Finite Group Databasebeta

← #13177975

Subgroups — #13177975

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
2C32×#1010C211yesmaximal, characteristic, complemented
3C3×#64157C211yesmaximal, characteristic, complemented
4#11723381C213yesmaximal, complemented
5C2×#88795C213yesmaximal, complemented
6#11723377C213yesmaximal, complemented
7#11723367C213yesmaximal, complemented
8#10596308—31nomaximal
9#10595448—34nomaximal
10C33×#41C2211yescharacteristic
11#89080—21no—
12D4×C3⋊(C3⋊D3)—23no—
13#89076C2213yes—
14C2×C4⋊(C3⋊(C3⋊D3))C2213yes—
15C32×D3×Q8—21no—
16C32×#237—23no—
17C32×#235C2213yescomplemented
18C2×C32×Dic6C2213yescomplemented
19C3×#3748C2213yescomplemented
20C2×C3×C3⋊(C3⋊C8)C2213yescomplemented
21#88799C2213yescomplemented
22#88795C2216yescomplemented
23#88795C2213yescomplemented
24#88794C2211yescharacteristic, complemented
25#88790C2213yescomplemented
26C32×2−1+4—31no—
27#64366—31no—
28#64366—34no—
29#64366—64no—
30C3×#1010D314yescomplemented
31#64159—33no—
32#64158—33no—
33#64157D311yescharacteristic, complemented
34#64151—33no—
35C2×#3755—33no—
36C3×#948—34no—
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
50C32×#67—23no—
51C2×C32×Dic3—23no—
52C32×D12C2311yescharacteristic
53C32×C4×D3—23no—
54C32×Dic6—23no—
55C32×Dic6C2313yes—
56C3×C3⋊(C3⋊C8)C2311yescharacteristic
57C3×C3⋊(C3⋊C8)C2313yes—
58#5837—94no—
59#5751—91no—
60C32×#41—24no—
61C32×#41—33no—
62C32×#41—63no—
63C32×#41C2×D311yescharacteristic
64C32×#41C2×D314yes—
65C2×C32×Q8—33no—
66C2×C32×Q8—61no—
67#3832—68no—
68#3832—61no—
69#3832—64no—
70D4×C3⋊D3—624no—
71D4×C3⋊D3—63no—
72D4×C3⋊D3—612no—
73#3828—33no—
74#3828—312no—
75#3828—612no—
76C2×C4⋊(C3⋊D3)—33no—
77C2×C4⋊(C3⋊D3)—312no—
78C2×C4⋊(C3⋊D3)—612no—
79C3×D3×Q8—24no—
80C3×#237—212no—
81C3×#235C2×D3112yescomplemented
82C2×C3×Dic6C2×D3112yescomplemented
83#3756—33no—
84#3755—33no—
85#3755—36no—
86#3754—31no—
87#3753—33no—
88#3748C2×D313yescomplemented
89C2×C3⋊(C3⋊C8)C2×D313yescomplemented
90C3×#208—312no—
91C2×C3×C3⋊C8—312no—
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×C32×D3—21no—
101C33×C4C2×D413yes—
102C33×C4C2411yescharacteristic
103C3⋊(C3⋊Dic3)—21no—
104C32×Dic3—23no—
105C3×2−1+4—34no—
106#1010C3⋊D311yescharacteristic, complemented
107#1009—94no—
108#1009—91no—
109#1009—184no—
110C3×#131—91no—
111#950—912no—
112#949—912no—
113#948—34no—
114#942—912no—
115C2×#215—912no—
116#909—93no—
117#908—93no—
118#904—93no—
119C2×#204—93no—
120C22×C3⋊D3—624no—
121C22×C3⋊D3—63no—
122C22×C3⋊D3—612no—
123C32×Q8—24no—
124C32×Q8—33no—
125C32×Q8—63no—
126C32×Q8D1211yescharacteristic
127C32×Q8#6714yes—
128C32×D4—212no—
129C32×D4—31no—
130C32×D4—63no—
131C32×D4D1213yes—
132C32×D4#67112yes—
133C2×C32×C4—212no—
