Finite Group Databasebeta

← #13159172

Subgroups — #13159172

Order 1728.

780 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C32×#892C211yesmaximal, characteristic, complemented
2#11723515C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic, complemented
4C3×#63636C211yesmaximal, characteristic, complemented
5#11722057C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592702—31nomaximal
9#10590505—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×#6290—21no—
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3—21no—
20#88803—21no—
21C32×#117—31no—
22#64139—31no—
23C3×#892D314yescomplemented
24#63908—34no—
25#63657—31no—
26#63636—31no—
27#63636D311yescharacteristic, complemented
28#63635—31no—
29#63635—34no—
30#63635—64no—
31#63634—31no—
32C3×#809—34no—
33#63575—34no—
34#63562—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3×C3⋊D3—21no—
37C22×C32×D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C32×C4×D3—21no—
46C32×C4×D3—21no—
47#6290—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50#5003—94no—
51#4784—91no—
52C22×C32×C4—31no—
53C2×#638—61no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C3×D12—34no—
56C2×C3×C4×D3C2×D314yes—
57C2×#625—64no—
58C32×C4⋊C4—24no—
59C32×C4⋊C4—31no—
60C32×C4⋊C4C2×D314yes—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C22⋊C4—61no—
63C32×C42—61no—
64#3752—61no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—64no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×(C2×D3)⋊C4—24no—
73C3×C4⋊Dic3C2×D314yes—
74C3×C4×Dic3—24no—
75C2×C3×C3⋊C8—34no—
76(C2×D3)⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C2×C32×D3—21no—
82C2×C32×D3—21no—
83C2×C32×D3—21no—
84C33×C4—41no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C32×Dic3—21no—
89C32×Dic3—41no—
90C3×#117—34no—
91#938—91no—
92#930—94no—
93#892C3⋊D311yescharacteristic, complemented
94C3×#102—91no—
95#832—91no—
96#830—94no—
97(C2×D3)⋊C8—91no—
98#818—91no—
99#809—34no—
100#809—94no—
101#808—94no—
102#808—91no—
103#808—184no—
104#807—94no—
105C23×C32—31no—
106C22×C3⋊D3—21no—
107C22×C3×D3—64no—
108C22×C3×D3#6714yes—
109C2×C32×C4—21no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—31no—
113C2×C32×C4—44no—
114C2×C32×C4—61no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119#638—122no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—124no—
123C4⋊(C3⋊D3)—21no—
124C4⋊(C3⋊D3)—21no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3#6714yes—
127C3×D12—64no—
128C3×D12—64no—
129C3×C4×D3—24no—
130C3×C4×D3—24no—
131#625—128no—
132C3⋊(C3⋊C8)—21no—
133C3⋊(C3⋊C8)—68no—
134C3⋊(C3⋊C8)—64no—
135C3⋊(C3⋊C8)—61no—
136C3×C3⋊C8—64no—
137#482—271noSylow
138C2×C33#12311yescharacteristic
139C2×C33#13511yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C3×C3⋊D3—81no—
142C32×D3—41no—
143C2×C3×D4—91no—
144C22×C3×C4—34no—
145C2×#67—184no—
146C2×#67—181no—
147C2×D12—34no—
148C2×C4×D3C2×C3⋊D311yescharacteristic
149C2×C3×C8—91no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—34no—
152C3×C4⋊C4D3214yes—
153C3×C4⋊C4C2×C3⋊D311yescharacteristic
154C3×C22⋊C4—64no—
155C3×C42—64no—
156#217—181no—
157(C2×D3)⋊C4—21no—
158(C2×D3)⋊C4—184no—
159C4⋊Dic3—94no—
160C4⋊Dic3—91no—
161C4⋊Dic3—184no—
162C4⋊Dic3C2×C3⋊D311yescharacteristic
163C4×Dic3—21no—
164C2×C3⋊C8—34no—
165C2×C3⋊C8—94no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—184no—
168C22×C32—24no—
169C22×C32—61no—
170C22×C32—61no—
171C22×C32—61no—
172C22×C32D3×D411yescharacteristic
173C22×C32C2×#6714yes—
174C2×C3⋊D3—41no—
175C2×C3⋊D3—41no—
176C2×C3⋊D3—41no—
177C2×C3×D3—24no—
178C2×C3×D3—24no—
179C2×C3×D3—24no—
180C2×C3×D3—124no—
181C2×C3×D3—124no—
182C2×C3×D3—124no—
183C32×C4—24no—
184C32×C4—24no—
185C32×C4—41no—
186C32×C4—44no—
187C32×C4—61no—
188C32×C4—84no—
