Finite Group Databasebeta

← #13177972

Subgroups — #13177972

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
2C32×#1009C211yesmaximal, characteristic, complemented
3C3×#64157C211yesmaximal, characteristic, complemented
4#11723378C213yesmaximal, complemented
5#11723377C213yesmaximal, complemented
6#11723367C213yesmaximal, complemented
7C2×#88790C213yesmaximal, complemented
8#10596306—31nomaximal
9#10595449—34nomaximal
10C33×#41C2211yescharacteristic
11#89080—21no—
12D4×C3⋊(C3⋊D3)—23no—
13#89076C2213yescomplemented
14C2×C4⋊(C3⋊(C3⋊D3))C2213yescomplemented
15C32×#239—21no—
16C32×D3×D4—23no—
17C32×#235C2213yescomplemented
18C2×C32×D12C2213yescomplemented
19C3×#3748C2213yescomplemented
20C2×C3×C3⋊(C3⋊C8)C2213yescomplemented
21#88799C2211yescharacteristic, complemented
22#88795C2213yescomplemented
23#88794C2213yescomplemented
24#88790C2213yescomplemented
25#88790C2216yescomplemented
26C32×H(5,𝔽2)—31no—
27#64366—31no—
28#64366—34no—
29#64366—64no—
30C3×#1009D314yescomplemented
31#64159—33no—
32#64158—33no—
33#64157D311yescharacteristic, complemented
34#64141—33no—
35C2×#3753—33no—
36C3×#948—34no—
37#63830—312no—
38#63829—312no—
39#63825—312no—
40C2×#3714—312no—
41C22×C3⋊(C3⋊D3)—23no—
42C22×C32×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
51C32×#67—23no—
52C32×D12—23no—
53C32×D12C2313yes—
54C32×C4×D3—23no—
55C32×Dic6C2311yescharacteristic
56C3×C3⋊(C3⋊C8)C2311yescharacteristic
57C3×C3⋊(C3⋊C8)C2313yes—
58#5835—94no—
59#5752—91no—
60C32×#41—24no—
61C32×#41—33no—
62C32×#41—61no—
63C32×#41C2×D311yescharacteristic
64C32×#41C2×D314yes—
65C2×C32×D4—33no—
66C2×C32×D4—63no—
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×#239—24no—
80C3×D3×D4—212no—
81C3×#235C2×D3112yescomplemented
82C2×C3×D12C2×D3112yescomplemented
83#3756—31no—
84#3755—33no—
85#3754—33no—
86#3753—33no—
87#3753—36no—
88#3748C2×D313yescomplemented
89C2×C3⋊(C3⋊C8)C2×D313yescomplemented
90C3×#208—312no—
91C2×C3×C3⋊C8—312no—
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×C32×D3—23no—
101C2×C32×D3—83no—
102C33×C4C2×D413yes—
103C33×C4C2411yescharacteristic
104C3⋊(C3⋊Dic3)—21no—
105C32×Dic3—21no—
106C3×H(5,𝔽2)—34no—
107#1009—94no—
108#1009—91no—
109#1009—184no—
110#1009C3⋊D311yescharacteristic, complemented
111C3×#131—91no—
112#950—912no—
113#949—912no—
114#948—34no—
115#932—912no—
116C2×#213—912no—
117#908—93no—
118#908—93no—
119#904—93no—
120C2×D24—93no—
121C23×C32—63no—
122C22×C3⋊D3—624no—
123C22×C3⋊D3—63no—
124C22×C3⋊D3—612no—
125C22×C3×D3—212no—
126C32×Q8—24no—
127C32×Q8—31no—
128C32×Q8D1211yescharacteristic
129C32×Q8#6714yes—
130C32×D4—212no—
131C32×D4—33no—
132C32×D4—63no—
133C32×D4—63no—
134C32×D4D1213yes—
135C32×D4#67112yes—
136C2×C32×C4—212no—
137C2×C32×C4—63no—
138C2×C32×C4C22×D3112yes—
139C2×C32×C4C22×D313yes—
