Finite Group Databasebeta

← #13177977

Subgroups — #13177977

Order 1728.

2212 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 2212 conjugacy classes of subgroups — too many to draw. Only its 332 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#11724684C211yesmaximal, characteristic, complemented
2C3×#64367C211yesmaximal, characteristic, complemented
3C32×#948C211yesmaximal, characteristic, complemented
4#11723387C213yesmaximal, complemented
5C2×#88797C213yesmaximal, complemented
6#11723371C213yesmaximal, complemented
7#11723369C213yesmaximal, complemented
8#10596222—31nomaximal
9#10595448—34nomaximal
10C33×#41C2211yescharacteristic
11#89080—21no—
12D4×C3⋊(C3⋊D3)—23no—
13#89076C2213yes—
14C2×C4⋊(C3⋊(C3⋊D3))C2213yes—
15C3×Q8×C3⋊D3—21no—
16C3×#3830—23no—
17C3×#3828C2213yescomplemented
18C2×C3×#634C2213yescomplemented
19C32×#208C2213yescomplemented
20C2×C32×C3⋊C8C2213yescomplemented
21#88800C2213yescomplemented
22#88797C2213yescomplemented
23#88797C2216yescomplemented
24#88796C2211yescharacteristic, complemented
25#88791C2213yescomplemented
26#64367D311yescharacteristic, complemented
27#64366—31no—
28#64366—34no—
29#64366—64no—
30C3×#1010—34no—
31C32×#131—31no—
32#64118—33no—
33#64117—33no—
34#64113—33no—
35C2×#3744—33no—
36C3×#948D314yescomplemented
37#63833—312no—
38C2×#3717—312no—
39#63829—312no—
40#63825—312no—
41C22×C3⋊(C3⋊D3)—23no—
42C33×Q8D411yescharacteristic
43C33×D4D413yes—
44C2×C33×C4C2313yes—
45#6312—23no—
46C4⋊(C3⋊(C3⋊D3))—23no—
47C4⋊(C3⋊(C3⋊D3))C2313yes—
48C4×C3⋊(C3⋊D3)—23no—
49#6308C2311yescharacteristic
50C3×#638—23no—
51C2×C3×C3⋊Dic3—23no—
52C3×C4⋊(C3⋊D3)C2311yescharacteristic
53C3×C4×C3⋊D3—23no—
54C3×#634—23no—
55C3×#634C2313yes—
56C32×C3⋊C8C2313yes—
57C32×C3⋊C8C2311yescharacteristic
58#5837—91no—
59#5751—94no—
60C32×#41—24no—
61C32×#41C2×D314yes—
62C32×#41C2×D311yescharacteristic
63#3832—68no—
64#3832—64no—
65#3832—61no—
66Q8×C3⋊D3—21no—
67#3830—23no—
68D4×C3⋊D3—624no—
69D4×C3⋊D3—612no—
70D4×C3⋊D3—63no—
71#3828—33no—
72#3828—312no—
73#3828—612no—
74#3828C2×D313yescomplemented
75C2×C4⋊(C3⋊D3)—33no—
76C2×C4⋊(C3⋊D3)—312no—
77C2×C4⋊(C3⋊D3)—612no—
78C2×#634C2×D313yescomplemented
79C3×D3×Q8—64no—
80C3×#237—612no—
81C3×#235—312no—
82C2×C3×Dic6—312no—
83C32×M16—33no—
84C2×C32×C8—33no—
85#3746—33no—
86#3745—33no—
87#3744—31no—
88#3744—33no—
89#3744—36no—
90C3×#208C2×D3112yescomplemented
91C2×C3×C3⋊C8C2×D3112yescomplemented
92#3719—312no—
93#3717—324no—
94#3717—312no—
95#3716—34no—
96#3714—312no—
97C22×C33C2×D413yes—
98C2×C3⋊(C3⋊D3)—23no—
99C2×C3⋊(C3⋊D3)—83no—
100C2×C3×C3⋊D3—21no—
101C33×C4C2×D413yes—
102C33×C4C2411yescharacteristic
103C3⋊(C3⋊Dic3)—21no—
104C3×C3⋊Dic3—23no—
105C3×2−1+4—91no—
106#1010—34no—
107#1009—94no—
108#1009—91no—
109#1009—184no—
110C3×#131—34no—
111#950—93no—
112#949—93no—
113#948C3⋊D311yescharacteristic, complemented
114#942—93no—
115C2×#215—93no—
116#909—912no—
117#908—912no—
118#904—912no—
119C2×#204—912no—
