Finite Group Databasebeta

← C3×#87085 #11910361

Subgroups — C3×#87085 #11910361

Order 1152.

2246 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2C3×#5684C216yesmaximal, complemented
3C3×#5680C216yesmaximal, complemented
4C3×#5672C213yesmaximal, complemented
5C3×#5668C213yesmaximal, complemented
6C3×#5595C216yesmaximal, complemented
7C3×#5586C216yesmaximal, complemented
8C3×#3529—31nomaximal
9#87085C311yesmaximal, characteristic, direct product factor
10C23×C3×Dic3C2212yescomplemented
11C2×C3×#239C2211yescharacteristic, complemented
12C2×C3×#237C2213yes—
13C32×#124C2213yes—
14C32×#124C2216yes—
15C32×#120C2216yes—
16C3×#953C2216yescomplemented
17C3×#946C2216yescomplemented
18C3×Q8×Dic3C2212yescomplemented
19C3×#944C2213yescomplemented
20C3×#940C2213yes—
21C3×#939C22112yes—
22C3×#938C2216yes—
23C3×#935C2216yes—
24C3×D4×Dic3C2216yes—
25C3×#886C2216yescomplemented
26C3×#884C22112yescomplemented
27C3×#883C22112yescomplemented
28C3×#67⋊C4C22112yescomplemented
29C3×#879C22112yescomplemented
30C3×#878C22112yescomplemented
31C3×#877C2216yescomplemented
32C3×#875C2213yescomplemented
33C3×C42×D3C2213yescomplemented
34C3×#870C2216yescomplemented
35C3×#563—33no—
36C3×#562—36no—
37C3×H(3,𝔽4)—21no—
38C3×H(3,𝔽4)D311yescharacteristic, complemented
39C3×H(3,𝔽4)C2×C311yescharacteristic, complemented
40C3×#560—36no—
41C3×#532—33no—
42C3×#529—36no—
43C3×#525—36no—
44#5684C2×C316yescomplemented
45#5680C2×C316yescomplemented
46#5672C2×C313yescomplemented
47#5668C2×C313yescomplemented
48#5595C2×C316yescomplemented
49#5586C2×C316yescomplemented
50C24×C32C2312yes—
51C2×C32×Q8C2311yescharacteristic
52C2×C32×Q8C2312yes—
53C2×C32×D4C2316yes—
54C2×C32×D4C2313yes—
55C2×C3×#67C23112yes—
56C2×C3×#67C2316yes—
57C22×C3×Dic3—44no—
58C22×C3×Dic3C2316yes—
59C22×C3×Dic3C2316yes—
60C3×#239—42no—
61C3×#237—46no—
62C2×C3×D12C2313yes—
63C2×C3×C4×D3C2313yes—
64C2×C3×C4×D3C2316yes—
65C2×C3×Dic6C2313yes—
66C32×C22⋊C4C2316yes—
67C32×C22⋊C4C23112yes—
68C32×C42C2313yes—
69C3×#217C2316yes—
70C3×#217C23112yes—
71C3×(C2×D3)⋊C4C23112yes—
72C3×C4⋊Dic3C2316yes—
73C3×C4⋊Dic3C2316yes—
74C3×Dic3⋊C4C23112yes—
75C3×Dic3⋊C4C23112yes—
76C3×C4×Dic3C2316yes—
77C3×C4×Dic3C2313yes—
78#3529—31noSylow
79C2×C3×#41—31no—
80C2×C3×#41—33no—
81C23×C3×C4—32no—
82C23×Dic3C22×C312yescomplemented
83C2×#239C22×C311yescharacteristic, complemented
84C2×#237C22×C313yes—
85C3×C4⋊Q8—33no—
86C3×#127—33no—
87C3×#126—312no—
88C3×#125—36no—
89C3×#124—26no—
90C3×#124—23no—
91C3×#124—33no—
92C3×#124C2×D316yes—
93C3×#124C2×D313yes—
94C3×#124C22×C313yes—
95C3×#124C22×C316yes—
96C3×#123—36no—
97C3×#123—312no—
98C3×#123—36no—
99C3×C22⋊Q8—36no—
100C3×C22⋊Q8—312no—
101C3×#121—36no—
