Finite Group Databasebeta

← #13135168

Subgroups — #13135168

Order 1728.

354 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 63 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#11721824C211yesmaximal, characteristic
2#63658⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#10589153—31nomaximal
5#10589152—31nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C2×#6229C411yescharacteristic, complemented
8C22×(C3⋊Dic3)⋊C3—21no—
9#88552—21no—
10C4⋊#1084C411yescharacteristic, complemented
11#88540—21no—
12C2×(C3⋊(C3⋊C8))⋊C3—21no—
13C3×#924—31no—
14C3×#924—31no—
15C3×#924—61no—
16#63658D311yescharacteristic
17#63657D311yescharacteristic
18C3×C3⋊#95—31no—
19C3×#830—31no—
20C23×H(3,𝔽3)D411yescharacteristic
21C2×C4×H(3,𝔽3)—41no—
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×#1084—21no—
25C2×#1084—21no—
26#6229—41no—
27C2×(C3⋊Dic3)⋊C3—22no—
28C2×(C3⋊Dic3)⋊C3—41no—
29C2×(C3⋊Dic3)⋊C3—41no—
30(C3⋊(C3⋊C8))⋊C3—41no—
31#4538—91no—
32#4467—91no—
33#4452—91no—
34C22×C32×C4—21no—
35C22×C32×C4C2×D311yescharacteristic
36C22×C32×C4C2×D311yescharacteristic
37C22×C3⋊Dic3—21no—
38C22×C3×Dic3—61no—
39C2×C3×Dic6—31no—
40C2×C3×Dic6—31no—
41C2×C3×Dic6—61no—
42C2×C3⋊(C3⋊C8)—21no—
43C3×#217—62no—
44C3×#217—61no—
45C3×#217—61no—
46C3×C4⋊Dic3—31no—
47C3×C4⋊Dic3—31no—
48C3×C4⋊Dic3—61no—
49C3×Dic3⋊C4—62no—
50C3×Dic3⋊C4—61no—
51C3×Dic3⋊C4—61no—
52C2×C3×C3⋊C8—61no—
53C22×H(3,𝔽3)—21no—
54C22×H(3,𝔽3)—21no—
55C22×H(3,𝔽3)C22⋊C411yescharacteristic
56C22×H(3,𝔽3)SD1611yescharacteristic
57C22×H(3,𝔽3)Q1611yescharacteristic
58C4×H(3,𝔽3)—21no—
59C4×H(3,𝔽3)—21no—
60#1084—41no—
61#1084—41no—
62(C3⋊Dic3)⋊C3—42no—
63C3×C22⋊Q8—91no—
64#924—31no—
65#924—31no—
66#924—61no—
67C3×C22⋊C8—91no—
68C3×#95—91no—
69C3⋊#95—31no—
70C3⋊#95—91no—
71#830—31no—
72#830—91no—
73C23×C32—21no—
74C23×C32D1211yescharacteristic
75C23×C32#6711yescharacteristic
76C2×C32×C4—21no—
77C2×C32×C4—21no—
78C2×C32×C4—41no—
79C2×C32×C4—41no—
80C2×C32×C4—42no—
81C2×C32×C4C4×D311yescharacteristic
82C2×C32×C4C4×D311yescharacteristic
83C2×C32×C4D1211yescharacteristic
84C2×C32×C4#6711yescharacteristic
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—41no—
87C2×C3⋊Dic3—41no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—121no—
96C2×C3×Dic3—121no—
97C3×Dic6—122no—
98C3×Dic6—121no—
99C3×Dic6—121no—
100C3⋊(C3⋊C8)—41no—
101C3×C3⋊C8—121no—
102#328—271noSylow
103C2×H(3,𝔽3)—21no—
104C2×H(3,𝔽3)—21no—
105C2×H(3,𝔽3)#9911yescharacteristic
106C2×H(3,𝔽3)#10311yescharacteristic
107C2×H(3,𝔽3)#10411yescharacteristic
108C2×C3×Q8—91no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—61no—
112C22×C3×C4—181no—
113C22×C3×C4D3211yescharacteristic
114C22×Dic3—61no—
115C22×Dic3—181no—
116C2×Dic6—31no—
117C2×Dic6—31no—
118C2×Dic6—61no—
119C2×C3×C8—181no—
120C3×C4⋊C4—91no—
121C3×C4⋊C4—181no—
122C3×C22⋊C4—181no—
123#217—61no—
124#217—61no—
125#217—62no—
126C4⋊Dic3—31no—
127C4⋊Dic3—31no—
128C4⋊Dic3—61no—
129Dic3⋊C4—61no—
130Dic3⋊C4—61no—
131Dic3⋊C4—62no—
132C2×C3⋊C8—61no—
133C2×C3⋊C8—181no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32#20411yescharacteristic
144C22×C32Dic1211yescharacteristic
145C22×C32(C2×D3)⋊C411yescharacteristic
146C22×C32(C2×D3)⋊C411yescharacteristic
147C22×C32#21411yescharacteristic
148C22×C32#21611yescharacteristic
149C32×C4—22no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—41no—
155C3⋊Dic3—42no—
156C3×Dic3—122no—