120C22×C3⋊D3—624no—
121C22×C3⋊D3—612no—
122C22×C3⋊D3—63no—
123C32×Q8—24no—
124C32×Q8D1214yes—
125C32×Q8#6711yescharacteristic
126C32×D4—212no—
127C32×D4D12112yes—
128C32×D4#6713yes—
129C2×C32×C4—212no—
130C2×C32×C4C22×D3112yes—
131C2×C32×C4C22×D313yes—
132#638—23no—
133#638—624no—
134#638—612no—
135#638—63no—
136C2×C3⋊Dic3—23no—
137C4⋊(C3⋊D3)—33no—
138C4⋊(C3⋊D3)—312no—
139C4⋊(C3⋊D3)—612no—
140C4⋊(C3⋊D3)—624no—
141C4⋊(C3⋊D3)—612no—
142C4⋊(C3⋊D3)—63no—
143C4⋊(C3⋊D3)C22×D311yescharacteristic
144C4×C3⋊D3—23no—
145C4×C3⋊D3—624no—
146C4×C3⋊D3—612no—
147C4×C3⋊D3—63no—
148#634—23no—
149#634—31no—
150#634—34no—
151#634—64no—
152#634C22×D313yes—
153C3×#67—612no—
154C2×C3×Dic3—612no—
155C3×D12—34no—
156C3×C4×D3—612no—
157C3×Dic6—312no—
158C3×Dic6—612no—
159C32×C8—31no—
160C32×C8—33no—
161C3×C3⋊C8C22×D3112yes—
162C3×C3⋊C8C22×D314yes—
163#577—271noSylow
164C2×C33—23no—
165C2×C33C22×D411yescharacteristic
166C3⋊(C3⋊D3)—43no—
167C3×C3⋊D3—41no—
168C3×#41—24no—
169C3×#41—93no—
170C3×#41—183no—
171C3×#41D3214yes—
172C3×#41C2×C3⋊D311yescharacteristic
173C2×C3×Q8—93no—
174C2×C3×Q8—181no—
175#239—188no—
176#239—184no—
177#239—181no—
178D3×Q8—64no—
179#237—612no—
180D3×D4—1824no—
181D3×D4—1812no—
182D3×D4—183no—
183#235—312no—
184#235—912no—
185#235—93no—
186#235—1812no—
187C2×D12—912no—
188C2×D12—93no—
189C2×D12—1812no—
190C2×Dic6—312no—
191C3×M16—312no—
192C2×C3×C8—312no—
193#216—93no—
194#215—93no—
195#215—96no—
196#214—91no—
197#213—93no—
198#208C2×C3⋊D313yescomplemented
199C2×C3⋊C8C2×C3⋊D313yescomplemented
200Dic12—912no—
201D24—912no—
202#204—912no—
203#204—924no—
204#204—94no—
205C22×C32—212no—
206C22×C32C2×D12112yes—
207C22×C32C2×#6713yes—
208C2×C3⋊D3—21no—
209C2×C3⋊D3—624no—
210C2×C3⋊D3—612no—
211C2×C3⋊D3—63no—
212C2×C3⋊D3—243no—
213C2×C3⋊D3—2412no—
214C2×C3⋊D3—2424no—
215C2×C3×D3—64no—
216C32×C4—24no—
217C32×C4—212no—
218C32×C4C2×D12112yes—
219C32×C4C2×#6713yes—
220C32×C4C23×D314yes—
221C32×C4C23×D311yescharacteristic
222C3⋊Dic3—23no—
223C3⋊Dic3—68no—
224C3⋊Dic3—64no—
225C3⋊Dic3—61no—
226C3×Dic3—612no—
2272−1+4—91no—
228H(5,𝔽2)—271no—
229#137—273no—
230AGL(1,ℤ/8)—273no—
231#135—273no—
232C2×SD16—273no—
233#131—31no—
234C33#57711yescharacteristic, complemented, Sylow
235C22×D3—1824no—
236C22×D3—1812no—
237C22×D3—183no—
238C3×Q8—24no—
239C3×Q8—93no—
240C3×Q8—183no—
241C3×Q8#62614yes—
242C3×Q8C4⋊(C3⋊D3)11yescharacteristic
243C3×D4—212no—
244C3×D4—91no—
245C3×D4—183no—
246C3×D4#626112yes—
247C3×D4C4⋊(C3⋊D3)13yes—
248C2×C3×C4—212no—
249C2×C3×C4—183no—
250C2×C3×C4—183no—
251C2×C3×C4C2×D32112yes—
252C2×C3×C4C22×C3⋊D313yes—
253#67—612no—
254#67—1824no—
255#67—1812no—
256#67—183no—
257C2×Dic3—612no—
258D12—34no—
259D12—912no—
260D12—93no—
261D12—1812no—
262D12—1824no—
263D12—1812no—
264D12—183no—
265C4×D3—612no—
266C4×D3—1824no—
267C4×D3—1812no—
268C4×D3—183no—
269Dic6—312no—
270Dic6—612no—
271Dic6—94no—
272Dic6—91no—
273Dic6—184no—
274C3×C8—34no—
275C3×C8—312no—
