Finite Group Databasebeta

← #13140005

Subgroups — #13140005

Order 1728.

561 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 117 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×#88540C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
3#63658⋊C3C211yesmaximal, characteristic
4#11720334C212yesmaximal
5#11720334C212yesmaximal
6#10591226—31nomaximal
7#10591225—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13#88540—22no—
14C4⋊((C3⋊Dic3)⋊C3)—22no—
15C2×C4⋊(C3⋊Dic3)D311yescharacteristic
16C2×C3×C4⋊Dic3—31no—
17C2×C3×Dic3⋊C4—31no—
18C2×C3×Dic3⋊C4—31no—
19C2×C3×Dic3⋊C4—61no—
20#63658D311yescharacteristic
21C3×C3⋊#95—31no—
22#63579—32no—
23#63579—32no—
24#63579—32no—
25#63579—32no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)D412yes—
29C2×#1084—21no—
30C2×#1084—21no—
31C2×#1084—21no—
32C2×#1084—22no—
33C2×#1084Q812yes—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—24no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39#4986—91no—
40#4985—91no—
41#4659—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3C2×D312yes—
46C22×C3⋊Dic3C2×D312yes—
47C22×C3×Dic3—31no—
48C22×C3×Dic3—31no—
49C22×C3×Dic3—31no—
50C22×C3×Dic3—31no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—61no—
54C22×C3×Dic3—61no—
55C4⋊(C3⋊Dic3)—22no—
56C3×C4⋊Dic3—62no—
57C3×Dic3⋊C4—64no—
58C3×Dic3⋊C4—62no—
59C3×Dic3⋊C4—62no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C22×H(3,𝔽3)C2×Q811yescharacteristic
62C22×H(3,𝔽3)#4111yescharacteristic
63C22×H(3,𝔽3)#4112yes—
64C22×H(3,𝔽3)#4112yes—
65C4×H(3,𝔽3)—22no—
66#1084—22no—
67#1084—41no—
68(C3⋊Dic3)⋊C3—42no—
69(C3⋊Dic3)⋊C3—42no—
70C2×C3×C4⋊C4—91no—
71C2×C3×C4⋊C4—91no—
72C2×C4⋊Dic3—31no—
73C2×C4⋊Dic3—91no—
74C2×Dic3⋊C4—31no—
75C2×Dic3⋊C4—31no—
76C2×Dic3⋊C4—61no—
77C3×#95—91no—
78C3⋊#95—31no—
79C3⋊#95—91no—
80C3⋊#95—92no—
81C3⋊#95—92no—
82C3⋊#95—92no—
83C3⋊#95—92no—
84C3⋊#95—92no—
85C3⋊#95—92no—
86C23×C32—21no—
87C23×C32C22×D311yescharacteristic
88C23×C32C22×D311yescharacteristic
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—42no—
93C2×C32×C4D1212yes—
94C2×C32×C4#6712yes—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—24no—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3—22no—
100C2×C3×Dic3—32no—
101C2×C3×Dic3—32no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—64no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—122no—
120#400—271noSylow
121C2×H(3,𝔽3)C22⋊Q811yescharacteristic
122C2×H(3,𝔽3)#12311yescharacteristic
123C2×H(3,𝔽3)#12411yescharacteristic
124C2×H(3,𝔽3)#12512yes—
125C2×H(3,𝔽3)#12612yes—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—61no—
129C22×C3×C4—91no—
130C22×C3×C4—92no—
131C22×C3×C4—92no—
132C22×C3×C4—91no—
133C22×C3×C4D3211yescharacteristic
134C22×Dic3—31no—
135C22×Dic3—31no—
136C22×Dic3—31no—
137C22×Dic3—31no—
138C22×Dic3—32no—
139C22×Dic3—32no—
140C22×Dic3—61no—
141C22×Dic3—61no—
142C22×Dic3—92no—
143C22×Dic3—92no—
144C3×C4⋊C4—182no—
145C3×C4⋊C4—182no—
146C4⋊Dic3—62no—
147C4⋊Dic3—182no—
148Dic3⋊C4—62no—
149Dic3⋊C4—62no—
150Dic3⋊C4—64no—
151C22×C32—22no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—22no—
155C22×C32—21no—
156C22×C32C2×D1211yescharacteristic
157C22×C32#23512yes—
158C22×C32#23512yes—
159C22×C32#23712yes—
160C22×C32#23711yescharacteristic
161C22×C32#23712yes—
162C22×C32D3×Q811yescharacteristic
163C22×C32D3×Q811yescharacteristic
164C22×C32#23911yescharacteristic
165C22×C32C2×#6711yescharacteristic
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—42no—
169C3⋊Dic3—42no—
170C3⋊Dic3—42no—
171C3×Dic3—64no—
172C3×Dic3—62no—
173C3×Dic3—62no—
