Finite Group Databasebeta

← C3×#87166 #11918516

Subgroups — C3×#87166 #11918516

Order 1152.

2977 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2C32×#568C211yesmaximal, characteristic, complemented
3C2×C3×#948C211yesmaximal, characteristic, complemented
4C3×#5750C218yesmaximal, complemented
5C3×D3×#131C218yesmaximal, complemented
6C3×#5745C216yesmaximal, complemented
7C2×C3×#907C216yesmaximal, complemented
8C3×#3573—31nomaximal
9#87166C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1010C414yes—
12C3×#1009C414yes—
13C3×D3×#41C2218yes—
14C3×#1007C412yes—
15C3×#1004C416yes—
16C2×C3×#235C2216yes—
17C32×#131C2218yescomplemented
18C2×C32×M16C2216yescomplemented
19C3×#948C2214yescomplemented
20C3×#948C2214yescomplemented
21C2×C3×#208C2213yescomplemented
22C22×C3×C3⋊C8C2213yescomplemented
23C3×#907C2214yescomplemented
24C3×#907C22112yescomplemented
25C3×#907C22112yescomplemented
26C3×#907C22112yescomplemented
27C3×D3×M16C22124yescomplemented
28C3×#901C22124yescomplemented
29C2×C3×ΓL(1,𝔽25)C22112yescomplemented
30C2×C3×D3×C8C22112yescomplemented
31C3×#585—31no—
32#5964C2×C311yescharacteristic
33C3×#568—21no—
34C3×#568—38no—
35C3×#568—36no—
36C3×#568D311yescharacteristic, complemented
37C3×#568C2×C311yescharacteristic, complemented
38C2×C3×#131—31no—
39C2×C3×#131—38no—
40C2×C3×#131—36no—
41C2×#948C2×C311yescharacteristic, complemented
42#5750C2×C318yescomplemented
43D3×#131C2×C318yescomplemented
44#5745C2×C316yescomplemented
45C2×#907C2×C316yescomplemented
46C32×#41C2×C414yes—
47C32×#41C2314yes—
48C2×C32×Q8C2×C411yescharacteristic
49C2×C32×D4C2×C413yes—
50C22×C32×C4C2313yes—
51C2×C3×#67C2×C4112yes—
52C3×#239C2×C418yes—
53C3×D3×Q8C2×C418yes—
54C3×#237C2×C4124yes—
55C3×D3×D4C2×C4124yes—
56C3×#235C2×C4112yes—
57C3×#235C2×C4112yes—
58C3×#235C23112yes—
59C3×#235C2314yes—
60C2×C3×D12C2×C416yes—
61C2×C3×C4×D3C23112yes—
62C2×C3×Dic6C2×C416yes—
63C32×M16C23112yes—
64C32×M16C2314yes—
65C2×C32×C8C23112yes—
66C3×#208C23112yes—
67C2×C3×C3⋊C8C2316yes—
68C2×C3×C3⋊C8C2316yes—
69C2×C3×C3⋊C8C2311yescharacteristic
70C2×C3×C3⋊C8C2313yes—
71C3×ΓL(1,𝔽25)C2318yes—
72C3×ΓL(1,𝔽25)C23124yes—
73C3×D3×C8C2318yes—
74C3×D3×C8C23124yes—
75#3573—31noSylow
76C3×2−1+4—34no—
77C3×2−1+4—32no—
78C3×H(5,𝔽2)—34no—
79C3×H(5,𝔽2)—36no—
80C2×C3×#41—21no—
81C2×C3×#41—38no—
82C2×C3×#41—36no—
83C2×C3×#41C2×D311yescharacteristic
84C2×C3×#41C22×C311yescharacteristic
85#1010C3×C414yes—
86#1009C3×C414yes—
87D3×#41C22×C318yes—
88#1007C3×C412yes—
89#1004C3×C416yes—
90C2×#235C22×C316yes—
91C3×#131—28no—
92C3×#131—34no—
93C3×#131—34no—
94C3×#131—312no—
95C3×#131—34no—
96C3×#131—312no—
97C3×#131—312no—
98C3×#131—324no—
99C3×#131C2×D318yescomplemented
100C3×#131C22×C318yescomplemented
101C2×C3×M16—26no—
102C2×C3×M16—33no—
103C2×C3×M16—324no—
104C2×C3×M16—312no—
105C2×C3×M16C2×D316yescomplemented
106C2×C3×M16C22×C316yescomplemented
107C22×C3×C8—33no—
108C22×C3×C8—312no—
109#948C22×C314yescomplemented
110#948C22×C314yescomplemented
111C2×#208C22×C313yescomplemented
112C22×C3⋊C8C22×C313yescomplemented
113#907C22×C314yescomplemented
114#907C22×C3112yescomplemented
115#907C22×C3112yescomplemented
116#907C22×C3112yescomplemented
117D3×M16C22×C3124yescomplemented
