Finite Group Databasebeta

← #13154270

Subgroups — #13154270

Order 1728.

1837 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1837 conjugacy classes of subgroups — too many to draw. Only its 213 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×#88744C211yesmaximal, characteristic, complemented
2C2×D4×#272C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
4#11721850C211yesmaximal, characteristic, complemented
5#11721849C214yesmaximal, complemented
6#11721847C214yesmaximal, complemented
7#11721846C211yesmaximal, characteristic, complemented
8#11721721C211yesmaximal, characteristic, complemented
9#11721718C211yesmaximal, characteristic, complemented
10C3×#5715—34nomaximal
11C23×#272C2212yes—
12C2×D4×H(3,𝔽3)C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×D4×H(3,𝔽3)C2214yescomplemented
15C2×#6235C2212yes—
16C2×#6235C2212yes—
17C2×#6235C2214yescomplemented
18C22×#1084C2212yescomplemented
19#88744—24no—
20D4×#272—24no—
21C2×C4⋊#272C2211yescharacteristic
22C2×C4×#272C2211yescharacteristic
23C2×#6229C2211yescharacteristic, complemented
24C42×H(3,𝔽3)C2211yescharacteristic, complemented
25#88552C2214yescomplemented
26#88542C2212yes—
27#88542C2214yescomplemented
28#88540C2212yescomplemented
29C4×#1084C2211yescharacteristic, complemented
30C2×C3×#237—34no—
31C2×C3×D3×D4—34no—
32C32×#127D314yescomplemented
33C3×#940—34no—
34C3×#939—316no—
35C3×#937—316no—
36C3×#936—34no—
37C3×#875—34no—
38C3×#872—34no—
39C23×H(3,𝔽3)C2312yes—
40C23×H(3,𝔽3)C2312yes—
41C22×#272—24no—
42C22×#272—44no—
43C22×#272C2312yes—
44C22×#272C2311yescharacteristic
45D4×H(3,𝔽3)—22no—
46D4×H(3,𝔽3)—22no—
47D4×H(3,𝔽3)—44no—
48C2×C4×H(3,𝔽3)C2312yes—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50#6235—24no—
51#6235—24no—
52#6235—44no—
53C2×#1084—24no—
54C2×#1084C2312yes—
55C2×#1084C2312yes—
56C2×#1084C2311yescharacteristic
57C4⋊#272—22no—
58C4×#272D414yescomplemented
59#6229—22no—
60C3×#535—91no—
61#5715—34no—
62C23×C3×D3—38no—
63C2×C32×D4C2×D314yes—
64C2×C32×D4C2×D314yescomplemented
65C2×C32×D4C2×D3116yescomplemented
66C2×C3×#67—38no—
67C2×C3×#67—38no—
68C2×C3×#67—316no—
69C22×C3×Dic3—38no—
70C3×#237—616no—
71C3×D3×D4—616no—
72C2×C3×D12—34no—
73C2×C3×C4×D3—34no—
74C2×C3×Dic6—34no—
75C32×C42C2×D314yescomplemented
76C3×#217—316no—
77C3×(C2×D3)⋊C4—38no—
78C3×(C2×D3)⋊C4—316no—
79C3×Dic3⋊C4—38no—
80C3×C4×Dic3—34no—
81C22×H(3,𝔽3)—22no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—24no—
84C22×H(3,𝔽3)—24no—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×#272—22no—
87C2×#272—22no—
88C2×#272—24no—
89C2×#272—48no—
90C2×#272C2×D412yes—
91C4×H(3,𝔽3)—22no—
92C4×H(3,𝔽3)C2×D412yes—
93#1084—22no—
94#1084—22no—
95#1084C2×D412yes—
96C2×C3×#41—91no—
97C22×C3×D4—91no—
98C2×#237—34no—
99C2×D3×D4—34no—
100C3×#127—34no—
101C3×#127—91no—
102C3×#127C3⋊D311yescharacteristic
103C3×#124—91no—
104C3×#124—91no—
105C3×#121—94no—