134C2×C32×C4—63no—
135C2×C32×C4—63no—
136C2×C32×C4C22×D3112yes—
137C2×C32×C4C22×D313yes—
138#638—624no—
139#638—63no—
140#638—612no—
141C4⋊(C3⋊D3)—33no—
142C4⋊(C3⋊D3)—312no—
143C4⋊(C3⋊D3)—612no—
144C4⋊(C3⋊D3)—624no—
145C4⋊(C3⋊D3)—63no—
146C4⋊(C3⋊D3)—612no—
147C4×C3⋊D3—624no—
148C4×C3⋊D3—63no—
149C4×C3⋊D3—612no—
150#634—31no—
151#634—34no—
152#634—64no—
153C3×#67—212no—
154C2×C3×Dic3—212no—
155C3×D12C22×D314yes—
156C3×C4×D3—212no—
157C3×Dic6—212no—
158C3×Dic6C22×D3112yes—
159C3⋊(C3⋊C8)C22×D311yescharacteristic
160C3⋊(C3⋊C8)C22×D313yes—
161C3×C3⋊C8—312no—
162C3×C3⋊C8—34no—
163#577—271noSylow
164C2×C33—23no—
165C2×C33C22×D411yescharacteristic
166C3⋊(C3⋊D3)—43no—
167C32×D3—41no—
168C3×#41—24no—
169C3×#41—312no—
170C3×#41—612no—
171C3×#41D3214yes—
172C3×#41C2×C3⋊D311yescharacteristic
173C2×C3×Q8—312no—
174C2×C3×Q8—64no—
175#239—188no—
176#239—184no—
177#239—181no—
178D3×Q8—21no—
179#237—23no—
180D3×D4—1824no—
181D3×D4—1812no—
182D3×D4—183no—
183#235—912no—
184#235—93no—
185#235—1812no—
186#235C2×C3⋊D313yescomplemented
187C2×D12—912no—
188C2×D12—93no—
189C2×D12—1812no—
190C2×Dic6C2×C3⋊D313yescomplemented
191C3×M16—93no—
192C2×C3×C8—93no—
193#216—912no—
194#215—912no—
195#215—924no—
196#214—94no—
197#213—912no—
198#208—312no—
199C2×C3⋊C8—312no—
200Dic12—93no—
201D24—93no—
202#204—91no—
203#204—93no—
204#204—96no—
205C22×C32—212no—
206C22×C32—61no—
207C22×C32C2×D1213yes—
208C22×C32C2×#67112yes—
209C2×C3⋊D3—624no—
210C2×C3⋊D3—63no—
211C2×C3⋊D3—612no—
212C2×C3⋊D3—243no—
213C2×C3⋊D3—2412no—
214C2×C3⋊D3—2424no—
215C2×C3×D3—24no—
216C32×C4—24no—
217C32×C4—212no—
218C32×C4—63no—
219C32×C4C2×D1213yes—
220C32×C4C2×#67112yes—
221C32×C4C23×D314yes—
222C32×C4C23×D311yescharacteristic
223C3⋊Dic3—68no—
224C3⋊Dic3—61no—
225C3⋊Dic3—64no—
226C3×Dic3—212no—
2272−1+4—31no—
228H(5,𝔽2)—271no—
229#137—273no—
230AGL(1,ℤ/8)—273no—
231#135—273no—
232C2×SD16—273no—
233#131—91no—
234C33#57711yescharacteristic, complemented, Sylow
235C22×D3—1824no—
236C22×D3—1812no—
237C22×D3—183no—
238C3×Q8—24no—
239C3×Q8—312no—
240C3×Q8—612no—
241C3×Q8#62614yes—
242C3×Q8#63811yescharacteristic
243C3×D4—212no—
244C3×D4—34no—
245C3×D4—612no—
246C3×D4#626112yes—
247C3×D4#63813yes—
248C2×C3×C4—212no—
249C2×C3×C4—612no—
250C2×C3×C4—612no—
251C2×C3×C4C2×D32112yes—
252C2×C3×C4C22×C3⋊D313yes—
253#67—23no—
254#67—1824no—
255#67—1812no—
256#67—183no—
257C2×Dic3—23no—
258D12—912no—
259D12—93no—
260D12—1812no—
261D12—1824no—
262D12—1812no—
263D12—183no—
264D12C22×C3⋊D311yescharacteristic
265C4×D3—23no—
266C4×D3—1824no—
267C4×D3—1812no—
268C4×D3—183no—
269Dic6—23no—
270Dic6—94no—
271Dic6—91no—
272Dic6—184no—
273Dic6C22×C3⋊D313yes—
274C3×C8—91no—
275C3×C8—93no—
276C3⋊C8—34no—
277C3⋊C8—312no—