189C32×C4—121no—
190C32×C4#23514yes—
191C32×C4#23514yes—
192C32×C4#23511yescharacteristic
193C32×C4#23711yescharacteristic
194C3⋊Dic3—121no—
195C3⋊Dic3—124no—
196C3⋊Dic3—244no—
197C3×Dic3—24no—
198C3×Dic3—44no—
199#121—271no—
200#117—31no—
201#107—271no—
202#106—271no—
203#102—91no—
204#102—271no—
205C22⋊C8—271no—
206C33#48211yescharacteristic, complemented, Sylow
207C23×C3—34no—
208C23×C3—181no—
209C22×D3—64no—
210C22×D3#63811yescharacteristic
211C3×D4—181no—
212C3×D4—181no—
213C2×C3×C4—24no—
214C2×C3×C4—21no—
215C2×C3×C4—24no—
216C2×C3×C4—34no—
217C2×C3×C4—44no—
218C2×C3×C4—64no—
219C2×C3×C4—64no—
220C2×C3×C4—64no—
221C2×C3×C4C2×D3214yes—
222C2×C3×C4C22×C3⋊D311yescharacteristic
223#67—362no—
224#67—368no—
225C2×Dic3—21no—
226C2×Dic3—184no—
227C2×Dic3—181no—
228C2×Dic3—364no—
229C2×Dic3#63811yescharacteristic
230D12—64no—
231D12—64no—
232C4×D3—21no—
233C4×D3—21no—
234C3×C8—181no—
235C3⋊C8—64no—
236C3⋊C8—188no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32—121no—
243C2×C32#88311yescharacteristic
244C2×C32#91211yescharacteristic
245C2×C32#91411yescharacteristic
246C2×C32#92914yes—
247C2×C32#94914yes—
248C2×C32#95014yes—
249C3⋊D3—81no—
250C3×D3—44no—
251C3×D3—244no—
252C2×D4—91no—
253C2×D4—541no—
254C22×C4—31no—
255C2×C8—91no—
256C2×C8—271no—
257C4⋊C4—31no—
258C4⋊C4—271no—
259C4⋊C4D3×C3⋊D311yescharacteristic
260C22⋊C4—61no—
261C22⋊C4—541no—
262C42—61no—
263C22×C3—24no—
264C22×C3—64no—
265C22×C3—64no—
266C22×C3—64no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3—361no—
270C22×C3D3×#6714yes—
271C22×C3C2×#63811yescharacteristic
272C2×D3—21no—
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—124no—
276C2×D3—124no—
277C2×D3—124no—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—41no—
281C3×C4—44no—
282C3×C4—64no—
283C3×C4—84no—
284C3×C4—124no—
285C3×C4#379514yes—
286C3×C4#379814yes—
287C3×C4#382811yescharacteristic
288C3×C4#382811yescharacteristic
289Dic3—21no—
290Dic3—41no—
291Dic3—361no—
292Dic3—364no—
293Dic3—724no—
294C32—24no—
295C32#478411yescharacteristic, complemented
296C32#500314yescomplemented
297C23—31no—
298C23—181no—
299D4—181no—
300D4—181no—
301D4—1082no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—541no—
308C2×C4C2×D3×C3⋊D311yescharacteristic
309C8—181no—
310C8—541no—
311C2×C3—24no—
312C2×C3—24no—
313C2×C3—24no—
314C2×C3—124no—
315C2×C3—721no—
316C2×C3#6390514yes—
317C2×C3#6393314yes—
318C2×C3#6393914yes—
319C2×C3#6413811yescharacteristic
320C2×C3#6415811yescharacteristic
321C2×C3#6415911yescharacteristic
322D3—41no—
323D3—244no—
324C22—61no—
325C22—61no—
326C22—61no—
327C22—361no—
328C22—361no—
329C22—361no—
330C22D3×#63811yescharacteristic
331C4—41no—
332C4—61no—
333C4—121no—
334C4—1081no—
335C4#8902011yescharacteristic
336C4#8902111yescharacteristic
337C3—24no—
338C3#1059050514yescomplemented
339C3#1059270211yescharacteristic, complemented
340C2—121no—
341C2—721no—
342C2#1172351311yescharacteristic
343C2#1172355611yescharacteristic
344C2#1172356811yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723515C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#482
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#892C2semidirect
3. #11722178C2semidirect
4. C3×#63636C2semidirect
5. #11722057C2semidirect
6. #11722033C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
23. C3×#892D3semidirect
27. #63636D3semidirect
93. #892C3⋊D3semidirect
206. C33#482semidirect
295. C32#4784semidirect
296. C32#5003semidirect
338. C3#10590505semidirect
339. C3#10592702semidirect

Maximal subgroups

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

subgroupindex
C32×#8922
#117235152
#117221782
C3×#636362
#117220572
#117220332
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927023
#105905053