174C3×Dic3—122no—
175C3×Dic3—122no—
176C3×Dic3—121no—
177C3×Dic3—122no—
178C3×Dic3—121no—
179C2×C4⋊C4—91no—
180C2×C4⋊C4—91no—
181#95—91no—
182#95—272no—
183#95—272no—
184H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3C2×D3211yescharacteristic
189C2×C3×C4—22no—
190C2×C3×C4—32no—
191C2×C3×C4—32no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—92no—
196C2×C3×C4—122no—
197C2×C3×C4—181no—
198C2×C3×C4—181no—
199C2×C3×C4—182no—
200C2×C3×C4—182no—
201C2×C3×C4—184no—
202C2×C3×C4—182no—
203C2×C3×C4—182no—
204C2×C3×C4—182no—
205C2×C3×C4—181no—
206C2×C3×C4#62612yes—
207C2×Dic3—32no—
208C2×Dic3—32no—
209C2×Dic3—62no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—64no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C2×Dic3—122no—
227C2×Dic3—182no—
228C2×Dic3—184no—
229C2×Dic3—182no—
230C2×Dic3—182no—
231C2×Dic3—182no—
232C2×C32—22no—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32#87511yescharacteristic
238C2×C32#87912yes—
239C2×C32#88611yescharacteristic
240C2×C32#88912yes—
241C2×C32#88912yes—
242C2×C32#89711yescharacteristic
243C2×C32#89812yes—
244C2×C32#92911yescharacteristic
245C2×C32#93611yescharacteristic
246C2×C32#94611yescharacteristic
247C22×C4—91no—
248C22×C4—92no—
249C22×C4—91no—
250C22×C4—92no—
251C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
252C4⋊C4—182no—
253C4⋊C4—182no—
254C22×C3—32no—
255C22×C3—32no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—31no—
259C22×C3—31no—
260C22×C3—32no—
261C22×C3—32no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—62no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—62no—
268C22×C3—61no—
269C22×C3#379512yes—
270C22×C3#379611yescharacteristic
271C22×C3#379711yescharacteristic
272C22×C3#380412yes—
273C22×C3C2×#62611yescharacteristic
274C3×C4—22no—
275C3×C4—62no—
276C3×C4—62no—
277C3×C4—122no—
278C3×C4—182no—
279C3×C4—362no—
280C3×C4—362no—
281C3×C4—361no—
282Dic3—62no—
283Dic3—62no—
284Dic3—64no—
285Dic3—121no—
286Dic3—121no—
287Dic3—122no—
288Dic3—122no—
289Dic3—122no—
290Dic3—362no—
291Dic3—362no—
292C32—21no—
293C32#465911yescharacteristic, complemented
294C32#498511yescharacteristic, complemented
295C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C2×C4—22no—
297C2×C4—92no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—182no—
301C2×C4—184no—
302C2×C4—182no—
303C2×C4—182no—
304C2×C4—182no—
305C2×C4—181no—
306C2×C4—182no—
307C2×C4#620012yes—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—32no—
311C2×C3—32no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—32no—
315C2×C3—32no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—61no—
319C2×C3—62no—
320C2×C3—61no—
321C2×C3—62no—
322C2×C3—61no—
323C2×C3#6384712yes—
324C2×C3#6385812yes—
325C2×C3#6386011yescharacteristic
326C2×C3#6387111yescharacteristic
327C2×C3#6396611yescharacteristic
328C22#8864812yes—
329C22#8864911yescharacteristic
330C22#8865111yescharacteristic
331C22#8867112yes—
332C22C2×#620011yescharacteristic
333C4—22no—
334C4—182no—
335C4—361no—
336C4—362no—
337C4—362no—
338C3—31no—
339C3—31no—
340C3—61no—
341C3#1059122711yescharacteristic
342C2#1172108612yes—
343C2#1172109411yescharacteristic
344C2#1172110812yes—
345C2#1172111011yescharacteristic
346C2#1172130111yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#400
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
184. H(3,𝔽3)#400semidirect
293. C32#4659semidirect
294. C32#4985semidirect

Maximal subgroups

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

subgroupindex
C2×#885402
C2×C4⋊((C3⋊Dic3)⋊C3)2
#63658⋊C32
#117203342
#105912263
#105912253