Finite Group Databasebeta

← #13159555

Subgroups — #13159555

Order 1728.

980 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 180 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×#88850C211yesmaximal, characteristic
2#11723503C211yesmaximal, characteristic
3#11723413C211yesmaximal, characteristic
4C3×#63645C211yesmaximal, characteristic, complemented
5C32×#817C211yesmaximal, characteristic, complemented
6#11722056C211yesmaximal, characteristic, complemented
7#11722053C211yesmaximal, characteristic, complemented
8#10592560—31nomaximal
9#10590688—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×#6289—21no—
12C2×C33×C8C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14#88850C414yescomplemented
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88806—21no—
20#88804—21no—
21Dic3×C3⋊Dic3—21no—
22C2×#3743—31no—
23C2×#3743—34no—
24C2×#3743—64no—
25#64101—31no—
26C4⋊C4×C3⋊D3—31no—
27#63900—34no—
28#63849—34no—
29C32×#106—31no—
30#63645D311yescharacteristic, complemented
31#63635—31no—
32#63634—31no—
33C3×#818—34no—
34C3×#817D314yescomplemented
35#63576—34no—
36#63574—34no—
37C22×C3⋊(C3⋊D3)D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C4×C3⋊(C3⋊D3)C2×C412yes—
41C4×C3⋊(C3⋊D3)Q812yes—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46#6289—42no—
47C33×C8C2×C412yescomplemented
48C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
49#4881—91no—
50#4861—94no—
51C2×C4×C3⋊D3—31no—
52C2×C4×C3⋊D3—34no—
53C2×C4×C3⋊D3—64no—
54C2×C4×C3⋊D3—61no—
55C2×#624—64no—
56C2×C32×C8—24no—
57C2×C32×C8C2×D314yes—
58C2×C32×C8C2×D311yescharacteristic
59C32×C4⋊C4—31no—
60C32×C4⋊C4—31no—
61#3752—61no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—61no—
65C4×C3⋊Dic3—61no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69#3743—34no—
70#3743—316no—
71#3743—616no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3C2×D314yes—
75C3×C4⋊Dic3C2×D314yes—
76#3724—64no—
77#3722—64no—
78Dic32—64no—
79C22×C33C2×D411yescharacteristic
80C2×C3⋊(C3⋊D3)—21no—
81C2×C3⋊(C3⋊D3)—21no—
82C2×C3⋊(C3⋊D3)C4⋊C412yes—
83C33×C4C22×C411yescharacteristic
84C33×C4C2×Q811yescharacteristic
85C3⋊(C3⋊Dic3)C4⋊C412yes—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3—41no—
88C32×Dic3—41no—
89C32×Dic3—41no—
90C2×ΓL(1,𝔽25)—94no—
91C2×ΓL(1,𝔽25)—91no—
92C2×ΓL(1,𝔽25)—184no—
93#892—94no—
94#892—91no—
95D3×C4⋊C4—94no—
96D3×C4⋊C4—91no—
97C3×#107—91no—
98C3×#106—34no—
99#818—34no—
100#817C3⋊D311yescharacteristic, complemented
101#808—94no—
102#808—91no—
103#807—94no—
104#807—91no—
105C22×C3⋊D3—31no—
106C22×C3⋊D3—34no—
107C22×C3⋊D3—64no—
108C2×C32×C4—24no—
109C2×C32×C4—61no—
110C2×C32×C4—61no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—34no—
117C2×C3⋊Dic3—64no—
118C4×C3⋊D3—32no—
119C4×C3⋊D3—32no—
120C4×C3⋊D3—38no—
121C4×C3⋊D3—38no—
122C4×C3⋊D3—68no—
123C4×C3⋊D3—68no—
124C4×C3⋊D3—122no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—64no—
129#624—128no—
130C32×C8—28no—
131C32×C8C4×D318yescomplemented
132C32×C8C4×D312yescomplemented
133C3⋊(C3⋊C8)—32no—
134C3⋊(C3⋊C8)—38no—
135C3⋊(C3⋊C8)—68no—
136#428—271noSylow
137C2×C33C2×C4⋊C411yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C2×C33#13711yescharacteristic
140C3⋊(C3⋊D3)—22no—
141C2×C4×D3—94no—
142C2×C4×D3—91no—
143C2×C4×D3—184no—
144C2×C4×D3—181no—
145C2×C4×D3—184no—
146C2×C3×C8—24no—
147C2×C3×C8D3214yes—
148C2×C3×C8C2×C3⋊D311yescharacteristic
149C3×C4⋊C4—34no—
150C3×C4⋊C4—34no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—91no—
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—184no—
155C4⋊Dic3—34no—
156C4⋊Dic3—34no—
157C4⋊Dic3C2×C3⋊D311yescharacteristic
158C4⋊Dic3C2×C3⋊D311yescharacteristic
159Dic3⋊C4—181no—
160Dic3⋊C4—184no—
161C4×Dic3—181no—
162C4×Dic3—184no—
163C2×C3⋊C8—94no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—184no—
166ΓL(1,𝔽25)—916no—
167ΓL(1,𝔽25)—94no—
168ΓL(1,𝔽25)—1816no—
169C22×C32—24no—
170C22×C32D3×D414yes—
171C22×C32D3×D411yescharacteristic
172C2×C3⋊D3—32no—
173C2×C3⋊D3—38no—
