Finite Group Databasebeta

← C3×#87160 #11918510

Subgroups — C3×#87160 #11918510

Order 1152.

2981 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2C2×C32×#131C211yesmaximal, characteristic, complemented
3C3×#5819C211yesmaximal, characteristic, complemented
4C3×#5750C218yesmaximal, complemented
5C3×D3×#131C218yesmaximal, complemented
6C3×#5745C216yesmaximal, complemented
7C2×C3×#901C216yesmaximal, complemented
8C3×#3573—31nomaximal
9#87160C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C3×#1010C414yes—
12C3×#1009C414yes—
13C3×D3×#41C2218yes—
14C3×#1007C412yes—
15C3×#1004C416yes—
16C2×C3×#235C2216yes—
17C32×#131C2214yescomplemented
18C32×#131C2214yescomplemented
19C2×C32×M16C2213yescomplemented
20C22×C32×C8C2213yescomplemented
21C3×#948C2218yescomplemented
22C2×C3×#208C2216yescomplemented
23C3×#907C22124yescomplemented
24C3×D3×M16C22124yescomplemented
25C3×#901C22112yescomplemented
26C3×#901C22112yescomplemented
27C3×#901C22112yescomplemented
28C3×#901C2214yescomplemented
29C2×C3×ΓL(1,𝔽25)C22112yescomplemented
30C2×C3×D3×C8C22112yescomplemented
31C3×#585—31no—
32#5964C2×C311yescharacteristic
33C3×#568—31no—
34C3×#568—38no—
35C3×#568—36no—
36C2×C3×#131—21no—
37C2×C3×#131—38no—
38C2×C3×#131—36no—
39C2×C3×#131D311yescharacteristic, complemented
40C2×C3×#131C2×C311yescharacteristic, complemented
41#5819C2×C311yescharacteristic, complemented
42#5750C2×C318yescomplemented
43D3×#131C2×C318yescomplemented
44#5745C2×C316yescomplemented
45C2×#901C2×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×#235C2314yes—
59C3×#235C23112yes—
60C2×C3×D12C2×C416yes—
61C2×C3×C4×D3C23112yes—
62C2×C3×Dic6C2×C416yes—
63C32×M16C23112yes—
64C2×C32×C8C2316yes—
65C2×C32×C8C2316yes—
66C2×C32×C8C2311yescharacteristic
67C2×C32×C8C2313yes—
68C3×#208C2314yes—
69C3×#208C23112yes—
70C2×C3×C3⋊C8C23112yes—
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—24no—
92C3×#131—24no—
93C3×#131—38no—
94C3×#131—324no—
95C3×#131—312no—
96C3×#131—34no—
97C3×#131—312no—
98C3×#131—312no—
99C3×#131C2×D314yescomplemented
100C3×#131C2×D314yescomplemented
101C3×#131C22×C314yescomplemented
102C3×#131C22×C314yescomplemented
103C2×C3×M16—23no—
104C2×C3×M16—36no—
105C2×C3×M16—324no—
106C2×C3×M16—312no—
107C2×C3×M16C2×D313yescomplemented
108C2×C3×M16C22×C313yescomplemented
109C22×C3×C8—23no—
110C22×C3×C8—312no—
111C22×C3×C8C2×D313yescomplemented
112C22×C3×C8C22×C313yescomplemented
113#948C22×C318yescomplemented
114C2×#208C22×C316yescomplemented
115#907C22×C3124yescomplemented
116D3×M16C22×C3124yescomplemented
117#901C22×C314yescomplemented
118#901C22×C3112yescomplemented
119#901C22×C3112yescomplemented
120#901C22×C3112yescomplemented
121C2×ΓL(1,𝔽25)C22×C3112yescomplemented
122C2×D3×C8C22×C3112yescomplemented
123C23×C32C22×C413yes—
124C22×C3×D3C22×C4112yes—
125C32×Q8C22×C414yes—
126C32×D4C22×C4112yes—
127C2×C32×C4C22×C416yes—
128C2×C32×C4C22×C413yes—
129C2×C32×C4C2416yes—
130C2×C32×C4C2411yescharacteristic
131C3×#67C22×C418yes—
132C3×#67C22×C4124yes—
133C2×C3×Dic3C22×C4112yes—
134C3×D12C22×C414yes—
135C3×D12C22×C4112yes—
136C3×C4×D3C22×C4124yes—
