Finite Group Databasebeta

← #13177976

Subgroups — #13177976

Order 1728.

2132 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 2132 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#11724685C211yesmaximal, characteristic, complemented
2C3×#64366C211yesmaximal, characteristic, complemented
3C32×#948C211yesmaximal, characteristic, complemented
4#11723388C213yesmaximal, complemented
5C2×#88796C213yesmaximal, complemented
6#11723370C213yesmaximal, complemented
7#11723369C213yesmaximal, complemented
8#10596222—31nomaximal
9#10595447—34nomaximal
10C33×#41C2211yescharacteristic
11Q8×C3⋊(C3⋊D3)—21no—
12#89078—23no—
13#89076C2213yescomplemented
14C2×#6308C2213yescomplemented
15C3×#3832—21no—
16C3×D4×C3⋊D3—23no—
17C3×#3828C2213yes—
18C2×C3×C4⋊(C3⋊D3)C2213yes—
19C32×#208C2213yescomplemented
20C2×C32×C3⋊C8C2213yescomplemented
21#88800C2213yescomplemented
22#88797C2211yescharacteristic, complemented
23#88796C2213yescomplemented
24#88796C2216yescomplemented
25#88791C2213yescomplemented
26#64367—31no—
27#64367—34no—
28#64367—64no—
29#64366D311yescharacteristic, complemented
30C3×#1009—34no—
31C32×#131—31no—
32#64118—33no—
33#64117—33no—
34#64113—33no—
35C2×#3744—33no—
36C3×#948D314yescomplemented
37#63831—312no—
38#63830—312no—
39#63829—312no—
40C2×#3716—312no—
41C22×C3×C3⋊D3—23no—
42C33×Q8D411yescharacteristic
43C33×D4D413yes—
44C2×C33×C4C2313yes—
45#6312—23no—
46C2×C3⋊(C3⋊Dic3)—23no—
47C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
48C4×C3⋊(C3⋊D3)—23no—
49#6308—23no—
50#6308C2313yes—
51C3×#638—23no—
52C3×C4⋊(C3⋊D3)—23no—
53C3×C4⋊(C3⋊D3)C2313yes—
54C3×C4×C3⋊D3—23no—
55C3×#634C2311yescharacteristic
56C32×C3⋊C8C2313yes—
57C32×C3⋊C8C2311yescharacteristic
58#5836—91no—
59#5751—94no—
60C32×#41—24no—
61C32×#41C2×D314yes—
62C32×#41C2×D311yescharacteristic
63#3832—21no—
64Q8×C3⋊D3—68no—
65Q8×C3⋊D3—64no—
66Q8×C3⋊D3—61no—
67#3830—624no—
68#3830—612no—
69#3830—63no—
70D4×C3⋊D3—23no—
71#3828—33no—
72#3828—312no—
73#3828—612no—
74#3828C2×D313yes—
75C2×C4⋊(C3⋊D3)C2×D313yes—
76C2×#634—33no—
77C2×#634—312no—
78C2×#634—612no—
79C3×#239—64no—
80C3×D3×D4—612no—
81C3×#235—312no—
82C2×C3×D12—312no—
83C32×M16—33no—
84C2×C32×C8—33no—
85#3746—33no—
86#3745—33no—
87#3744—33no—
88#3744—36no—
89#3744—31no—
90C3×#208C2×D3112yescomplemented
91C2×C3×C3⋊C8C2×D3112yescomplemented
92#3719—312no—
93#3717—34no—
94#3716—312no—
95#3716—324no—
96#3714—312no—
97C22×C33C2×D413yes—
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3×C3⋊D3—23no—
100C2×C3×C3⋊D3—83no—
101C33×C4C2×D413yes—
102C33×C4C2411yescharacteristic
103C3⋊(C3⋊Dic3)—23no—
104C3×C3⋊Dic3—21no—
105C3×H(5,𝔽2)—91no—
106#1010—94no—
107#1010—91no—
108#1010—184no—
109#1009—34no—
110C3×#131—34no—
111#951—93no—
112#950—93no—
113#948C3⋊D311yescharacteristic, complemented
114C2×#214—93no—
115#932—93no—