174C2×C3⋊D3—68no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—61no—
177C2×C3⋊D3—64no—
178C2×C3⋊D3—64no—
179C2×C3⋊D3—124no—
180C2×C3⋊D3—124no—
181C32×C4—24no—
182C32×C4—24no—
183C32×C4—121no—
184C32×C4—121no—
185C32×C4C2×C4×D314yes—
186C32×C4C2×C4×D311yescharacteristic
187C32×C4D3×Q814yes—
188C32×C4D3×Q811yescharacteristic
189C3⋊Dic3—32no—
190C3⋊Dic3—38no—
191C3⋊Dic3—41no—
192C3⋊Dic3—41no—
193C3⋊Dic3—68no—
194C3×Dic3—44no—
195C3×Dic3—44no—
196C3×Dic3—124no—
197C3×Dic3—124no—
198C2×M16—271no—
199#117—271no—
200C2×C4⋊C4—271no—
201#107—91no—
202#107—271no—
203#106—31no—
204#106—271no—
205C33#42811yescharacteristic, complemented, Sylow
206C22×D3—94no—
207C22×D3—91no—
208C22×D3—184no—
209C2×C3×C4—24no—
210C2×C3×C4—64no—
211C2×C3×C4—64no—
212C2×C3×C4—181no—
213C2×C3×C4—181no—
214C2×C3×C4C2×D3214yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—64no—
219C2×Dic3—64no—
220C2×Dic3—94no—
221C2×Dic3—91no—
222C2×Dic3—184no—
223C4×D3—98no—
224C4×D3—98no—
225C4×D3—92no—
226C4×D3—92no—
227C4×D3—188no—
228C4×D3—188no—
229C4×D3—362no—
230C4×D3—368no—
231C3×C8—28no—
232C3×C8#62418yescomplemented
233C3×C8C4×C3⋊D312yescomplemented
234C3⋊C8—98no—
235C3⋊C8—92no—
236C3⋊C8—188no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32D3×C4⋊C414yes—
241C2×C32D3×C4⋊C411yescharacteristic
242C2×C32#91111yescharacteristic
243C2×C32#91414yes—
244C2×C32#91514yes—
245C2×C32#91811yescharacteristic
246C3⋊D3—62no—
247C3⋊D3—68no—
248C3⋊D3—128no—
249C22×C4—271no—
250C22×C4—541no—
251M16—274no—
252C2×C8—271no—
253C2×C8D3×C3⋊D311yescharacteristic
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—91no—
257C4⋊C4—91no—
258C4⋊C4—541no—
259C22⋊C4—541no—
260C42—541no—
261C22×C3—24no—
262C22×C3#380214yes—
263C22×C3D4×C3⋊D311yescharacteristic
264C2×D3—92no—
265C2×D3—98no—
266C2×D3—188no—
267C2×D3—184no—
268C2×D3—184no—
269C2×D3—181no—
270C2×D3—181no—
271C2×D3—364no—
272C2×D3—364no—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—124no—
276C3×C4—124no—
277C3×C4—361no—
278C3×C4—361no—
279C3×C4#379714yes—
280C3×C4C2×#62414yes—
281C3×C4C2×C4×C3⋊D311yescharacteristic
282C3×C4Q8×C3⋊D311yescharacteristic
283Dic3—41no—
284Dic3—41no—
285Dic3—92no—
286Dic3—98no—
287Dic3—124no—
288Dic3—124no—
289Dic3—188no—
290C32—24no—
291C32#486114yescomplemented
292C32#488111yescharacteristic, complemented
293C23—271no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4—271no—
299C2×C4—272no—
300C2×C4—272no—
301C2×C4—1082no—
302C2×C4C2×D3×C3⋊D311yescharacteristic
303C8—272no—
304C8#628912yescomplemented
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3#6379814yes—
309C2×C3#6381114yes—
310C2×C3#6390014yes—
311C2×C3C4⋊C4×C3⋊D311yescharacteristic
312C2×C3#6412311yescharacteristic
313C2×C3#6412411yescharacteristic
314D3—182no—
315D3—188no—
316D3—368no—
317C22—272no—
318C22—541no—
319C22—541no—
320C22#8902611yescharacteristic
321C4—121no—
322C4—121no—
323C4—272no—
324C4—361no—
325C4—361no—
326C4#8901911yescharacteristic
327C4C2×#628911yescharacteristic
328C3—24no—
329C3#1059068814yescomplemented
330C3#1059256011yescharacteristic, complemented
331C2—542no—
332C2#1172332311yescharacteristic
333C2#1172334411yescharacteristic
334C2#1172350311yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#428
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C3×#63645C2semidirect
5. C32×#817C2semidirect
6. #11722056C2semidirect
7. #11722053C2semidirect
14. #88850C4semidirect
30. #63645D3semidirect
34. C3×#817D3semidirect
47. C33×C8C2×C4semidirect
48. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
100. #817C3⋊D3semidirect
131. C32×C8C4×D3semidirect
132. C32×C8C4×D3semidirect
205. C33#428semidirect
232. C3×C8#624semidirect
233. C3×C8C4×C3⋊D3semidirect
291. C32#4861semidirect
292. C32#4881semidirect
304. C8#6289semidirect
329. C3#10590688semidirect
330. C3#10592560semidirect

Maximal subgroups

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

subgroupindex
C2×#888502
#117235032
#117234132
C3×#636452
C32×#8172
#117220562
#117220532
#105925603
#105906883