Finite Group Databasebeta

← #13139946

Subgroups — #13139946

Order 1728.

491 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 85 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, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic, complemented
4#11720308C212yesmaximal, complemented
5#11720307C212yesmaximal, complemented
6#10591125—31nomaximal
7#10591091—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084—21no—
10C2×#638⋊C3—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C4⋊#1084—21no—
13C4⋊#1084C2212yes—
14#3752⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#64139D311yescharacteristic
18C3×#930—31no—
19C2×C3×C4⋊Dic3—31no—
20C2×C3×C4⋊Dic3—31no—
21C2×C3×C4⋊Dic3—61no—
22#63657D311yescharacteristic, complemented
23C3×#830—31no—
24#63574—32no—
25#63573—32no—
26#63564—32no—
27#63562—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C22×(C3⋊D3)⋊C3—21no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—22no—
34C2×#1084—42no—
35#638⋊C3—42no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38(C3⋊(C3⋊C8))⋊C3—22no—
39#5034—91no—
40#4968—91no—
41#4735—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C2×#638—21no—
46C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
47C2×C3×#67—61no—
48C22×C3×Dic3—62no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51C2×C3×D12—31no—
52#3752—21no—
53C4⋊(C3⋊Dic3)C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)C2×D312yes—
55C3×(C2×D3)⋊C4—61no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—62no—
60C3×C4⋊Dic3—62no—
61C3×C4⋊Dic3—61no—
62C3×C4⋊Dic3—61no—
63C2×C3×C3⋊C8—32no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)—21no—
66C22×H(3,𝔽3)SD1612yes—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C2×(C3⋊D3)⋊C3—41no—
69C2×(C3⋊D3)⋊C3—41no—
70C2×(C3⋊D3)⋊C3—41no—
71C4×H(3,𝔽3)—21no—
72C4×H(3,𝔽3)#4112yes—
73#1084—42no—
74(C3⋊Dic3)⋊C3—41no—
75C3×#121—91no—
76C2×C3×C4⋊C4—91no—
77#930—31no—
78#930—91no—
79C2×C4⋊Dic3—31no—
80C2×C4⋊Dic3—31no—
81C2×C4⋊Dic3—61no—
82C3×C22⋊C8—91no—
83#832—92no—
84#830—31no—
85#830—91no—
86#821—92no—
87#817—92no—
88#809—92no—
89#808—92no—
90#808—92no—
91C23×C32—21no—
92C23×C32D1211yescharacteristic
93C23×C32#6711yescharacteristic
94C22×C3⋊D3—21no—
95C22×C3×D3—61no—
96C2×C32×C4—24no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4D1211yescharacteristic
102C2×C32×C4#6711yescharacteristic
103C2×C32×C4C22×D311yescharacteristic
104C2×C32×C4C22×D311yescharacteristic
105#638—42no—
106C2×C3⋊Dic3—21no—
107C4⋊(C3⋊D3)—22no—
108C3×#67—122no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—124no—
114C2×C3×Dic3—122no—
115C2×C3×Dic3—122no—
116C3×D12—62no—
117C3⋊(C3⋊C8)—22no—
118C3×C3⋊C8—62no—
119#481—271noSylow
120C2×H(3,𝔽3)—22no—
121C2×H(3,𝔽3)#12311yescharacteristic
122C2×H(3,𝔽3)C2×SD1611yescharacteristic
123C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
124(C3⋊D3)⋊C3—81no—
125C2×C3×D4—91no—
126C2×C3×D4—181no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—61no—
130C22×C3×C4—181no—
131C22×C3×C4D3211yescharacteristic
132C2×#67—61no—
133C2×#67—181no—
134C22×Dic3—61no—
135C22×Dic3—61no—
136C22×Dic3—62no—
137C2×D12—31no—
138C2×D12—91no—
139C2×C3×C8—92no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—92no—
142C3×C4⋊C4—181no—
143C3×C22⋊C4—181no—
144(C2×D3)⋊C4—61no—
145(C2×D3)⋊C4—181no—
146C4⋊Dic3—32no—
147C4⋊Dic3—32no—
148C4⋊Dic3—31no—
149C4⋊Dic3—62no—
150C4⋊Dic3—61no—
151C4⋊Dic3—61no—
152C4⋊Dic3—62no—
153C4⋊Dic3—91no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—92no—
156C22×C32—22no—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32#20412yes—
165C22×C32#21412yes—
166C22×C32C2×D1211yescharacteristic
167C22×C32C2×#6711yescharacteristic
168C2×C3⋊D3—41no—
169C2×C3⋊D3—41no—
