Finite Group Databasebeta

← #13177978

Subgroups — #13177978

Order 1728.

2272 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 2272 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×#64366C211yesmaximal, characteristic, complemented
3C32×#948C211yesmaximal, characteristic, complemented
4#11723371C213yesmaximal, complemented
5#11723370C213yesmaximal, complemented
6#11723369C213yesmaximal, complemented
7C2×#88791C213yesmaximal, complemented
8#10596223—31nomaximal
9#10595449—34nomaximal
10C33×#41C2211yescharacteristic
11#89080—21no—
12D4×C3⋊(C3⋊D3)—23no—
13#89076C2213yescomplemented
14C2×C4⋊(C3⋊(C3⋊D3))C2213yescomplemented
15C3×#3832—21no—
16C3×D4×C3⋊D3—23no—
17C3×#3828C2213yescomplemented
18C2×C3×C4⋊(C3⋊D3)C2213yescomplemented
19C32×#208C2213yescomplemented
20C2×C32×C3⋊C8C2213yescomplemented
21#88800C2211yescharacteristic, complemented
22#88797C2213yescomplemented
23#88796C2213yescomplemented
24#88791C2213yescomplemented
25#88791C2216yescomplemented
26#64366—31no—
27#64366—34no—
28#64366—64no—
29#64366D311yescharacteristic, complemented
30C3×#1009—34no—
31C32×#131—31no—
32#64117—33no—
33#64117—33no—
34#64113—33no—
35C2×#3745—33no—
36C3×#948D314yescomplemented
37#63830—312no—
38#63829—312no—
39#63825—312no—
40C2×#3714—312no—
41C22×C3⋊(C3⋊D3)—23no—
42C22×C3×C3⋊D3—23no—
43C33×Q8D411yescharacteristic
44C33×D4D413yes—
45C2×C33×C4C2313yes—
46#6312—23no—
47C4⋊(C3⋊(C3⋊D3))—23no—
48C4⋊(C3⋊(C3⋊D3))C2313yes—
49C4×C3⋊(C3⋊D3)—23no—
50#6308C2311yescharacteristic
51C3×#638—23no—
52C3×C4⋊(C3⋊D3)—23no—
53C3×C4⋊(C3⋊D3)C2313yes—
54C3×C4×C3⋊D3—23no—
55C3×#634C2311yescharacteristic
56C32×C3⋊C8C2311yescharacteristic
57C32×C3⋊C8C2313yes—
58#5835—91no—
59#5752—94no—
60C32×#41—24no—
61C32×#41C2×D314yes—
62C32×#41C2×D311yescharacteristic
63#3832—21no—
64#3832—68no—
65#3832—64no—
66#3832—61no—
67D4×C3⋊D3—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×C4⋊(C3⋊D3)C2×D313yescomplemented
79C3×#239—64no—
80C3×D3×D4—612no—
81C3×#235—312no—
82C2×C3×D12—312no—
83C32×M16—33no—
84C2×C32×C8—33no—
85#3746—31no—
86#3745—33no—
87#3745—36no—
88#3744—33no—
89#3744—33no—
90C3×#208C2×D3112yescomplemented
91C2×C3×C3⋊C8C2×D3112yescomplemented
92#3719—34no—
93#3717—312no—
94#3716—312no—
95#3714—312no—
96#3714—324no—
97C22×C33C2×D413yes—
98C2×C3⋊(C3⋊D3)—23no—
99C2×C3⋊(C3⋊D3)—83no—
100C2×C3×C3⋊D3—23no—
101C2×C3×C3⋊D3—83no—
102C33×C4C2×D413yes—
103C33×C4C2411yescharacteristic
104C3⋊(C3⋊Dic3)—21no—
105C3×C3⋊Dic3—21no—
106C3×H(5,𝔽2)—91no—
107#1009—34no—
108#1009—94no—
109#1009—91no—
110#1009—184no—
111C3×#131—34no—
112#950—93no—
113#949—93no—
114#948C3⋊D311yescharacteristic, complemented
115#932—93no—
116C2×#213—93no—
117#908—912no—
118#908—912no—
119#904—912no—
120C2×D24—912no—
121C22×C3⋊D3—23no—
122C22×C3⋊D3—624no—
123C22×C3⋊D3—612no—
124C22×C3⋊D3—63no—
125C22×C3×D3—612no—
126C32×Q8—24no—
127C32×Q8D1214yes—
128C32×Q8#6711yescharacteristic
129C32×D4—212no—
130C32×D4D12112yes—