137C3×C4×D3C2418yes—
138C3×Dic6C22×C414yes—
139C3×Dic6C22×C4112yes—
140C32×C8C2416yes—
141C32×C8C2411yescharacteristic
142C32×C8C2411yescharacteristic
143C3×C3⋊C8C2418yes—
144#585—31no—
145#568—31no—
146#568—38no—
147#568—36no—
148C2×#131—38no—
149C2×#131—36no—
150C2×#131C3×D311yescharacteristic, complemented
151C3×#41—24no—
152C3×#41—24no—
153C3×#41—38no—
154C3×#41—324no—
155C3×#41—312no—
156C3×#41—312no—
157C3×#41—312no—
158C3×#41—34no—
159C3×#41C4×D314yes—
160C3×#41C2×C3×C414yes—
161C3×#41C22×D314yes—
162C3×#41C23×C314yes—
163C2×C3×Q8—21no—
164C2×C3×Q8—38no—
165C2×C3×Q8—36no—
166C2×C3×Q8C4×D311yescharacteristic
167C2×C3×Q8C2×C3×C411yescharacteristic
168C2×C3×D4—23no—
169C2×C3×D4—312no—
170C2×C3×D4—324no—
171C2×C3×D4—36no—
172C2×C3×D4C4×D313yes—
173C2×C3×D4C2×C3×C413yes—
174C22×C3×C4—23no—
175C22×C3×C4—312no—
176C22×C3×C4C22×D313yes—
177C22×C3×C4C23×C313yes—
178C2×#67C2×C3×C4112yes—
179#239C2×C3×C418yes—
180D3×Q8C2×C3×C418yes—
181#237C2×C3×C4124yes—
182D3×D4C2×C3×C4124yes—
183#235C2×C3×C4112yes—
184#235C2×C3×C4112yes—
185#235C23×C314yes—
186#235C23×C3112yes—
187C2×D12C2×C3×C416yes—
188C2×C4×D3C23×C3112yes—
189C2×Dic6C2×C3×C416yes—
190C3×M16—212no—
191C3×M16—312no—
192C3×M16—34no—
193C3×M16—324no—
194C3×M16—38no—
195C3×M16C22×D3112yes—
196C3×M16C23×C3112yes—
197C2×C3×C8—21no—
198C2×C3×C8—23no—
199C2×C3×C8—26no—
200C2×C3×C8—26no—
201C2×C3×C8—312no—
202C2×C3×C8—324no—
203C2×C3×C8—38no—
204C2×C3×C8C22×D316yes—
205C2×C3×C8C22×D313yes—
206C2×C3×C8C22×D316yes—
207C2×C3×C8C22×D311yescharacteristic
208C2×C3×C8C23×C316yes—
209C2×C3×C8C23×C316yes—
210C2×C3×C8C23×C311yescharacteristic
211C2×C3×C8C23×C313yes—
212#208C23×C314yes—
213#208C23×C3112yes—
214C2×C3⋊C8C23×C3112yes—
215ΓL(1,𝔽25)C23×C318yes—
216ΓL(1,𝔽25)C23×C3124yes—
217D3×C8C23×C318yes—
218D3×C8C23×C3124yes—
219C22×C32—43no—
220C22×C32C23×C416yes—
221C22×C32C23×C411yescharacteristic
222C2×C3×D3—412no—
223C2×C3×D3C23×C418yes—
224C32×C4C23×C411yescharacteristic
225C32×C4C23×C416yes—
226C32×C4C2511yescharacteristic
227C3×Dic3C23×C418yes—
2282−1+4—34no—
2292−1+4—32no—
230H(5,𝔽2)—34no—
231H(5,𝔽2)—36no—
232C2×#41—38no—
233C2×#41—36no—
234C2×#41C2×C3×D311yescharacteristic
235#131—38no—
236#131—324no—
237#131—312no—
238#131—34no—
239#131—312no—
240#131—312no—
241#131C2×C3×D314yescomplemented
242#131C2×C3×D314yescomplemented
243C2×M16—36no—
244C2×M16—324no—
245C2×M16—312no—
246C2×M16C2×C3×D313yescomplemented
247C22×C8—312no—
248C22×C8C2×C3×D313yescomplemented
249C23×C3—23no—
250C23×C3—312no—
251C23×C3C2×C4×D313yes—
252C23×C3C22×C3×C413yes—
253C22×D3C22×C3×C4112yes—
254C3×Q8—24no—
255C3×Q8—312no—
256C3×Q8—34no—
257C3×Q8C2×C4×D314yes—
258C3×Q8C22×C3×C414yes—
259C3×D4—212no—
260C3×D4—324no—
261C3×D4—38no—
262C3×D4—312no—
263C3×D4—34no—
264C3×D4C2×C4×D3112yes—
265C3×D4C22×C3×C4112yes—
266C2×C3×C4—21no—
267C2×C3×C4—26no—
268C2×C3×C4—23no—
269C2×C3×C4—26no—
270C2×C3×C4—312no—
271C2×C3×C4—324no—
272C2×C3×C4—38no—
273C2×C3×C4C2×C4×D316yes—
274C2×C3×C4C2×C4×D313yes—
275C2×C3×C4C22×C3×C416yes—
276C2×C3×C4C22×C3×C413yes—
277C2×C3×C4C23×D316yes—
278C2×C3×C4C23×D311yescharacteristic