118#901C22×C3124yescomplemented
119C2×ΓL(1,𝔽25)C22×C3112yescomplemented
120C2×D3×C8C22×C3112yescomplemented
121C23×C32C22×C413yes—
122C22×C3×D3C22×C4112yes—
123C32×Q8C22×C414yes—
124C32×D4C22×C4112yes—
125C2×C32×C4C22×C416yes—
126C2×C32×C4C22×C413yes—
127C2×C32×C4C2411yescharacteristic
128C2×C32×C4C2416yes—
129C3×#67C22×C418yes—
130C3×#67C22×C4124yes—
131C2×C3×Dic3C22×C4112yes—
132C3×D12C22×C414yes—
133C3×D12C22×C4112yes—
134C3×C4×D3C22×C4124yes—
135C3×C4×D3C2418yes—
136C3×Dic6C22×C414yes—
137C3×Dic6C22×C4112yes—
138C32×C8C2418yes—
139C3×C3⋊C8C2411yescharacteristic
140C3×C3⋊C8C2416yes—
141C3×C3⋊C8C2411yescharacteristic
142#585—31no—
143#568—38no—
144#568—36no—
145#568C3×D311yescharacteristic, complemented
146C2×#131—31no—
147C2×#131—38no—
148C2×#131—36no—
149C3×#41—24no—
150C3×#41—24no—
151C3×#41—38no—
152C3×#41—324no—
153C3×#41—312no—
154C3×#41—312no—
155C3×#41—34no—
156C3×#41—312no—
157C3×#41C4×D314yes—
158C3×#41C2×C3×C414yes—
159C3×#41C22×D314yes—
160C3×#41C23×C314yes—
161C2×C3×Q8—21no—
162C2×C3×Q8—38no—
163C2×C3×Q8—36no—
164C2×C3×Q8C4×D311yescharacteristic
165C2×C3×Q8C2×C3×C411yescharacteristic
166C2×C3×D4—23no—
167C2×C3×D4—312no—
168C2×C3×D4—324no—
169C2×C3×D4—36no—
170C2×C3×D4C4×D313yes—
171C2×C3×D4C2×C3×C413yes—
172C22×C3×C4—23no—
173C22×C3×C4—312no—
174C22×C3×C4C22×D313yes—
175C22×C3×C4C23×C313yes—
176C2×#67C2×C3×C4112yes—
177#239C2×C3×C418yes—
178D3×Q8C2×C3×C418yes—
179#237C2×C3×C4124yes—
180D3×D4C2×C3×C4124yes—
181#235C2×C3×C4112yes—
182#235C2×C3×C4112yes—
183#235C23×C3112yes—
184#235C23×C314yes—
185C2×D12C2×C3×C416yes—
186C2×C4×D3C23×C3112yes—
187C2×Dic6C2×C3×C416yes—
188C3×M16—24no—
189C3×M16—212no—
190C3×M16—312no—
191C3×M16—38no—
192C3×M16—324no—
193C3×M16C22×D3112yes—
194C3×M16C22×D314yes—
195C3×M16C23×C3112yes—
196C3×M16C23×C314yes—
197C2×C3×C8—212no—
198C2×C3×C8—31no—
199C2×C3×C8—36no—
200C2×C3×C8—33no—
201C2×C3×C8—36no—
202C2×C3×C8—38no—
203C2×C3×C8—324no—
204C2×C3×C8C22×D3112yes—
205C2×C3×C8C23×C3112yes—
206#208C23×C3112yes—
207C2×C3⋊C8C23×C316yes—
208C2×C3⋊C8C23×C316yes—
209C2×C3⋊C8C23×C311yescharacteristic
210C2×C3⋊C8C23×C313yes—
211ΓL(1,𝔽25)C23×C318yes—
212ΓL(1,𝔽25)C23×C3124yes—
213D3×C8C23×C318yes—
214D3×C8C23×C3124yes—
215C22×C32—43no—
216C22×C32C23×C416yes—
217C22×C32C23×C411yescharacteristic
218C2×C3×D3—412no—
219C2×C3×D3C23×C418yes—
220C32×C4C23×C411yescharacteristic
221C32×C4C23×C416yes—
222C32×C4C2511yescharacteristic
223C3×Dic3C23×C418yes—
2242−1+4—34no—
2252−1+4—32no—
226H(5,𝔽2)—34no—
227H(5,𝔽2)—36no—
228C2×#41—38no—
229C2×#41—36no—
230C2×#41C2×C3×D311yescharacteristic
231#131—34no—
232#131—34no—
233#131—312no—
234#131—34no—
235#131—312no—
236#131—312no—
237#131—324no—
238#131C2×C3×D318yescomplemented
239C2×M16—33no—
240C2×M16—324no—
241C2×M16—312no—
242C2×M16C2×C3×D316yescomplemented
243C22×C8—33no—
244C22×C8—312no—
245C23×C3—23no—
246C23×C3—312no—
247C23×C3C2×C4×D313yes—
248C23×C3C22×C3×C413yes—
249C22×D3C22×C3×C4112yes—
250C3×Q8—24no—
251C3×Q8—312no—
252C3×Q8—34no—
253C3×Q8C2×C4×D314yes—
254C3×Q8C22×C3×C414yes—
255C3×D4—212no—
256C3×D4—324no—
257C3×D4—38no—
258C3×D4—312no—
259C3×D4—34no—
260C3×D4C2×C4×D3112yes—
261C3×D4C22×C3×C4112yes—
262C2×C3×C4—26no—
263C2×C3×C4—21no—