102C3×#121—312no—
103C3×#121—312no—
104C3×#120—26no—
105C3×#120—36no—
106C3×#120C2×D316yes—
107C3×#120C22×C316yes—
108C3×C4×Q8—32no—
109C3×C4×D4—36no—
110C3×C4×D4—312no—
111C3×#117—36no—
112C2×C3×C42—33no—
113#953C22×C316yescomplemented
114#946C22×C316yescomplemented
115Q8×Dic3C22×C312yescomplemented
116#944C22×C313yescomplemented
117#940C22×C313yes—
118#939C22×C3112yes—
119#938C22×C316yes—
120#935C22×C316yes—
121D4×Dic3C22×C316yes—
122#886C22×C316yescomplemented
123#884C22×C3112yescomplemented
124#883C22×C3112yescomplemented
125#67⋊C4C22×C3112yescomplemented
126#879C22×C3112yescomplemented
127#878C22×C3112yescomplemented
128#877C22×C316yescomplemented
129#875C22×C313yescomplemented
130C42×D3C22×C313yescomplemented
131#870C22×C316yescomplemented
132C23×C32—44no—
133C23×C32—42no—
134C23×C32C2416yes—
135C22×C3×D3C2413yes—
136C32×Q8—41no—
137C32×Q8—42no—
138C32×D4—43no—
139C32×D4—46no—
140C2×C32×C4C2413yes—
141C2×C32×C4C2416yes—
142C3×#67—46no—
143C3×#67—412no—
144C2×C3×Dic3—212no—
145C2×C3×Dic3—46no—
146C2×C3×Dic3C2416yes—
147C2×C3×Dic3C2416yes—
148C2×C3×Dic3C2411yescharacteristic
149C3×D12—43no—
150C3×C4×D3—26no—
151C3×C4×D3—46no—
152C3×Dic6—43no—
153#563—33no—
154#562—36no—
155H(3,𝔽4)C3×D311yescharacteristic, complemented
156#560—36no—
157#532—33no—
158#529—36no—
159#525—36no—
160C24×C3—22no—
161C24×C3C22×D312yes—
162C24×C3C23×C312yes—
163C3×#41—126no—
164C3×#41—122no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—33no—
168C2×C3×Q8C22×D312yes—
169C2×C3×Q8C22×D311yescharacteristic
170C2×C3×Q8C23×C311yescharacteristic
171C2×C3×Q8C23×C312yes—
172C2×C3×D4—26no—
173C2×C3×D4—23no—
174C2×C3×D4—312no—
175C2×C3×D4—36no—
176C2×C3×D4—33no—
177C2×C3×D4C22×D316yes—
178C2×C3×D4C22×D313yes—
179C2×C3×D4C23×C316yes—
180C2×C3×D4C23×C313yes—
181C22×C3×C4—36no—
182C22×C3×C4—36no—
183C22×C3×C4—33no—
184C22×C3×C4—36no—
185C22×C3×C4—124no—
186C2×#67C23×C3112yes—
187C2×#67C23×C316yes—
188C22×Dic3—44no—
189C22×Dic3C23×C316yes—
190C22×Dic3C23×C316yes—
191#239—42no—
192#237—46no—
193C2×D12C23×C313yes—
194C2×C4×D3C23×C313yes—
195C2×C4×D3C23×C316yes—
196C2×Dic6C23×C313yes—
197C3×C4⋊C4—312no—
198C3×C4⋊C4—312no—
199C3×C4⋊C4—36no—
200C3×C4⋊C4—36no—
201C3×C22⋊C4—26no—
202C3×C22⋊C4—212no—
203C3×C22⋊C4—36no—
204C3×C22⋊C4—312no—
205C3×C22⋊C4—312no—
206C3×C22⋊C4C22×D316yes—
207C3×C22⋊C4C22×D3112yes—
208C3×C22⋊C4C23×C316yes—
209C3×C22⋊C4C23×C3112yes—
210C3×C42—23no—
211C3×C42—36no—
212C3×C42—33no—
213C3×C42C22×D313yes—
214C3×C42C23×C313yes—
215#217C23×C316yes—
216#217C23×C3112yes—
217(C2×D3)⋊C4C23×C3112yes—
218C4⋊Dic3C23×C316yes—
219C4⋊Dic3C23×C316yes—
220Dic3⋊C4C23×C3112yes—
221Dic3⋊C4C23×C3112yes—
222C4×Dic3C23×C316yes—