106C3×#120—94no—
107C3×#117—91no—
108#940—34no—
109#939—316no—
110#937—316no—
111#936—34no—
112#875—34no—
113#872—34no—
114C23×C32C22×D318yes—
115C23×C32C22×D318yes—
116C22×C3×D3—34no—
117C22×C3×D3—38no—
118C22×C3×D3—616no—
119C22×C3×D3—1216no—
120C32×D4—28no—
121C32×D4—28no—
122C32×D4—416no—
123C2×C32×C4C22×D318yes—
124C2×C32×C4C22×D314yes—
125C3×#67—616no—
126C3×#67—616no—
127C3×#67—1216no—
128C2×C3×Dic3—34no—
129C2×C3×Dic3—38no—
130C2×C3×Dic3—38no—
131C2×C3×Dic3—616no—
132C3×D12—68no—
133C3×C4×D3—316no—
134C3×Dic6—68no—
135#535—91noSylow
136C2×H(3,𝔽3)—42no—
137C2×H(3,𝔽3)—42no—
138C2×H(3,𝔽3)C22×D411yescharacteristic
139C2×H(3,𝔽3)H(5,𝔽2)12yes—
140#272—22no—
141#272—42no—
142C24×C3—92no—
143C23×D3—38no—
144C3×#41—184no—
145C2×C3×Q8—91no—
146C2×C3×D4—34no—
147C2×C3×D4—34no—
148C2×C3×D4—316no—
149C2×C3×D4—92no—
150C2×C3×D4—92no—
151C2×C3×D4—94no—
152C2×C3×D4—91no—
153C2×C3×D4—184no—
154C2×C3×D4C2×C3⋊D311yescharacteristic
155C2×C3×D4C2×C3⋊D311yescharacteristic
156C2×C3×D4C2×C3⋊D314yes—
157C22×C3×C4—91no—
158C22×C3×C4—92no—
159C2×#67—38no—
160C2×#67—38no—
161C2×#67—316no—
162C22×Dic3—38no—
163#237—616no—
164D3×D4—616no—
165C2×D12—34no—
166C2×C4×D3—34no—
167C2×Dic6—34no—
168C3×C4⋊C4—92no—
169C3×C22⋊C4—94no—
170C3×C22⋊C4—92no—
171C3×C22⋊C4—94no—
172C3×C42—34no—
173C3×C42—91no—
174C3×C42C2×C3⋊D311yescharacteristic
175#217—316no—
176(C2×D3)⋊C4—38no—
177(C2×D3)⋊C4—316no—
178Dic3⋊C4—38no—
179C4×Dic3—34no—
180C22×C32—28no—
181C22×C32—28no—
182C22×C32—216no—
183C22×C32—216no—
184C22×C32C23×D314yes—
185C2×C3×D3—38no—
186C2×C3×D3—68no—
187C2×C3×D3—68no—
188C2×C3×D3—616no—
189C2×C3×D3—1232no—
190C32×C4—28no—
191C32×C4D3×D418yes—
192C3×Dic3—38no—
193C3×Dic3—68no—
194C3×Dic3—68no—
195C2×#41—91no—
196C22×D4—91no—
197#127—91no—
198#127#27211yescharacteristic, complemented
199#124—91no—
200#124—91no—
201#121—94no—
202#120—94no—
203#117—91no—
204H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
205C23×C3—38no—
206C23×C3—38no—
207C23×C3—91no—
208C23×C3—92no—
209C23×C3—184no—
210C23×C3—364no—
211C23×C3C22×C3⋊D312yes—
212C23×C3C22×C3⋊D312yes—
213C22×D3—34no—
214C22×D3—38no—
215C22×D3—616no—
216C22×D3—1216no—
217C3×Q8—182no—
218C3×D4—22no—
219C3×D4—22no—
220C3×D4—44no—
221C3×D4—68no—
222C3×D4—68no—
223C3×D4—1216no—
224C3×D4—184no—
225C3×D4—184no—
226C3×D4—182no—
227C3×D4—364no—
228C2×C3×C4—34no—
229C2×C3×C4—38no—
230C2×C3×C4—91no—
231C2×C3×C4—94no—
232C2×C3×C4—92no—
233C2×C3×C4—92no—
234C2×C3×C4—184no—
235C2×C3×C4C22×C3⋊D312yes—
236C2×C3×C4C22×C3⋊D311yescharacteristic
237#67—616no—
238#67—616no—
239#67—1216no—
240C2×Dic3—34no—
241C2×Dic3—38no—
242C2×Dic3—38no—
243C2×Dic3—616no—
244D12—68no—
245C4×D3—316no—
246Dic6—68no—
247C2×C32—48no—
248C2×C32—48no—
249C2×C32C2×D3×D414yes—
250C2×C32#100418yes—
251C3×D3—68no—
252C3×D3—128no—
253C24—92no—
254#41—184no—
255C2×Q8—91no—
256C2×D4—92no—
257C2×D4—92no—
258C2×D4—94no—
259C2×D4—91no—