264C2×C3×C4—23no—
265C2×C3×C4—26no—
266C2×C3×C4—312no—
267C2×C3×C4—324no—
268C2×C3×C4—38no—
269C2×C3×C4C2×C4×D316yes—
270C2×C3×C4C2×C4×D313yes—
271C2×C3×C4C22×C3×C416yes—
272C2×C3×C4C22×C3×C413yes—
273C2×C3×C4C23×D311yescharacteristic
274C2×C3×C4C23×D316yes—
275C2×C3×C4C24×C311yescharacteristic
276C2×C3×C4C24×C316yes—
277#67C22×C3×C418yes—
278#67C22×C3×C4124yes—
279C2×Dic3C22×C3×C4112yes—
280D12C22×C3×C414yes—
281D12C22×C3×C4112yes—
282C4×D3C22×C3×C4124yes—
283C4×D3C24×C318yes—
284Dic6C22×C3×C414yes—
285Dic6C22×C3×C4112yes—
286C3×C8—28no—
287C3×C8—31no—
288C3×C8—36no—
289C3×C8—31no—
290C3×C8C23×D318yes—
291C3×C8C24×C318yes—
292C3⋊C8C24×C311yescharacteristic
293C3⋊C8C24×C316yes—
294C3⋊C8C24×C311yescharacteristic
295C2×C32—26no—
296C2×C32—21no—
297C2×C32C24×C411yescharacteristic
298C3×D3—28no—
299#41—38no—
300#41—324no—
301#41—34no—
302#41—312no—
303#41—312no—
304#41—312no—
305#41C3×C4×D314yes—
306#41C22×C3×D314yes—
307C2×Q8—38no—
308C2×Q8—36no—
309C2×Q8C3×C4×D311yescharacteristic
310C2×D4—312no—
311C2×D4—324no—
312C2×D4—36no—
313C2×D4C3×C4×D313yes—
314C22×C4—312no—
315C22×C4C22×C3×D313yes—
316M16—312no—
317M16—324no—
318M16—38no—
319M16C22×C3×D3112yes—
320M16C22×C3×D314yes—
321C2×C8—31no—
322C2×C8—36no—
323C2×C8—33no—
324C2×C8—36no—
325C2×C8—38no—
326C2×C8—324no—
327C2×C8C22×C3×D3112yes—
328C22×C3—21no—
329C22×C3—26no—
330C22×C3—38no—
331C22×C3—43no—
332C22×C3—43no—
333C22×C3—83no—
334C22×C3—1212no—
335C22×C3C22×C4×D316yes—
336C22×C3C22×C4×D311yescharacteristic
337C22×C3C23×C3×C416yes—
338C22×C3C23×C3×C411yescharacteristic
339C2×D3—412no—
340C2×D3C23×C3×C418yes—
341C3×C4—21no—
342C3×C4—26no—
343C3×C4—21no—
344C3×C4—38no—
345C3×C4C22×C4×D316yes—
346C3×C4C22×C4×D311yescharacteristic
347C3×C4C23×C3×C411yescharacteristic
348C3×C4C23×C3×C416yes—
349C3×C4C24×D311yescharacteristic
350C3×C4C25×C311yescharacteristic
351Dic3C23×C3×C418yes—
352C32#357311yescharacteristic, complemented, Sylow
353C23—312no—
354C23C2×C3×C4×D313yes—
355Q8—312no—
356Q8—34no—
357Q8C2×C3×C4×D314yes—
358D4—324no—
359D4—38no—
360D4—312no—
361D4—34no—
362D4C2×C3×C4×D3112yes—
363C2×C4—312no—
364C2×C4—324no—
365C2×C4—38no—
366C2×C4C2×C3×C4×D316yes—
367C2×C4C2×C3×C4×D313yes—
368C2×C4C23×C3×D311yescharacteristic
369C2×C4C23×C3×D316yes—
370C8—31no—
371C8—31no—
372C8—36no—
373C8C23×C3×D318yes—
374C2×C3—21no—
375C2×C3—26no—
376C2×C3—26no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—41no—
380C2×C3—46no—
381C2×C3—68no—
382C2×C3C23×C4×D311yescharacteristic
383C2×C3C24×C3×C411yescharacteristic
384D3—28no—
385C22—38no—
386C22—43no—
387C22—1212no—
388C22C22×C3×C4×D316yes—
389C22C22×C3×C4×D311yescharacteristic
390C4—38no—
391C4C22×C3×C4×D316yes—
392C4C22×C3×C4×D311yescharacteristic
393C4C24×C3×D311yescharacteristic
394C3—21no—
395C3#8716611yescharacteristic, direct product factor
396C3C3×#357311yescharacteristic, complemented
397C2—21no—
398C2—26no—
399C2—68no—
400C2C23×C3×C4×D311yescharacteristic
subgroupG/H
centerC3×C4C24×D3
commutatorC2×C3C24×C3×C4
FrattiniC4C24×C3×D3
FittingC32×#568C2
socleC2×C32C24×C4
prime radicalC3×#5964C2
cosocleC3×C4C25×C3
hypercenterC3×#568D3
hypocenterC3C3×#3573
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C3×#568