170C2×C3⋊D3—41no—
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C32×C4—22no—
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—22no—
178C32×C4#23712yes—
179C32×C4#23712yes—
180C3⋊Dic3—41no—
181C3×Dic3—124no—
182C3×Dic3—121no—
183C3×Dic3—122no—
184C3×Dic3—122no—
185#121—91no—
186C2×C4⋊C4—91no—
187#106—272no—
188#102—272no—
189C22⋊C8—91no—
190H(3,𝔽3)#48111yescharacteristic, complemented, Sylow
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—61no—
194C23×C3—181no—
195C23×C3#62611yescharacteristic
196C22×D3—61no—
197C22×D3—181no—
198C3×D4—182no—
199C3×D4—362no—
200C2×C3×C4—22no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—64no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—181no—
211C2×C3×C4—182no—
212C2×C3×C4—362no—
213C2×C3×C4#62611yescharacteristic
214C2×C3×C4C2×D3211yescharacteristic
215#67—122no—
216#67—362no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—64no—
220C2×Dic3—61no—
221C2×Dic3—122no—
222C2×Dic3—122no—
223C2×Dic3—124no—
224C2×Dic3—181no—
225D12—62no—
226D12—182no—
227C3×C8—182no—
228C3⋊C8—62no—
229C3⋊C8—182no—
230C2×C32—24no—
231C2×C32—22no—
232C2×C32—22no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32#88611yescharacteristic
237C2×C32C2×#20411yescharacteristic
238C2×C32#90811yescharacteristic
239C2×C32C2×#21411yescharacteristic
240C2×C32#93511yescharacteristic
241C2×C32#94911yescharacteristic
242C3⋊D3—81no—
243C3×D3—241no—
244C2×D4—91no—
245C2×D4—181no—
246C22×C4—181no—
247C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
248C2×C8—92no—
249C4⋊C4—92no—
250C4⋊C4—91no—
251C4⋊C4—181no—
252C22⋊C4—181no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—32no—
258C22×C3—32no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—62no—
264C22×C3—62no—
265C22×C3—61no—
266C22×C3—62no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3—361no—
270C22×C3#371612yes—
271C22×C3C2×#62611yescharacteristic
272C2×D3—121no—
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—361no—
276C2×D3—361no—
277C2×D3—361no—
278C3×C4—21no—
279C3×C4—32no—
280C3×C4—32no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—62no—
284C3×C4—62no—
285C3×C4—361no—
286C3×C4—362no—
287C3×C4#380512yes—
288Dic3—122no—
289Dic3—122no—
290Dic3—124no—
291Dic3—121no—
292Dic3—361no—
293C32—21no—
294C32#473511yescharacteristic, complemented
295C32#503411yescharacteristic, complemented
296C23—181no—
297C23#620011yescharacteristic
298D4—182no—
299D4—362no—
300C2×C4—22no—
301C2×C4—181no—
302C2×C4—182no—
303C2×C4—362no—
304C2×C4#620011yescharacteristic
305C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C8—182no—
307C2×C3—22no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—62no—
315C2×C3—62no—
316C2×C3—64no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3—61no—
320C2×C3—721no—
321C2×C3#6382511yescharacteristic
322C2×C3C2×#371611yescharacteristic
323C2×C3#6396211yescharacteristic
324D3—241no—
325D3—721no—
326C22—21no—
327C22—21no—
328C22—361no—
329C22—361no—
330C22—361no—
331C22#8843812yes—
332C22C2×#620011yescharacteristic
333C4—21no—
334C4—362no—
335C4—361no—
336C4#8867212yes—
337C3—31no—
338C3—31no—
339C3—61no—
340C3#1059109211yescharacteristic
341C2—22no—
342C2—721no—
343C2#1172106811yescharacteristic
344C2C2×#8843811yescharacteristic
345C2#1172130711yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#481
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1084C2semidirect
3. #63657⋊C3C2semidirect
4. #11720308C2semidirect
5. #11720307C2semidirect
22. #63657D3semidirect
190. H(3,𝔽3)#481semidirect
294. C32#4735semidirect
295. C32#5034semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
#64139⋊C32
#63657⋊C32
#117203082
#117203072
#105911253
#105910913