140#638—624no—
141#638—63no—
142#638—612no—
143C4⋊(C3⋊D3)—33no—
144C4⋊(C3⋊D3)—312no—
145C4⋊(C3⋊D3)—612no—
146C4⋊(C3⋊D3)—624no—
147C4⋊(C3⋊D3)—63no—
148C4⋊(C3⋊D3)—612no—
149C4×C3⋊D3—624no—
150C4×C3⋊D3—63no—
151C4×C3⋊D3—612no—
152#634—31no—
153#634—34no—
154#634—64no—
155C3×#67—212no—
156C3×D12—212no—
157C3×D12C22×D3112yes—
158C3×C4×D3—212no—
159C3×Dic6C22×D314yes—
160C3⋊(C3⋊C8)C22×D311yescharacteristic
161C3⋊(C3⋊C8)C22×D313yes—
162C3×C3⋊C8—34no—
163C3×C3⋊C8—312no—
164#576—271noSylow
165C2×C33—23no—
166C2×C33C22×D411yescharacteristic
167C3⋊(C3⋊D3)—43no—
168C32×D3—43no—
169C3×#41—24no—
170C3×#41—312no—
171C3×#41—64no—
172C3×#41D3214yes—
173C3×#41C2×C3⋊D311yescharacteristic
174C2×C3×D4—312no—
175C2×C3×D4—612no—
176#239—21no—
177#239—188no—
178#239—184no—
179#239—181no—
180D3×D4—23no—
181D3×D4—1824no—
182D3×D4—1812no—
183D3×D4—183no—
184#235—912no—
185#235—93no—
186#235—1812no—
187#235C2×C3⋊D313yescomplemented
188C2×D12—912no—
189C2×D12—93no—
190C2×D12—1812no—
191C2×D12C2×C3⋊D313yescomplemented
192C3×M16—93no—
193C2×C3×C8—93no—
194#216—94no—
195#215—912no—
196#214—912no—
197#213—912no—
198#213—924no—
199#208—312no—
200C2×C3⋊C8—312no—
201Dic12—91no—
202D24—93no—
203D24—96no—
204#204—93no—
205#204—93no—
206C22×C32—212no—
207C22×C32—63no—
208C22×C32—243no—
209C22×C32C2×D1213yes—
210C22×C32C2×#67112yes—
211C2×C3⋊D3—624no—
212C2×C3⋊D3—63no—
213C2×C3⋊D3—612no—
214C2×C3⋊D3—243no—
215C2×C3⋊D3—2412no—
216C2×C3⋊D3—2424no—
217C2×C3×D3—212no—
218C2×C3×D3—812no—
219C32×C4—24no—
220C32×C4—212no—
221C32×C4—61no—
222C32×C4C2×D1213yes—
223C32×C4C2×#67112yes—
224C32×C4C23×D314yes—
225C32×C4C23×D311yescharacteristic
226C3⋊Dic3—68no—
227C3⋊Dic3—61no—
228C3⋊Dic3—64no—
229C3×Dic3—24no—
230H(5,𝔽2)—31no—
231H(5,𝔽2)—271no—
232AGL(1,ℤ/8)—273no—
233AGL(1,ℤ/8)—273no—
234#135—273no—
235C2×D8—273no—
236#131—91no—
237C33#57611yescharacteristic, complemented, Sylow
238C23×C3—612no—
239C22×D3—23no—
240C22×D3—1824no—
241C22×D3—1812no—
242C22×D3—183no—
243C3×Q8—24no—
244C3×Q8—34no—
245C3×Q8#62614yes—
246C3×Q8#63811yescharacteristic
247C3×D4—212no—
248C3×D4—312no—
249C3×D4—612no—
250C3×D4—612no—
251C3×D4#626112yes—
252C3×D4#63813yes—
253C2×C3×C4—212no—
254C2×C3×C4—612no—
255C2×C3×C4C2×D32112yes—
256C2×C3×C4C22×C3⋊D313yes—
257#67—23no—
258#67—1824no—
259#67—1812no—
260#67—183no—
261D12—23no—
262D12—912no—
263D12—93no—
264D12—1812no—
265D12—1824no—
266D12—1812no—
267D12—183no—
268D12C22×C3⋊D313yes—
269C4×D3—23no—
270C4×D3—1824no—
271C4×D3—1812no—
272C4×D3—183no—
273Dic6—94no—
274Dic6—91no—
275Dic6—184no—
276Dic6C22×C3⋊D311yescharacteristic
277C3×C8—93no—
278C3×C8—91no—
279C3⋊C8—34no—
280C3⋊C8—312no—
281C2×C32—23no—
282C2×C32—24no—
283C2×C32—212no—
284C2×C32—412no—
285C2×C32—123no—
286C2×C32C22×D1211yescharacteristic
287C2×C32C22×#6714yes—