279C2×C3×C4C24×C316yes—
280C2×C3×C4C24×C311yescharacteristic
281#67C22×C3×C418yes—
282#67C22×C3×C4124yes—
283C2×Dic3C22×C3×C4112yes—
284D12C22×C3×C414yes—
285D12C22×C3×C4112yes—
286C4×D3C22×C3×C4124yes—
287C4×D3C24×C318yes—
288Dic6C22×C3×C414yes—
289Dic6C22×C3×C4112yes—
290C3×C8—21no—
291C3×C8—21no—
292C3×C8—26no—
293C3×C8—38no—
294C3×C8C23×D316yes—
295C3×C8C23×D311yescharacteristic
296C3×C8C23×D311yescharacteristic
297C3×C8C24×C311yescharacteristic
298C3×C8C24×C316yes—
299C3×C8C24×C311yescharacteristic
300C3⋊C8C24×C318yes—
301C2×C32—21no—
302C2×C32—26no—
303C2×C32C24×C411yescharacteristic
304C3×D3—28no—
305#41—38no—
306#41—324no—
307#41—312no—
308#41—34no—
309#41—312no—
310#41—312no—
311#41C3×C4×D314yes—
312#41C22×C3×D314yes—
313C2×Q8—38no—
314C2×Q8—36no—
315C2×Q8C3×C4×D311yescharacteristic
316C2×D4—312no—
317C2×D4—324no—
318C2×D4—36no—
319C2×D4C3×C4×D313yes—
320C22×C4—312no—
321C22×C4C22×C3×D313yes—
322M16—312no—
323M16—34no—
324M16—324no—
325M16—38no—
326M16C22×C3×D3112yes—
327C2×C8—312no—
328C2×C8—38no—
329C2×C8—324no—
330C2×C8C22×C3×D316yes—
331C2×C8C22×C3×D311yescharacteristic
332C2×C8C22×C3×D316yes—
333C2×C8C22×C3×D313yes—
334C22×C3—21no—
335C22×C3—26no—
336C22×C3—38no—
337C22×C3—43no—
338C22×C3—43no—
339C22×C3—46no—
340C22×C3—1212no—
341C22×C3C22×C4×D316yes—
342C22×C3C22×C4×D311yescharacteristic
343C22×C3C23×C3×C416yes—
344C22×C3C23×C3×C411yescharacteristic
345C2×D3—412no—
346C2×D3C23×C3×C418yes—
347C3×C4—21no—
348C3×C4—26no—
349C3×C4—21no—
350C3×C4—38no—
351C3×C4C22×C4×D316yes—
352C3×C4C22×C4×D311yescharacteristic
353C3×C4C23×C3×C411yescharacteristic
354C3×C4C23×C3×C416yes—
355C3×C4C24×D311yescharacteristic
356C3×C4C25×C311yescharacteristic
357Dic3C23×C3×C418yes—
358C32#357311yescharacteristic, complemented, Sylow
359C23—312no—
360C23C2×C3×C4×D313yes—
361Q8—312no—
362Q8—34no—
363Q8C2×C3×C4×D314yes—
364D4—324no—
365D4—38no—
366D4—312no—
367D4—34no—
368D4C2×C3×C4×D3112yes—
369C2×C4—312no—
370C2×C4—324no—
371C2×C4—38no—
372C2×C4C2×C3×C4×D316yes—
373C2×C4C2×C3×C4×D313yes—
374C2×C4C23×C3×D316yes—
375C2×C4C23×C3×D311yescharacteristic
376C8—38no—
377C8C23×C3×D311yescharacteristic
378C8C23×C3×D316yes—
379C8C23×C3×D311yescharacteristic
380C2×C3—22no—
381C2×C3—21no—
382C2×C3—26no—
383C2×C3—26no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—46no—
387C2×C3—68no—
388C2×C3C23×C4×D311yescharacteristic
389C2×C3C24×C3×C411yescharacteristic
390D3—28no—
391C22—38no—
392C22—43no—
393C22—1212no—
394C22C22×C3×C4×D316yes—
395C22C22×C3×C4×D311yescharacteristic
396C4—38no—
397C4C22×C3×C4×D316yes—
398C4C22×C3×C4×D311yescharacteristic
399C4C24×C3×D311yescharacteristic
400C3—21no—
401C3#8716011yescharacteristic, direct product factor
402C3C3×#357311yescharacteristic, complemented
403C2—21no—
404C2—26no—
405C2—68no—
406C2C23×C3×C4×D311yescharacteristic
subgroupG/H
centerC3×C8C23×D3
commutatorC2×C3C24×C3×C4
FrattiniC4C24×C3×D3
FittingC2×C32×#131C2
socleC2×C32C24×C4
prime radicalC3×#5964C2
cosocleC3×C4C25×C3
hypercenterC2×C3×#131D3
hypocenterC3C3×#3573
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C8 ◁ C2×C3×#131

