Finite Group Databasebeta

← #13140123

Subgroups — #13140123

Order 1728.

703 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 139 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×C4×#1084C211yesmaximal, characteristic
2C2×#88449C211yesmaximal, characteristic
3C2×#88449C211yesmaximal, characteristic
4C2×#88448C212yesmaximal
5#63658⋊C3C212yesmaximal
6#10591327—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C4×#1084—22no—
13#88449—22no—
14#88449C414yescomplemented
15#88448—24no—
16C2×C3×C4×Dic3—32no—
17C2×C3×C4×Dic3—61no—
18C2×#3724—32no—
19C2×Dic3⋊Dic3—32no—
20C2×Dic3⋊Dic3—32no—
21C2×Dic3⋊Dic3—32no—
22#63658D312yes—
23C3×C3⋊#95—32no—
24C23×H(3,𝔽3)C2311yescharacteristic
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)Q812yes—
28C2×#1084—21no—
29C2×#1084—21no—
30C2×#1084C2×C412yes—
31C2×#1084D412yes—
32C2×#1084D412yes—
33C2×#1084Q812yes—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3C2×C414yes—
40#4928—91no—
41#4647—92no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D312yes—
44C22×C3⋊Dic3C2×D312yes—
45C22×C3⋊Dic3C2×D312yes—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C3×C4×Dic3—64no—
53C3×C4×Dic3—64no—
54#3724—64no—
55Dic3⋊Dic3—38no—
56Dic3⋊Dic3—64no—
57Dic3⋊Dic3—64no—
58C22×H(3,𝔽3)C22×C411yescharacteristic
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C22×H(3,𝔽3)C2×Q811yescharacteristic
62C22×H(3,𝔽3)C2×Q811yescharacteristic
63C22×H(3,𝔽3)#4111yescharacteristic
64C22×H(3,𝔽3)#4111yescharacteristic
65C4×H(3,𝔽3)—22no—
66#1084—22no—
67#1084C4⋊C414yescomplemented
68(C3⋊Dic3)⋊C3—24no—
69(C3⋊Dic3)⋊C3—42no—
70C2×C3×C42—91no—
71C2×C4⋊Dic3—91no—
72C2×C4⋊Dic3—91no—
73C2×C4⋊Dic3—92no—
74C2×Dic3⋊C4—92no—
75C2×Dic3⋊C4—92no—
76C2×Dic3⋊C4—92no—
77C2×Dic3⋊C4—92no—
78C2×C4×Dic3—32no—
79C2×C4×Dic3—61no—
80C3×#95—92no—
81C3⋊#95—32no—
82C3⋊#95—92no—
83C23×C32—21no—
84C23×C32C22×D312yes—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4Dic614yes—
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3C4×D314yes—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—34no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115#389—271noSylow
116C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
117C2×H(3,𝔽3)C4×D411yescharacteristic
118C2×H(3,𝔽3)C4×Q811yescharacteristic
119C2×H(3,𝔽3)#12111yescharacteristic
120C2×H(3,𝔽3)C22⋊Q811yescharacteristic
121C2×H(3,𝔽3)#12511yescharacteristic
122C2×H(3,𝔽3)C4⋊Q811yescharacteristic
123C22×C3×C4—32no—
124C22×C3×C4—61no—
125C22×C3×C4—91no—
126C22×C3×C4—92no—
127C22×C3×C4—91no—
128C22×C3×C4—92no—
129C22×C3×C4D3211yescharacteristic
130C22×Dic3—32no—
131C22×Dic3—32no—
132C22×Dic3—32no—
133C22×Dic3—32no—
134C22×Dic3—61no—
135C22×Dic3—61no—
136C22×Dic3—92no—
137C22×Dic3—92no—
138C3×C42—182no—
139C4⋊Dic3—94no—
140C4⋊Dic3—184no—
141C4⋊Dic3—182no—
142Dic3⋊C4—98no—
143Dic3⋊C4—184no—
144Dic3⋊C4—184no—
145Dic3⋊C4—184no—
146C4×Dic3—64no—
147C4×Dic3—64no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32C2×Dic612yes—
156C22×C32C2×C4×D312yes—
157C22×C32#23512yes—
158C22×C32D3×D412yes—
159C22×C32D3×D412yes—
160C22×C32#23712yes—
161C22×C32D3×Q812yes—
162C32×C4—24no—
163C32×C4—24no—
164C3⋊Dic3—24no—
165C3⋊Dic3—42no—
166C3×Dic3—38no—
167C3×Dic3—64no—
168C3×Dic3—64no—
169C3×Dic3—64no—
170C3×Dic3—64no—
171C3×Dic3—122no—
172C2×C4⋊C4—271no—
173C2×C4⋊C4—271no—
174C2×C4⋊C4—272no—
175C2×C42—91no—
176#95—92no—
177H(3,𝔽3)#38911yescharacteristic, complemented, Sylow