278C2×C32—23no—
279C2×C32—24no—
280C2×C32—212no—
281C2×C32—412no—
282C2×C32—121no—
283C2×C32C22×D1211yescharacteristic
284C2×C32C22×#6714yes—
285C3⋊D3—1224no—
286C3⋊D3—123no—
287C3⋊D3—1212no—
288C3×D3—44no—
289#41—33no—
290#41—63no—
291#41—273no—
292#41—541no—
293#41D3×C3⋊D311yescharacteristic
294C2×Q8—33no—
295C2×Q8—61no—
296C2×D4—273no—
297C2×D4—543no—
298Q16—273no—
299SD16—271no—
300SD16—273no—
301SD16—276no—
302D8—273no—
303M16—93no—
304C2×C8—93no—
305C22×C3—212no—
306C22×C3—64no—
307C22×C3C2×#626112yes—
308C22×C3C2×#63813yes—
309C2×D3—21no—
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—612no—
319C3×C4C2×#626112yes—
320C3×C4C2×#63813yes—
321C3×C4C22×D3214yes—
322C3×C4C23×C3⋊D311yescharacteristic
323Dic3—23no—
324Dic3—188no—
325Dic3—184no—
326Dic3—181no—
327C32—24no—
328C32#575111yescharacteristic, complemented
329C32#583714yescomplemented
330C23—543no—
331Q8—33no—
332Q8—63no—
333Q8—271no—
334Q8#629111yescharacteristic
335D4—31no—
336D4—63no—
337D4—273no—
338D4—543no—
339D4—543no—
340D4#629113yes—
341C2×C4—63no—
342C2×C4—63no—
343C2×C4—543no—
344C2×C4C2×D3×C3⋊D313yes—
345C8—91no—
346C8—93no—
347C2×C3—212no—
348C2×C3—23no—
349C2×C3—24no—
350C2×C3—412no—
351C2×C3—124no—
352C2×C3C22×#62614yes—
353C2×C3C22×#63811yescharacteristic
354D3—41no—
355D3—3624no—
356D3—3612no—
357D3—363no—
358C22—61no—
359C22—543no—
360C22—2163no—
361C22C2×#629113yes—
362C4—63no—
363C4—541no—
364C4C2×#629113yes—
365C4C22×D3×C3⋊D311yescharacteristic
366C3—24no—
367C3#1059544814yescomplemented
368C3#1059630811yescharacteristic, complemented
369C2—23no—
370C2—121no—
371C2—1083no—
372C2C22×#629111yescharacteristic
subgroupG/H
centerC2C22×#6291
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. C32×#1010C2semidirect
3. C3×#64157C2semidirect
4. #11723381C2semidirect
5. C2×#88795C2semidirect
6. #11723377C2semidirect
7. #11723367C2semidirect
17. C32×#235C22semidirect
18. C2×C32×Dic6C22semidirect
19. C3×#3748C22semidirect
20. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. #88799C22semidirect
22. #88795C22semidirect
23. #88795C22semidirect
24. #88794C22semidirect
25. #88790C22semidirect
30. C3×#1010D3semidirect
33. #64157D3semidirect
81. C3×#235C2×D3semidirect
82. C2×C3×Dic6C2×D3semidirect
88. #3748C2×D3semidirect
89. C2×C3⋊(C3⋊C8)C2×D3semidirect
106. #1010C3⋊D3semidirect
186. #235C2×C3⋊D3semidirect
190. C2×Dic6C2×C3⋊D3semidirect
234. C33#577semidirect
328. C32#5751semidirect
329. C32#5837semidirect
367. C3#10595448semidirect
368. C3#10596308semidirect

Maximal subgroups

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

subgroupindex
#117246842
C32×#10102
C3×#641572
#117233812
C2×#887952
#117233772
#117233672
#105963083
#105954483