116#909—912no—
117#908—912no—
118#904—912no—
119C2×#204—912no—
120C22×C3⋊D3—23no—
121C22×C3×D3—612no—
122C32×Q8—24no—
123C32×Q8D1214yes—
124C32×Q8#6711yescharacteristic
125C32×D4—212no—
126C32×D4D12112yes—
127C32×D4#6713yes—
128C2×C32×C4—212no—
129C2×C32×C4C22×D3112yes—
130C2×C32×C4C22×D313yes—
131#638—23no—
132#638—624no—
133#638—612no—
134#638—63no—
135C2×C3⋊Dic3—624no—
136C2×C3⋊Dic3—612no—
137C2×C3⋊Dic3—63no—
138C4⋊(C3⋊D3)—23no—
139C4⋊(C3⋊D3)—31no—
140C4⋊(C3⋊D3)—34no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)C22×D313yes—
143C4×C3⋊D3—23no—
144C4×C3⋊D3—624no—
145C4×C3⋊D3—612no—
146C4×C3⋊D3—63no—
147#634—33no—
148#634—312no—
149#634—612no—
150#634—624no—
151#634—612no—
152#634—63no—
153#634C22×D311yescharacteristic
154C3×#67—612no—
155C3×D12—312no—
156C3×D12—612no—
157C3×C4×D3—612no—
158C3×Dic6—34no—
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)—41no—
167C3×C3⋊D3—43no—
168C3×#41—24no—
169C3×#41—93no—
170C3×#41—181no—
171C3×#41D3214yes—
172C3×#41C2×C3⋊D311yescharacteristic
173C2×C3×D4—93no—
174C2×C3×D4—183no—
175#239—64no—
176D3×Q8—188no—
177D3×Q8—184no—
178D3×Q8—181no—
179#237—1824no—
180#237—1812no—
181#237—183no—
182D3×D4—612no—
183#235—312no—
184#235—912no—
185#235—93no—
186#235—1812no—
187C2×D12—312no—
188C2×Dic6—912no—
189C2×Dic6—93no—
190C2×Dic6—1812no—
191C3×M16—312no—
192C2×C3×C8—312no—
193#216—93no—
194#215—91no—
195#214—96no—
196#214—93no—
197#213—93no—
198#208C2×C3⋊D313yescomplemented
199C2×C3⋊C8C2×C3⋊D313yescomplemented
200Dic12—912no—
201D24—912no—
202#204—94no—
203#204—912no—
204#204—924no—
205C22×C32—212no—
206C22×C32C2×D12112yes—
207C22×C32C2×#6713yes—
208C2×C3⋊D3—23no—
209C2×C3⋊D3—68no—
210C2×C3⋊D3—64no—
211C2×C3⋊D3—61no—
212C2×C3⋊D3—83no—
213C2×C3×D3—612no—
214C2×C3×D3—2412no—
215C32×C4—24no—
216C32×C4—212no—
217C32×C4C2×D12112yes—
218C32×C4C2×#6713yes—
219C32×C4C23×D314yes—
220C32×C4C23×D311yescharacteristic
221C3⋊Dic3—21no—
222C3⋊Dic3—624no—
223C3⋊Dic3—612no—
224C3⋊Dic3—63no—
225C3×Dic3—64no—
2262−1+4—271no—
227H(5,𝔽2)—91no—
228#137—273no—
229AGL(1,ℤ/8)—273no—
230#135—273no—
231C2×SD16—273no—
232#131—31no—
233C33#57711yescharacteristic, complemented, Sylow
234C23×C3—183no—
235C22×D3—612no—
236C3×Q8—24no—
237C3×Q8—91no—
238C3×Q8#62614yes—
239C3×Q8C4⋊(C3⋊D3)11yescharacteristic
240C3×D4—212no—
241C3×D4—93no—
242C3×D4—183no—
243C3×D4—183no—
244C3×D4#626112yes—
245C3×D4C4⋊(C3⋊D3)13yes—
246C2×C3×C4—212no—
247C2×C3×C4—183no—
248C2×C3×C4C2×D32112yes—
249C2×C3×C4C22×C3⋊D313yes—
250#67—612no—
251#67—1824no—
252#67—1812no—
253#67—183no—
254C2×Dic3—1824no—
255C2×Dic3—1812no—
256C2×Dic3—183no—
257D12—312no—
258D12—612no—
259D12—94no—
260D12—91no—
261D12—184no—
262C4×D3—612no—
263C4×D3—1824no—
264C4×D3—1812no—
265C4×D3—183no—
266Dic6—34no—
267Dic6—912no—
268Dic6—93no—
269Dic6—1812no—
270Dic6—1824no—