260C2×D4—184no—
261C2×D4C2×#27211yescharacteristic
262C2×D4C2×#27211yescharacteristic, complemented
263C2×D4C2×#27214yescomplemented
264C22×C4—91no—
265C22×C4—92no—
266C4⋊C4—92no—
267C22⋊C4—94no—
268C22⋊C4—92no—
269C22⋊C4—94no—
270C42—91no—
271C42C2×#27211yescharacteristic, complemented
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—24no—
275C22×C3—24no—
276C22×C3—34no—
277C22×C3—68no—
278C22×C3—68no—
279C22×C3—616no—
280C22×C3—616no—
281C22×C3—92no—
282C22×C3—182no—
283C22×C3—182no—
284C22×C3—184no—
285C22×C3—368no—
286C22×C3C23×C3⋊D311yescharacteristic
287C2×D3—38no—
288C2×D3—68no—
289C2×D3—68no—
290C2×D3—616no—
291C2×D3—1232no—
292C3×C4—22no—
293C3×C4—38no—
294C3×C4—68no—
295C3×C4—92no—
296C3×C4—182no—
297C3×C4—182no—
298C3×C4D4×C3⋊D312yes—
299Dic3—38no—
300Dic3—68no—
301Dic3—68no—
302C32#571514yescomplemented
303C23—91no—
304C23—92no—
305C23—184no—
306C23—364no—
307C23C22×#27212yes—
308C23C22×#27212yes—
309Q8—182no—
310D4—22no—
311D4—22no—
312D4—44no—
313D4—184no—
314D4—184no—
315D4—182no—
316D4—364no—
317C2×C4—91no—
318C2×C4—94no—
319C2×C4—92no—
320C2×C4—92no—
321C2×C4—184no—
322C2×C4C22×#27212yes—
323C2×C4C22×#27211yescharacteristic
324C2×C3—34no—
325C2×C3—38no—
326C2×C3—42no—
327C2×C3—42no—
328C2×C3—128no—
329C2×C3—128no—
330C2×C3—182no—
331C2×C3—362no—
332C2×C3C2×D4×C3⋊D311yescharacteristic
333C2×C3#6436112yes—
334D3—68no—
335D3—128no—
336C22—22no—
337C22—22no—
338C22—24no—
339C22—24no—
340C22—92no—
341C22—182no—
342C22—182no—
343C22—184no—
344C22—368no—
345C22C23×#27211yescharacteristic
346C4—22no—
347C4—92no—
348C4—182no—
349C4—182no—
350C4D4×#27212yes—
351C3—34no—
352C3#1059618611yescharacteristic
353C2—42no—
354C2—42no—
355C2—182no—
356C2—362no—
357C2C2×D4×#27211yescharacteristic
358C2#1172266412yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#127C2
hypercenterC3×#127C3⋊D3
hypocenterH(3,𝔽3)#535
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
1. C2×#88744C2semidirect
2. C2×D4×#272C2semidirect
3. H(3,𝔽3)×#127C2semidirect
4. #11721850C2semidirect
5. #11721849C2semidirect
6. #11721847C2semidirect
7. #11721846C2semidirect
8. #11721721C2semidirect
9. #11721718C2semidirect
13. C2×D4×H(3,𝔽3)C22semidirect
14. C2×D4×H(3,𝔽3)C22semidirect
17. C2×#6235C22semidirect
18. C22×#1084C22semidirect
23. C2×#6229C22semidirect
24. C42×H(3,𝔽3)C22semidirect
25. #88552C22semidirect
27. #88542C22semidirect
28. #88540C22semidirect
29. C4×#1084C22semidirect
32. C32×#127D3semidirect
58. C4×#272D4semidirect
64. C2×C32×D4C2×D3semidirect
65. C2×C32×D4C2×D3semidirect
75. C32×C42C2×D3semidirect
198. #127#272semidirect
204. H(3,𝔽3)#535semidirect
262. C2×D4C2×#272semidirect
263. C2×D4C2×#272semidirect
271. C42C2×#272semidirect
302. C32#5715semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
C2×D4×#2722
H(3,𝔽3)×#1272
#117218502
#117218492
#117218472
#117218462
#117217212
#117217182
C3×#57153