288C3⋊D3—1224no—
289C3⋊D3—123no—
290C3⋊D3—1212no—
291C3×D3—412no—
292#41—33no—
293#41—61no—
294#41—273no—
295#41—541no—
296#41D3×C3⋊D311yescharacteristic
297C2×D4—33no—
298C2×D4—63no—
299C2×D4—273no—
300C2×D4—543no—
301Q16—271no—
302SD16—273no—
303SD16—273no—
304D8—273no—
305D8—276no—
306M16—93no—
307C2×C8—93no—
308C22×C3—212no—
309C22×C3—612no—
310C22×C3—2412no—
311C22×C3C2×#626112yes—
312C22×C3C2×#63813yes—
313C2×D3—23no—
314C2×D3—83no—
315C2×D3—1824no—
316C2×D3—1812no—
317C2×D3—183no—
318C2×D3—7212no—
319C2×D3—723no—
320C2×D3—7224no—
321C3×C4—24no—
322C3×C4—212no—
323C3×C4—64no—
324C3×C4C2×#626112yes—
325C3×C4C2×#63813yes—
326C3×C4C22×D3214yes—
327C3×C4C23×C3⋊D311yescharacteristic
328Dic3—21no—
329Dic3—188no—
330Dic3—184no—
331Dic3—181no—
332C32—24no—
333C32#575211yescharacteristic, complemented
334C32#583514yescomplemented
335C23—63no—
336C23—543no—
337Q8—31no—
338Q8—271no—
339Q8#629111yescharacteristic
340D4—33no—
341D4—63no—
342D4—63no—
343D4—273no—
344D4—543no—
345D4—543no—
346D4#629113yes—
347C2×C4—63no—
348C2×C4—543no—
349C2×C4C2×D3×C3⋊D313yes—
350C8—91no—
351C8—93no—
352C2×C3—212no—
353C2×C3—23no—
354C2×C3—24no—
355C2×C3—412no—
356C2×C3—1212no—
357C2×C3C22×#62614yes—
358C2×C3C22×#63811yescharacteristic
359D3—43no—
360D3—3624no—
361D3—3612no—
362D3—363no—
363C22—63no—
364C22—243no—
365C22—543no—
366C22—2163no—
367C22C2×#629113yes—
368C4—61no—
369C4—541no—
370C4C2×#629113yes—
371C4C22×D3×C3⋊D311yescharacteristic
372C3—24no—
373C3#1059544914yescomplemented
374C3#1059630611yescharacteristic, complemented
375C2—23no—
376C2—123no—
377C2—1083no—
378C2C22×#629111yescharacteristic
subgroupG/H
centerC2C22×#6291
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. C32×#1009C2semidirect
3. C3×#64157C2semidirect
4. #11723378C2semidirect
5. #11723377C2semidirect
6. #11723367C2semidirect
7. C2×#88790C2semidirect
13. #89076C22semidirect
14. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
17. C32×#235C22semidirect
18. C2×C32×D12C22semidirect
19. C3×#3748C22semidirect
20. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. #88799C22semidirect
22. #88795C22semidirect
23. #88794C22semidirect
24. #88790C22semidirect
25. #88790C22semidirect
30. C3×#1009D3semidirect
33. #64157D3semidirect
81. C3×#235C2×D3semidirect
82. C2×C3×D12C2×D3semidirect
88. #3748C2×D3semidirect
89. C2×C3⋊(C3⋊C8)C2×D3semidirect
110. #1009C3⋊D3semidirect
187. #235C2×C3⋊D3semidirect
191. C2×D12C2×C3⋊D3semidirect
237. C33#576semidirect
333. C32#5752semidirect
334. C32#5835semidirect
373. C3#10595449semidirect
374. C3#10596306semidirect

Maximal subgroups

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

subgroupindex
#117246842
C32×#10092
C3×#641572
#117233782
#117233772
#117233672
C2×#887902
#105963063
#105954493