Complements

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

NH ≅ G/Ntype
2. C32×#568C2semidirect
3. C2×C3×#948C2semidirect
4. C3×#5750C2semidirect
5. C3×D3×#131C2semidirect
6. C3×#5745C2semidirect
7. C2×C3×#907C2semidirect
9. #87166C3direct (N × H)
17. C32×#131C22semidirect
18. C2×C32×M16C22semidirect
19. C3×#948C22semidirect
20. C3×#948C22semidirect
21. C2×C3×#208C22semidirect
22. C22×C3×C3⋊C8C22semidirect
23. C3×#907C22semidirect
24. C3×#907C22semidirect
25. C3×#907C22semidirect
26. C3×#907C22semidirect
27. C3×D3×M16C22semidirect
28. C3×#901C22semidirect
29. C2×C3×ΓL(1,𝔽25)C22semidirect
30. C2×C3×D3×C8C22semidirect
36. C3×#568D3semidirect
37. C3×#568C2×C3semidirect
41. C2×#948C2×C3semidirect
42. #5750C2×C3semidirect
43. D3×#131C2×C3semidirect
44. #5745C2×C3semidirect
45. C2×#907C2×C3semidirect
99. C3×#131C2×D3semidirect
100. C3×#131C22×C3semidirect
105. C2×C3×M16C2×D3semidirect
106. C2×C3×M16C22×C3semidirect
109. #948C22×C3semidirect
110. #948C22×C3semidirect
111. C2×#208C22×C3semidirect
112. C22×C3⋊C8C22×C3semidirect
113. #907C22×C3semidirect
114. #907C22×C3semidirect
115. #907C22×C3semidirect
116. #907C22×C3semidirect
117. D3×M16C22×C3semidirect
118. #901C22×C3semidirect
119. C2×ΓL(1,𝔽25)C22×C3semidirect
120. C2×D3×C8C22×C3semidirect
145. #568C3×D3semidirect
238. #131C2×C3×D3semidirect
242. C2×M16C2×C3×D3semidirect
352. C32#3573semidirect
395. C3#87166direct (N × H)
396. C3C3×#3573semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C32×#5682
C2×C3×#9482
C3×#57502
C3×D3×#1312
C3×#57452
C2×C3×#9072
C3×#35733
#871663