178C23×C3—32no—
179C23×C3—61no—
180C23×C3C2×D3211yescharacteristic
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—34no—
184C2×C3×C4—62no—
185C2×C3×C4—62no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—92no—
190C2×C3×C4—92no—
191C2×C3×C4—92no—
192C2×C3×C4—94no—
193C2×C3×C4—92no—
194C2×C3×C4—182no—
195C2×C3×C4—182no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198C2×C3×C4—182no—
199C2×C3×C4—182no—
200C2×C3×C4—182no—
201C2×C3×C4#62712yes—
202C2×Dic3—34no—
203C2×Dic3—34no—
204C2×Dic3—34no—
205C2×Dic3—34no—
206C2×Dic3—34no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—94no—
221C2×Dic3—182no—
222C2×Dic3—182no—
223C2×Dic3—182no—
224C2×Dic3—182no—
225C2×Dic3—182no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32C4×Dic612yes—
234C2×C32#87812yes—
235C2×C32#67⋊C412yes—
236C2×C32#88412yes—
237C2×C32#88812yes—
238C2×C32#88912yes—
239C2×C32D3×C4⋊C412yes—
240C22×C4—91no—
241C22×C4—92no—
242C22×C4—91no—
243C22×C4—92no—
244C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
245C4⋊C4—274no—
246C4⋊C4—542no—
247C4⋊C4—544no—
248C42—182no—
249C22×C3—32no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—32no—
253C22×C3—32no—
254C22×C3—32no—
255C22×C3—32no—
256C22×C3—61no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3#379711yescharacteristic
264C22×C3#379811yescharacteristic
265C22×C3C4×D3211yescharacteristic
266C22×C3#380211yescharacteristic
267C22×C3#380511yescharacteristic
268C22×C3C2×#62711yescharacteristic
269C22×C3#381111yescharacteristic
270C3×C4—22no—
271C3×C4—64no—
272C3×C4—64no—
273C3×C4—94no—
274C3×C4—184no—
275C3×C4—182no—
276C3×C4—362no—
277Dic3—38no—
278Dic3—64no—
279Dic3—64no—
280Dic3—64no—
281Dic3—64no—
282Dic3—122no—
283Dic3—184no—
284Dic3—362no—
285C32—21no—
286C32#464712yescomplemented
287C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—92no—
291C2×C4—92no—
292C2×C4—92no—
293C2×C4—94no—
294C2×C4—92no—
295C2×C4—181no—
296C2×C4—182no—
297C2×C4—182no—
298C2×C4—181no—
299C2×C4—182no—
300C2×C4—182no—
301C2×C4—182no—
302C2×C4#619612yes—
303C2×C3—32no—
304C2×C3—32no—
305C2×C3—32no—
306C2×C3—32no—
307C2×C3—32no—
308C2×C3—32no—
309C2×C3—32no—
310C2×C3—61no—
311C2×C3—61no—
312C2×C3—61no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3—61no—
316C2×C3—61no—
317C2×C3#6387011yescharacteristic
318C2×C3#6388311yescharacteristic
319C2×C3#6390211yescharacteristic
320C2×C3#6391211yescharacteristic
321C2×C3C4×#62711yescharacteristic
322C2×C3#6391911yescharacteristic
323C2×C3#6398211yescharacteristic
324C22#8865011yescharacteristic
325C22#8865111yescharacteristic
326C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
327C22#8865511yescharacteristic
328C22C2×#619611yescharacteristic
329C22#8867211yescharacteristic
330C22#8867711yescharacteristic
331C4—22no—
332C4—94no—
333C4—184no—
334C4—182no—
335C4—362no—
336C3—32no—
337C3—61no—
338C3#1059132811yescharacteristic
339C2#1172109811yescharacteristic
340C2C4×#619611yescharacteristic
341C2#1172111711yescharacteristic
342C2#1172113811yescharacteristic
343C2#1172114211yescharacteristic
344C2#1172114711yescharacteristic
345C2#1172130911yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#389
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
14. #88449C4semidirect
67. #1084C4⋊C4semidirect
177. H(3,𝔽3)#389semidirect
286. C32#4647semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
C2×#884492
C2×#884482
#63658⋊C32
#105913273