157C3×Dic3—122no—
158C3×Dic3—122no—
159C3×Dic3—121no—
160C3×Dic3—121no—
161C3×Dic3—121no—
162C3×Dic3—121no—
163C22⋊Q8—91no—
164C22⋊C8—91no—
165#95—91no—
166H(3,𝔽3)#32811yescharacteristic, complemented, Sylow
167C23×C3—31no—
168C23×C3—31no—
169C23×C3—61no—
170C23×C3#62611yescharacteristic
171C3×Q8—361no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—41no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—121no—
180C2×C3×C4—121no—
181C2×C3×C4—122no—
182C2×C3×C4—182no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185C2×C3×C4—361no—
186C2×C3×C4—361no—
187C2×C3×C4#62411yescharacteristic
188C2×C3×C4#62611yescharacteristic
189C2×Dic3—62no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—121no—
197C2×Dic3—121no—
198C2×Dic3—182no—
199C2×Dic3—361no—
200C2×Dic3—361no—
201Dic6—122no—
202Dic6—121no—
203Dic6—121no—
204C3×C8—361no—
205C3⋊C8—121no—
206C3⋊C8—361no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32#80511yescharacteristic
217C2×C32#80611yescharacteristic
218C2×C32#80611yescharacteristic
219C2×C32#81011yescharacteristic
220C2×C32#81611yescharacteristic
221C2×C32#82511yescharacteristic
222C2×Q8—91no—
223C22×C4—181no—
224C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
225C2×C8—181no—
226C4⋊C4—91no—
227C4⋊C4—181no—
228C22⋊C4—181no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—31no—
232C22×C3—31no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—31no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—61no—
244C22×C3—121no—
245C22×C3—121no—
246C22×C3#371611yescharacteristic
247C22×C3#371911yescharacteristic
248C22×C3#372211yescharacteristic
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—62no—
256C3×C4—121no—
257C3×C4—362no—
258C3×C4—361no—
259C3×C4—361no—
260Dic3—122no—
261Dic3—121no—
262Dic3—121no—
263Dic3—121no—
264Dic3—121no—
265Dic3—122no—
266Dic3—122no—
267Dic3—362no—
268C32—21no—
269C32#445211yescharacteristic, complemented
270C32#446711yescharacteristic, complemented
271C23#620011yescharacteristic
272Q8—361no—
273C2×C4—41no—
274C2×C4—182no—
275C2×C4—181no—
276C2×C4—181no—
277C2×C4—361no—
278C2×C4—361no—
279C2×C4#619911yescharacteristic
280C2×C4#620011yescharacteristic
281C8—361no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—61no—
291C2×C3—61no—
292C2×C3—61no—
293C2×C3—61no—
294C2×C3—61no—
295C2×C3—61no—
296C2×C3—61no—
297C2×C3—121no—
298C2×C3—121no—
299C2×C3#6356711yescharacteristic
300C2×C3#6357011yescharacteristic
301C2×C3#6358111yescharacteristic
302C22—21no—
303C22—21no—
304C22#8843811yescharacteristic
305C22#8843911yescharacteristic
306C22#8845111yescharacteristic
307C4—21no—
308C4—21no—
309C4—362no—
310C4—361no—
311C4—361no—
312C3—31no—
313C3—31no—
314C3—61no—
315C3#1058915411yescharacteristic
316C2—21no—
317C2—21no—
318C2#1172032611yescharacteristic
319C2#1172032811yescharacteristic
320C2#1172033511yescharacteristic
subgroupG/H
centerC22#88451
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC22×C3×C4D32
Fitting, cosocleC22×C4×H(3,𝔽3)C22
socleC22×C3#3722
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#328
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
7. C2×#6229C4semidirect
10. C4⋊#1084C4semidirect
166. H(3,𝔽3)#328semidirect
269. C32#4452semidirect
270. C32#4467semidirect

Maximal subgroups

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

subgroupindex
#117218242
#63658⋊C32
#63657⋊C32
#105891533
#105891523