229Q814yescomplemented
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47#634⋊C3—44no—
48#6196—42no—
49#6196Q814yescomplemented
50#5722—92no—
51Q8×Dic6—91no—
52C2×#634C2×D314yes—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—34no—
55C2×C3×Dic6—61no—
56C2×#627—32no—
57C2×#627—34no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D312yes—
60C4⋊(C3⋊Dic3)C2×D312yes—
61#3750C2×D314yes—
62C4×C3⋊Dic3C2×D312yes—
63C3×C4⋊Dic3—32no—
64C3×Dic3⋊C4—34no—
65C3×Dic3⋊C4—34no—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—34no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—62no—
71C3×C4×Dic3—61no—
72#3724—32no—
73Dic3⋊Dic3—34no—
74Dic3⋊Dic3—34no—
75Dic3⋊Dic3—32no—
76Dic3⋊Dic3—34no—
77Dic3⋊Dic3—32no—
78Dic32—34no—
79Dic32—34no—
80C22×H(3,𝔽3)C2411yescharacteristic
81C4×H(3,𝔽3)—22no—
82C4×H(3,𝔽3)C2×Q812yes—
83#1084—21no—
84#1084C2×Q812yes—
85#1084C2×Q814yes—
86(C3⋊Dic3)⋊C3—24no—
87(C3⋊Dic3)⋊C3—22no—
88(C3⋊Dic3)⋊C3C2×Q814yes—
89C3×C4⋊Q8—92no—
90C3×C4×Q8—91no—
91Q8×Dic3—92no—
92#944—94no—
93#944—94no—
94#888—32no—
95#888—92no—
96#888—92no—
97#888—94no—
98#888—94no—
99#887—32no—
100#887—61no—
101#887—94no—
102#887—92no—
103#869—91no—
104#869—92no—
105C4×Dic6—91no—
106C4×Dic6—92no—
107C4×Dic6—92no—
108C4×Dic6—92no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D312yes—
113C2×C3⋊Dic3C22×D314yes—
114C2×C3⋊Dic3C22×D312yes—
115C2×C3⋊Dic3C22×D312yes—
116#634—44no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—61no—
126C3×Dic6—38no—
127C3×Dic6—64no—
128C3×Dic6—124no—
129#627—38no—
130#627—124no—
131Q82—271noSylow
132C2×H(3,𝔽3)C22×Q811yescharacteristic
133C2×H(3,𝔽3)C22×Q811yescharacteristic
134C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
135C2×C3×Q8—94no—
136C2×C3×Q8—91no—
137C2×Dic6—32no—
138C2×Dic6—34no—
139C2×Dic6—61no—
140C2×Dic6—94no—
141C2×Dic6—91no—
142C2×Dic6—92no—
143C2×Dic6—92no—
144C2×Dic6—94no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—61no—
147C3×C4⋊C4—94no—
148C3×C4⋊C4—92no—
149C3×C4⋊C4—92no—
150C3×C4⋊C4D3211yescharacteristic
151C3×C42—92no—
152C3×C42—92no—
153C3×C42—91no—
154C4⋊Dic3—32no—
155C4⋊Dic3—92no—
156C4⋊Dic3—91no—
157C4⋊Dic3—92no—
158C4⋊Dic3—94no—
159C4⋊Dic3—92no—
160Dic3⋊C4—34no—
161Dic3⋊C4—34no—
162Dic3⋊C4—62no—
163Dic3⋊C4—94no—
164Dic3⋊C4—92no—
165Dic3⋊C4—92no—
166Dic3⋊C4—94no—
167Dic3⋊C4—92no—
168Dic3⋊C4—94no—
169Dic3⋊C4—94no—
170C4×Dic3—32no—
171C4×Dic3—34no—
172C4×Dic3—32no—
173C4×Dic3—62no—
174C4×Dic3—61no—
175C4×Dic3—92no—
176C4×Dic3—94no—
177C4×Dic3—94no—
178C4×Dic3—94no—
179C22×C32—21no—
180C22×C32C23×D312yes—
181C32×C4—24no—
182C32×C4—22no—
183C32×C4—42no—
184C32×C4D3×Q814yes—
185C3⋊Dic3—24no—
186C3⋊Dic3—22no—
187C3⋊Dic3D3×Q814yes—
188C3×Dic3—34no—
189C3×Dic3—34no—
190C3×Dic3—38no—
191C3×Dic3—64no—
192C3×Dic3—62no—
193C3×Dic3—64no—
194C3×Dic3—62no—
195C3×Dic3—62no—