Complements

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

NH ≅ G/Ntype
2. C2×C32×#131C2semidirect
3. C3×#5819C2semidirect
4. C3×#5750C2semidirect
5. C3×D3×#131C2semidirect
6. C3×#5745C2semidirect
7. C2×C3×#901C2semidirect
9. #87160C3direct (N × H)
17. C32×#131C22semidirect
18. C32×#131C22semidirect
19. C2×C32×M16C22semidirect
20. C22×C32×C8C22semidirect
21. C3×#948C22semidirect
22. C2×C3×#208C22semidirect
23. C3×#907C22semidirect
24. C3×D3×M16C22semidirect
25. C3×#901C22semidirect
26. C3×#901C22semidirect
27. C3×#901C22semidirect
28. C3×#901C22semidirect
29. C2×C3×ΓL(1,𝔽25)C22semidirect
30. C2×C3×D3×C8C22semidirect
39. C2×C3×#131D3semidirect
40. C2×C3×#131C2×C3semidirect
41. #5819C2×C3semidirect
42. #5750C2×C3semidirect
43. D3×#131C2×C3semidirect
44. #5745C2×C3semidirect
45. C2×#901C2×C3semidirect
99. C3×#131C2×D3semidirect
100. C3×#131C2×D3semidirect
101. C3×#131C22×C3semidirect
102. C3×#131C22×C3semidirect
107. C2×C3×M16C2×D3semidirect
108. C2×C3×M16C22×C3semidirect
111. C22×C3×C8C2×D3semidirect
112. C22×C3×C8C22×C3semidirect
113. #948C22×C3semidirect
114. C2×#208C22×C3semidirect
115. #907C22×C3semidirect
116. D3×M16C22×C3semidirect
117. #901C22×C3semidirect
118. #901C22×C3semidirect
119. #901C22×C3semidirect
120. #901C22×C3semidirect
121. C2×ΓL(1,𝔽25)C22×C3semidirect
122. C2×D3×C8C22×C3semidirect
150. C2×#131C3×D3semidirect
241. #131C2×C3×D3semidirect
242. #131C2×C3×D3semidirect
246. C2×M16C2×C3×D3semidirect
248. C22×C8C2×C3×D3semidirect
358. C32#3573semidirect
401. C3#87160direct (N × H)
402. C3C3×#3573semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C2×C32×#1312
C3×#58192
C3×#57502
C3×D3×#1312
C3×#57452
C2×C3×#9012
C3×#35733
#871603