276C3⋊C8C22×C3⋊D311yescharacteristic
277C3⋊C8C22×C3⋊D313yes—
278C2×C32—23no—
279C2×C32—24no—
280C2×C32—212no—
281C2×C32—412no—
282C2×C32C22×D1214yes—
283C2×C32C22×#6711yescharacteristic
284C3⋊D3—41no—
285C3⋊D3—1224no—
286C3⋊D3—1212no—
287C3⋊D3—123no—
288C3×D3—124no—
289#41—93no—
290#41—183no—
291#41—273no—
292#41—541no—
293#41D3×C3⋊D311yescharacteristic
294C2×Q8—93no—
295C2×Q8—181no—
296C2×D4—273no—
297C2×D4—543no—
298Q16—273no—
299SD16—271no—
300SD16—273no—
301SD16—276no—
302D8—273no—
303M16—33no—
304C2×C8—33no—
305C22×C3—212no—
306C22×C3—181no—
307C22×C3C2×#626112yes—
308C22×C3C2×C4⋊(C3⋊D3)13yes—
309C2×D3—64no—
310C2×D3—1824no—
311C2×D3—1812no—
312C2×D3—183no—
313C2×D3—7212no—
314C2×D3—723no—
315C2×D3—7224no—
316C3×C4—24no—
317C3×C4—212no—
318C3×C4—183no—
319C3×C4C2×#626112yes—
320C3×C4C2×C4⋊(C3⋊D3)13yes—
321C3×C4C22×D3214yes—
322C3×C4C23×C3⋊D311yescharacteristic
323Dic3—612no—
324Dic3—188no—
325Dic3—184no—
326Dic3—181no—
327C32—24no—
328C32#575114yescomplemented
329C32#583711yescharacteristic, complemented
330C23—543no—
331Q8—93no—
332Q8—183no—
333Q8—271no—
334Q8#629211yescharacteristic
335D4—91no—
336D4—183no—
337D4—273no—
338D4—543no—
339D4—543no—
340D4#629213yes—
341C2×C4—183no—
342C2×C4—183no—
343C2×C4—543no—
344C2×C4C2×D3×C3⋊D313yes—
345C8—31no—
346C8—33no—
347C2×C3—212no—
348C2×C3—23no—
349C2×C3—24no—
350C2×C3—412no—
351C2×C3—361no—
352C2×C3C22×#62614yes—
353C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
354D3—124no—
355D3—3624no—
356D3—3612no—
357D3—363no—
358C22—181no—
359C22—543no—
360C22—2163no—
361C22C2×#629213yes—
362C4—183no—
363C4—541no—
364C4C2×#629213yes—
365C4C22×D3×C3⋊D311yescharacteristic
366C3—24no—
367C3#1059544814yescomplemented
368C3#1059622211yescharacteristic, complemented
369C2—23no—
370C2—361no—
371C2—1083no—
372C2C22×#629211yescharacteristic
subgroupG/H
centerC2C22×#6292
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#577
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11724684C2semidirect
2. C3×#64367C2semidirect
3. C32×#948C2semidirect
4. #11723387C2semidirect
5. C2×#88797C2semidirect
6. #11723371C2semidirect
7. #11723369C2semidirect
17. C3×#3828C22semidirect
18. C2×C3×#634C22semidirect
19. C32×#208C22semidirect
20. C2×C32×C3⋊C8C22semidirect
21. #88800C22semidirect
22. #88797C22semidirect
23. #88797C22semidirect
24. #88796C22semidirect
25. #88791C22semidirect
26. #64367D3semidirect
36. C3×#948D3semidirect
74. #3828C2×D3semidirect
78. C2×#634C2×D3semidirect
90. C3×#208C2×D3semidirect
91. C2×C3×C3⋊C8C2×D3semidirect
113. #948C3⋊D3semidirect
198. #208C2×C3⋊D3semidirect
199. C2×C3⋊C8C2×C3⋊D3semidirect
234. C33#577semidirect
328. C32#5751semidirect
329. C32#5837semidirect
367. C3#10595448semidirect
368. C3#10596222semidirect

Maximal subgroups

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

subgroupindex
#117246842
C3×#643672
C32×#9482
#117233872
C2×#887972
#117233712
#117233692
#105962223
#105954483