223C4×Dic3C23×C313yes—
224C22×C32—26no—
225C22×C32—212no—
226C22×C32—48no—
227C22×C32C2511yescharacteristic
228C2×C3×D3—23no—
229C2×C3×D3—26no—
230C32×C4—23no—
231C32×C4—26no—
232C3×Dic3—26no—
233C3×Dic3—26no—
234C3×Dic3H(5,𝔽2)12yes—
235C2×#41—31no—
236C2×#41—33no—
237C23×C4—32no—
238C4⋊Q8—33no—
239#127—33no—
240#126—312no—
241#125—36no—
242#124—33no—
243#124C2×C3×D316yes—
244#124C2×C3×D313yes—
245#123—36no—
246#123—36no—
247#123—312no—
248C22⋊Q8—36no—
249C22⋊Q8—312no—
250#121—312no—
251#121—36no—
252#121—312no—
253#120—36no—
254#120C2×C3×D316yes—
255C4×Q8—32no—
256C4×D4—36no—
257C4×D4—312no—
258#117—36no—
259C2×C42—33no—
260C23×C3—26no—
261C23×C3—33no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—44no—
265C23×C3—44no—
266C23×C3—82no—
267C23×C3—84no—
268C23×C3C23×D316yes—
269C23×C3C24×C316yes—
270C22×D3C24×C313yes—
271C3×Q8—41no—
272C3×Q8—42no—
273C3×Q8—41no—
274C3×Q8—42no—
275C3×Q8—81no—
276C3×Q8—82no—
277C3×Q8—123no—
278C3×D4—43no—
279C3×D4—46no—
280C3×D4—43no—
281C3×D4—46no—
282C3×D4—83no—
283C3×D4—86no—
284C3×D4—123no—
285C3×D4—1212no—
286C3×D4—126no—
287C2×C3×C4—26no—
288C2×C3×C4—23no—
289C2×C3×C4—31no—
290C2×C3×C4—36no—
291C2×C3×C4—36no—
292C2×C3×C4—66no—
293C2×C3×C4—612no—
294C2×C3×C4—126no—
295C2×C3×C4—126no—
296C2×C3×C4C23×D316yes—
297C2×C3×C4C23×D313yes—
298C2×C3×C4C24×C313yes—
299C2×C3×C4C24×C316yes—
300#67—46no—
301#67—412no—
302C2×Dic3—212no—
303C2×Dic3—46no—
304C2×Dic3C24×C316yes—
305C2×Dic3C24×C316yes—
306C2×Dic3C24×C311yescharacteristic
307D12—43no—
308C4×D3—26no—
309C4×D3—46no—
310Dic6—43no—
311C2×C32—46no—
312C2×C32C2×H(5,𝔽2)11yescharacteristic
313C2×C32#58512yes—
314C3×D3—43no—
315C24C22×C3×D312yes—
316#41—126no—
317#41—122no—
318C2×Q8—33no—
319C2×Q8C22×C3×D312yes—
320C2×Q8C22×C3×D311yescharacteristic
321C2×D4—312no—
322C2×D4—36no—
323C2×D4—33no—
324C2×D4C22×C3×D316yes—
325C2×D4C22×C3×D313yes—
326C22×C4—36no—
327C22×C4—36no—
328C22×C4—33no—
329C22×C4—36no—
330C22×C4—124no—
331C4⋊C4—312no—
332C4⋊C4—312no—
333C4⋊C4—36no—
334C4⋊C4—36no—
335C22⋊C4—36no—
336C22⋊C4—312no—
337C22⋊C4—312no—
338C22⋊C4C22×C3×D316yes—
339C22⋊C4C22×C3×D3112yes—
340C42—36no—
341C42—33no—
342C42C22×C3×D313yes—
343C22×C3—26no—
344C22×C3—212no—
345C22×C3—26no—
346C22×C3—212no—
347C22×C3—21no—
348C22×C3—46no—
349C22×C3—412no—
350C22×C3—48no—
351C22×C3—48no—
352C22×C3—63no—
353C22×C3—66no—
354C22×C3—88no—
355C22×C3C24×D311yescharacteristic
356C22×C3C25×C311yescharacteristic
357C2×D3—23no—
358C2×D3—26no—
359C3×C4—23no—
360C3×C4—26no—
361C3×C4—23no—
362C3×C4—26no—
363C3×C4—32no—
364C3×C4—43no—
365C3×C4—46no—
366C3×C4—66no—
367C3×C4—66no—
368Dic3—26no—
369Dic3—26no—
370Dic3C3×H(5,𝔽2)12yes—
371C32#352911yescharacteristic, complemented, Sylow