196C3×Dic3—121no—
197C4⋊Q8—92no—
198C4⋊Q8—271no—
199C4⋊Q8—272no—
200C4⋊Q8—274no—
201C4×Q8—91no—
202C4×Q8—271no—
203C4×Q8—272no—
204C4×Q8—272no—
205H(3,𝔽3)Q8211yescharacteristic, complemented, Sylow
206C3×Q8—94no—
207C3×Q8—364no—
208C2×C3×C4—32no—
209C2×C3×C4—34no—
210C2×C3×C4—61no—
211C2×C3×C4—62no—
212C2×C3×C4—92no—
213C2×C3×C4—92no—
214C2×C3×C4—91no—
215C2×C3×C4—94no—
216C2×C3×C4—92no—
217C2×C3×C4—91no—
218C2×C3×C4C2×D3212yes—
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—32no—
221C2×Dic3—34no—
222C2×Dic3—32no—
223C2×Dic3—32no—
224C2×Dic3—32no—
225C2×Dic3—34no—
226C2×Dic3—61no—
227C2×Dic3—62no—
228C2×Dic3—61no—
229C2×Dic3—92no—
230C2×Dic3—92no—
231C2×Dic3—94no—
232Dic6—38no—
233Dic6—64no—
234Dic6—94no—
235Dic6—98no—
236Dic6—124no—
237Dic6—364no—
238Dic6—362no—
239Dic6—364no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32#100412yes—
244C2×C32C2×D3×Q812yes—
245C2×C32C2×D3×Q812yes—
246C2×Q8—91no—
247C2×Q8—94no—
248C2×Q8—271no—
249C2×Q8—272no—
250C4⋊C4—92no—
251C4⋊C4—94no—
252C4⋊C4—92no—
253C4⋊C4—271no—
254C4⋊C4—272no—
255C4⋊C4—272no—
256C4⋊C4—274no—
257C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C42—92no—
259C42—91no—
260C42—92no—
261C42—274no—
262C22×C3—32no—
263C22×C3—61no—
264C22×C3C22×D3211yescharacteristic
265C3×C4—22no—
266C3×C4—34no—
267C3×C4—64no—
268C3×C4—62no—
269C3×C4—94no—
270C3×C4—94no—
271C3×C4—92no—
272C3×C4—122no—
273C3×C4—184no—
274C3×C4—182no—
275C3×C4—181no—
276C3×C4#379712yes—
277Dic3—38no—
278Dic3—34no—
279Dic3—34no—
280Dic3—64no—
281Dic3—62no—
282Dic3—62no—
283Dic3—62no—
284Dic3—64no—
285Dic3—94no—
286Dic3—121no—
287Dic3—182no—
288Dic3—184no—
289C32—21no—
290C32#572212yescomplemented
291Q8—94no—
292Q8—274no—
293Q8—364no—
294Q8—1082no—
295C2×C4—91no—
296C2×C4—92no—
297C2×C4—92no—
298C2×C4—91no—
299C2×C4—92no—
300C2×C4—94no—
301C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
302C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
303C2×C3—32no—
304C2×C3—32no—
305C2×C3—32no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3C2×#379711yescharacteristic
310C2×C3#6431311yescharacteristic
311C2×C3D32×Q811yescharacteristic
312C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
313C4—22no—
314C4—92no—
315C4—94no—
316C4—94no—
317C4—182no—
318C4—184no—
319C4—181no—
320C4#8865112yes—
321C3—32no—
322C3—61no—
323C3#1059431311yescharacteristic
324C2C2×#8865111yescharacteristic
325C2#1172250611yescharacteristic
326C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)Q82
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
43. #6229Q8semidirect
49. #6196Q8semidirect
205. H(3,𝔽3)Q82semidirect
290. C32#5722semidirect

Maximal subgroups

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

subgroupindex
#117217552
#64097⋊C32
#117211172
C4×#61962
#117210992
#117210832
#117210792
#117210782
#105943123