131C32×D4#6713yes—
132C2×C32×C4—212no—
133C2×C32×C4C22×D3112yes—
134C2×C32×C4C22×D313yes—
135#638—23no—
136#638—624no—
137#638—612no—
138#638—63no—
139C4⋊(C3⋊D3)—23no—
140C4⋊(C3⋊D3)—33no—
141C4⋊(C3⋊D3)—312no—
142C4⋊(C3⋊D3)—612no—
143C4⋊(C3⋊D3)—624no—
144C4⋊(C3⋊D3)—612no—
145C4⋊(C3⋊D3)—63no—
146C4⋊(C3⋊D3)C22×D313yes—
147C4×C3⋊D3—23no—
148C4×C3⋊D3—624no—
149C4×C3⋊D3—612no—
150C4×C3⋊D3—63no—
151#634—31no—
152#634—34no—
153#634—64no—
154#634C22×D311yescharacteristic
155C3×#67—612no—
156C3×D12—312no—
157C3×D12—612no—
158C3×C4×D3—612no—
159C3×Dic6—34no—
160C32×C8—33no—
161C32×C8—31no—
162C3×C3⋊C8C22×D314yes—
163C3×C3⋊C8C22×D3112yes—
164#576—271noSylow
165C2×C33—23no—
166C2×C33C22×D411yescharacteristic
167C3⋊(C3⋊D3)—43no—
168C3×C3⋊D3—43no—
169C3×#41—24no—
170C3×#41—93no—
171C3×#41—181no—
172C3×#41D3214yes—
173C3×#41C2×C3⋊D311yescharacteristic
174C2×C3×D4—93no—
175C2×C3×D4—183no—
176#239—64no—
177#239—188no—
178#239—184no—
179#239—181no—
180D3×D4—612no—
181D3×D4—1824no—
182D3×D4—1812no—
183D3×D4—183no—
184#235—312no—
185#235—912no—
186#235—93no—
187#235—1812no—
188C2×D12—312no—
189C2×D12—912no—
190C2×D12—93no—
191C2×D12—1812no—
192C3×M16—312no—
193C2×C3×C8—312no—
194#216—91no—
195#215—93no—
196#214—93no—
197#213—93no—
198#213—96no—
199#208C2×C3⋊D313yescomplemented
200C2×C3⋊C8C2×C3⋊D313yescomplemented
201Dic12—94no—
202D24—912no—
203D24—924no—
204#204—912no—
205#204—912no—
206C22×C32—212no—
207C22×C32C2×D12112yes—
208C22×C32C2×#6713yes—
209C2×C3⋊D3—23no—
210C2×C3⋊D3—624no—
211C2×C3⋊D3—612no—
212C2×C3⋊D3—63no—
213C2×C3⋊D3—83no—
214C2×C3⋊D3—243no—
215C2×C3⋊D3—2412no—
216C2×C3⋊D3—2424no—
217C2×C3×D3—612no—
218C2×C3×D3—2412no—
219C32×C4—24no—
220C32×C4—212no—
221C32×C4C2×D12112yes—
222C32×C4C2×#6713yes—
223C32×C4C23×D314yes—
224C32×C4C23×D311yescharacteristic
225C3⋊Dic3—21no—
226C3⋊Dic3—68no—
227C3⋊Dic3—64no—
228C3⋊Dic3—61no—
229C3×Dic3—64no—
230H(5,𝔽2)—91no—
231H(5,𝔽2)—271no—
232AGL(1,ℤ/8)—273no—
233AGL(1,ℤ/8)—273no—
234#135—273no—
235C2×D8—273no—
236#131—31no—
237C33#57611yescharacteristic, complemented, Sylow
238C23×C3—183no—
239C22×D3—612no—
240C22×D3—1824no—
241C22×D3—1812no—
242C22×D3—183no—
243C3×Q8—24no—
244C3×Q8—91no—
245C3×Q8#62614yes—
246C3×Q8C4⋊(C3⋊D3)11yescharacteristic
247C3×D4—212no—
248C3×D4—93no—
249C3×D4—183no—
250C3×D4—183no—
251C3×D4#626112yes—
252C3×D4C4⋊(C3⋊D3)13yes—
253C2×C3×C4—212no—
254C2×C3×C4—183no—
255C2×C3×C4C2×D32112yes—
256C2×C3×C4C22×C3⋊D313yes—
257#67—612no—
258#67—1824no—
259#67—1812no—
260#67—183no—
261D12—312no—
262D12—612no—
263D12—912no—
264D12—93no—
265D12—1812no—
266D12—1824no—
267D12—1812no—
268D12—183no—
269C4×D3—612no—
270C4×D3—1824no—
271C4×D3—1812no—
272C4×D3—183no—
273Dic6—34no—
274Dic6—94no—
275Dic6—91no—
276Dic6—184no—
277C3×C8—312no—
278C3×C8—34no—
279C3⋊C8C22×C3⋊D311yescharacteristic
280C3⋊C8C22×C3⋊D313yes—
281C2×C32—23no—
282C2×C32—24no—
283C2×C32—212no—
284C2×C32—412no—
285C2×C32C22×D1214yes—