271Dic6—1812no—
272Dic6—183no—
273C3×C8—34no—
274C3×C8—312no—
275C3⋊C8C22×C3⋊D311yescharacteristic
276C3⋊C8C22×C3⋊D313yes—
277C2×C32—23no—
278C2×C32—24no—
279C2×C32—212no—
280C2×C32—412no—
281C2×C32C22×D1214yes—
282C2×C32C22×#6711yescharacteristic
283C3⋊D3—43no—
284C3⋊D3—128no—
285C3⋊D3—124no—
286C3⋊D3—121no—
287C3×D3—1212no—
288#41—93no—
289#41—181no—
290#41—273no—
291#41—543no—
292#41D3×C3⋊D311yescharacteristic
293C2×Q8—273no—
294C2×Q8—541no—
295C2×D4—93no—
296C2×D4—183no—
297Q16—273no—
298SD16—273no—
299SD16—276no—
300SD16—271no—
301D8—273no—
302M16—33no—
303C2×C8—33no—
304C22×C3—212no—
305C22×C3—183no—
306C22×C3—723no—
307C22×C3C2×#626112yes—
308C22×C3C2×C4⋊(C3⋊D3)13yes—
309C2×D3—612no—
310C2×D3—188no—
311C2×D3—184no—
312C2×D3—181no—
313C2×D3—2412no—
314C3×C4—24no—
315C3×C4—212no—
316C3×C4—181no—
317C3×C4C2×#626112yes—
318C3×C4C2×C4⋊(C3⋊D3)13yes—
319C3×C4C22×D3214yes—
320C3×C4C23×C3⋊D311yescharacteristic
321Dic3—64no—
322Dic3—1824no—
323Dic3—1812no—
324Dic3—183no—
325C32—24no—
326C32#575114yescomplemented
327C32#583611yescharacteristic, complemented
328C23—183no—
329Q8—91no—
330Q8—273no—
331Q8—543no—
332Q8#629211yescharacteristic
333D4—93no—
334D4—183no—
335D4—183no—
336D4—271no—
337D4—543no—
338D4#629213yes—
339C2×C4—183no—
340C2×C4—543no—
341C2×C4—543no—
342C2×C4C2×D3×C3⋊D313yes—
343C8—31no—
344C8—33no—
345C2×C3—212no—
346C2×C3—23no—
347C2×C3—24no—
348C2×C3—412no—
349C2×C3—363no—
350C2×C3C22×#62614yes—
351C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
352D3—1212no—
353D3—368no—
354D3—364no—
355D3—361no—
356C22—183no—
357C22—541no—
358C22—723no—
359C22C2×#629213yes—
360C4—181no—
361C4—543no—
362C4C2×#629213yes—
363C4C22×D3×C3⋊D311yescharacteristic
364C3—24no—
365C3#1059544714yescomplemented
366C3#1059622211yescharacteristic, complemented
367C2—23no—
368C2—363no—
369C2—1081no—
370C2C22×#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. #11724685C2semidirect
2. C3×#64366C2semidirect
3. C32×#948C2semidirect
4. #11723388C2semidirect
5. C2×#88796C2semidirect
6. #11723370C2semidirect
7. #11723369C2semidirect
13. #89076C22semidirect
14. C2×#6308C22semidirect
19. C32×#208C22semidirect
20. C2×C32×C3⋊C8C22semidirect
21. #88800C22semidirect
22. #88797C22semidirect
23. #88796C22semidirect
24. #88796C22semidirect
25. #88791C22semidirect
29. #64366D3semidirect
36. C3×#948D3semidirect
90. C3×#208C2×D3semidirect
91. C2×C3×C3⋊C8C2×D3semidirect
113. #948C3⋊D3semidirect
198. #208C2×C3⋊D3semidirect
199. C2×C3⋊C8C2×C3⋊D3semidirect
233. C33#577semidirect
326. C32#5751semidirect
327. C32#5836semidirect
365. C3#10595447semidirect
366. C3#10596222semidirect

Maximal subgroups

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

subgroupindex
#117246852
C3×#643662
C32×#9482
#117233882
C2×#887962
#117233702
#117233692
#105962223
#105954473