372C23—33no—
373C23—42no—
374C23—44no—
375C23C23×C3×D316yes—
376Q8—41no—
377Q8—42no—
378Q8—123no—
379D4—43no—
380D4—46no—
381D4—123no—
382D4—126no—
383D4—1212no—
384C2×C4—31no—
385C2×C4—36no—
386C2×C4—36no—
387C2×C4—66no—
388C2×C4—612no—
389C2×C4—126no—
390C2×C4—126no—
391C2×C4C23×C3×D316yes—
392C2×C4C23×C3×D313yes—
393C2×C3—21no—
394C2×C3—22no—
395C2×C3—46no—
396C2×C3—46no—
397C2×C3—86no—
398C2×C3—123no—
399C2×C3D3×H(5,𝔽2)11yescharacteristic
400C2×C3#596612yes—
401C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
402C2×C3C3×#58512yes—
403D3—43no—
404C22—26no—
405C22—212no—
406C22—48no—
407C22—63no—
408C22—66no—
409C22C24×C3×D311yescharacteristic
410C4—23no—
411C4—26no—
412C4—32no—
413C4—66no—
414C4—66no—
415C3—21no—
416C3#8708511yescharacteristic, direct product factor
417C3C3×#352911yescharacteristic, complemented
418C2—46no—
419C2—123no—
420C2C3×D3×H(5,𝔽2)11yescharacteristic
421C2C3×#596612yes—
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×H(3,𝔽4)C2
socleC22×C32C25
hypercenterC3×H(3,𝔽4)D3
hypocenterC3C3×#3529
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C32×H(3,𝔽4)C2semidirect
2. C3×#5684C2semidirect
3. C3×#5680C2semidirect
4. C3×#5672C2semidirect
5. C3×#5668C2semidirect
6. C3×#5595C2semidirect
7. C3×#5586C2semidirect
9. #87085C3direct (N × H)
10. C23×C3×Dic3C22semidirect
11. C2×C3×#239C22semidirect
16. C3×#953C22semidirect
17. C3×#946C22semidirect
18. C3×Q8×Dic3C22semidirect
19. C3×#944C22semidirect
25. C3×#886C22semidirect
26. C3×#884C22semidirect
27. C3×#883C22semidirect
28. C3×#67⋊C4C22semidirect
29. C3×#879C22semidirect
30. C3×#878C22semidirect
31. C3×#877C22semidirect
32. C3×#875C22semidirect
33. C3×C42×D3C22semidirect
34. C3×#870C22semidirect
38. C3×H(3,𝔽4)D3semidirect
39. C3×H(3,𝔽4)C2×C3semidirect
44. #5684C2×C3semidirect
45. #5680C2×C3semidirect
46. #5672C2×C3semidirect
47. #5668C2×C3semidirect
48. #5595C2×C3semidirect
49. #5586C2×C3semidirect
82. C23×Dic3C22×C3semidirect
83. C2×#239C22×C3semidirect
113. #953C22×C3semidirect
114. #946C22×C3semidirect
115. Q8×Dic3C22×C3semidirect
116. #944C22×C3semidirect
122. #886C22×C3semidirect
123. #884C22×C3semidirect
124. #883C22×C3semidirect
125. #67⋊C4C22×C3semidirect
126. #879C22×C3semidirect
127. #878C22×C3semidirect
128. #877C22×C3semidirect
129. #875C22×C3semidirect
130. C42×D3C22×C3semidirect
131. #870C22×C3semidirect
155. H(3,𝔽4)C3×D3semidirect
371. C32#3529semidirect
416. C3#87085direct (N × H)
417. C3C3×#3529semidirect

Maximal subgroups

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

subgroupindex
C32×H(3,𝔽4)2
C3×#56842
C3×#56802
C3×#56722
C3×#56682
C3×#55952
C3×#55862
C3×#35293
#870853