286C2×C32C22×#6711yescharacteristic
287C3⋊D3—43no—
288C3⋊D3—1224no—
289C3⋊D3—1212no—
290C3⋊D3—123no—
291C3×D3—1212no—
292#41—93no—
293#41—181no—
294#41—273no—
295#41—541no—
296#41D3×C3⋊D311yescharacteristic
297C2×D4—93no—
298C2×D4—183no—
299C2×D4—273no—
300C2×D4—543no—
301Q16—271no—
302SD16—273no—
303SD16—273no—
304D8—273no—
305D8—276no—
306M16—33no—
307C2×C8—33no—
308C22×C3—212no—
309C22×C3—183no—
310C22×C3—723no—
311C22×C3C2×#626112yes—
312C22×C3C2×C4⋊(C3⋊D3)13yes—
313C2×D3—612no—
314C2×D3—1824no—
315C2×D3—1812no—
316C2×D3—183no—
317C2×D3—2412no—
318C2×D3—7212no—
319C2×D3—723no—
320C2×D3—7224no—
321C3×C4—24no—
322C3×C4—212no—
323C3×C4—181no—
324C3×C4C2×#626112yes—
325C3×C4C2×C4⋊(C3⋊D3)13yes—
326C3×C4C22×D3214yes—
327C3×C4C23×C3⋊D311yescharacteristic
328Dic3—64no—
329Dic3—188no—
330Dic3—184no—
331Dic3—181no—
332C32—24no—
333C32#575214yescomplemented
334C32#583511yescharacteristic, complemented
335C23—183no—
336C23—543no—
337Q8—91no—
338Q8—271no—
339Q8#629211yescharacteristic
340D4—93no—
341D4—183no—
342D4—183no—
343D4—273no—
344D4—543no—
345D4—543no—
346D4#629213yes—
347C2×C4—183no—
348C2×C4—543no—
349C2×C4C2×D3×C3⋊D313yes—
350C8—31no—
351C8—33no—
352C2×C3—212no—
353C2×C3—23no—
354C2×C3—24no—
355C2×C3—412no—
356C2×C3—363no—
357C2×C3C22×#62614yes—
358C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
359D3—1212no—
360D3—3624no—
361D3—3612no—
362D3—363no—
363C22—183no—
364C22—543no—
365C22—723no—
366C22—2163no—
367C22C2×#629213yes—
368C4—181no—
369C4—541no—
370C4C2×#629213yes—
371C4C22×D3×C3⋊D311yescharacteristic
372C3—24no—
373C3#1059544914yescomplemented
374C3#1059622311yescharacteristic, complemented
375C2—23no—
376C2—363no—
377C2—1083no—
378C2C22×#629211yescharacteristic
subgroupG/H
centerC2C22×#6292
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#576
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×#64366C2semidirect
3. C32×#948C2semidirect
4. #11723371C2semidirect
5. #11723370C2semidirect
6. #11723369C2semidirect
7. C2×#88791C2semidirect
13. #89076C22semidirect
14. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
17. C3×#3828C22semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C32×#208C22semidirect
20. C2×C32×C3⋊C8C22semidirect
21. #88800C22semidirect
22. #88797C22semidirect
23. #88796C22semidirect
24. #88791C22semidirect
25. #88791C22semidirect
29. #64366D3semidirect
36. C3×#948D3semidirect
74. #3828C2×D3semidirect
78. C2×C4⋊(C3⋊D3)C2×D3semidirect
90. C3×#208C2×D3semidirect
91. C2×C3×C3⋊C8C2×D3semidirect
114. #948C3⋊D3semidirect
199. #208C2×C3⋊D3semidirect
200. C2×C3⋊C8C2×C3⋊D3semidirect
237. C33#576semidirect
333. C32#5752semidirect
334. C32#5835semidirect
373. C3#10595449semidirect
374. C3#10596223semidirect

Maximal subgroups

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

subgroupindex
#117246842
C3×#643662
C32×#9482
#117233712
#117233702
#117233692
C2×#887912
#105962233
#105954493