Finite Group Databasebeta

← C3×#36165 #10654772

Subgroups — C3×#36165 #10654772

Order 768.

1854 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2859C211yesmaximal, characteristic, complemented
2C3×#2851C212yesmaximal, complemented
3C3×#2571C212yesmaximal, complemented
4C3×#2508C214yesmaximal, complemented
5C3×#2505C218yesmaximal, complemented
6C3×#2442C214yesmaximal, complemented
7C3×#2349C214yesmaximal
8C3×D4×C4⋊C4C212yesmaximal
9C3×C4×#125C211yesmaximal, characteristic, complemented
10C2×C3×#382C212yesmaximal, complemented
11C3×C42×D4C211yesmaximal, characteristic
12#36165C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×C3×#125C2212yes—
15C2×C3×C4×D4C2212yes—
16C2×C3×C4×D4C2214yes—
17C2×C3×C4×D4C2211yescharacteristic
18C22×C3×C4⋊C4C2214yes—
19C22×C3×C42C2212yes—
20C3×#400C2218yes—
21C3×#400C2218yes—
22C3×#398C2218yes—
23C3×#389C2212yes—
24C3×#389C2214yes—
25C3×#389C2214yes—
26C3×#389C2214yes—
27C3×#389C2214yes—
28C3×#389C2214yes—
29C3×#387C2214yes—
30C3×#387C2218yes—
31C3×#385C2218yes—
32C3×#385C2218yes—
33C3×#385C2218yes—
34C3×#384C2212yes—
35C3×#382C2214yes—
36C3×#382C2218yes—
37C3×#382C2214yes—
38C3×#382C2211yescharacteristic
39C3×#382C2212yes—
40C3×#380C2218yes—
41C3×#380C2214yes—
42C3×C4×C4⋊C4C2212yes—
43C3×C4×C4⋊C4C2214yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×C4×C4⋊C4C2212yes—
46C3×C4×C22⋊C4C2212yes—
47C3×C4×C22⋊C4C2214yes—
48C2×C3×#95C2218yes—
49C3×C43C2211yescharacteristic
50#2859C2×C311yescharacteristic, complemented
51#2851C2×C312yescomplemented
52#2571C2×C312yescomplemented
53#2508C2×C314yescomplemented
54#2505C2×C318yescomplemented
55#2442C2×C314yescomplemented
56#2349C2×C314yes—
57D4×C4⋊C4C2×C312yes—
58C4×#125C2×C311yescharacteristic, complemented
59C2×#382C2×C312yescomplemented
60C42×D4C2×C311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4C2314yes—
63C23×C3×C4C2312yes—
64C23×C3×C4C2318yes—
65C3×#125—24no—
66C3×#125—44no—
67C3×C4×D4—28no—
68C3×C4×D4—216no—
69C3×C4×D4Q818yes—
70C2×C3×C4⋊C4—48no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2314yes—
75C2×C3×C4⋊C4C2314yes—
76C2×C3×C4⋊C4C2314yes—
77C2×C3×C4⋊C4C2314yes—
78C2×C3×C4⋊C4C2318yes—
79C2×C3×C4⋊C4C2318yes—
80C2×C3×C4⋊C4C2318yes—
81C2×C3×C4⋊C4C2314yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4C2314yes—
84C2×C3×C22⋊C4C2318yes—
85C2×C3×C22⋊C4C2312yes—
86C2×C3×C42—28no—
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2312yes—
89C2×C3×C42C2314yes—
90C2×C3×C42C2312yes—
91C2×C3×C42C2314yes—
92C2×C3×C42C2312yes—
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C3×#95C2314yes—
96C3×#95C2318yes—
97C3×#95C2318yes—
98C3×#95C2318yes—
99C3×#95C2314yes—
100C3×#95C2312yes—
101C3×#95C2314yes—
102C3×#95C2314yes—
103C3×#95C2318yes—
104C3×#95C2318yes—
105C2×#125C22×C311yescharacteristic
106C2×#125C22×C312yes—
107C2×C4×D4C22×C314yes—
108C2×C4×D4C22×C312yes—
109C2×C4×D4C22×C311yescharacteristic
110C22×C4⋊C4C22×C314yes—
111C22×C42C22×C312yes—
112#400C22×C318yes—
113#400C22×C318yes—
114#398C22×C318yes—
115#389C22×C312yes—
116#389C22×C314yes—
117#389C22×C314yes—
118#389C22×C314yes—
119#389C22×C314yes—
120#389C22×C314yes—
121#387C22×C314yes—
122#387C22×C318yes—
123#385C22×C318yes—
124#385C22×C318yes—
125#385C22×C318yes—
126#384C22×C312yes—
127#382C22×C311yescharacteristic
128#382C22×C314yes—
129#382C22×C314yes—
130#382C22×C312yes—
131#382C22×C318yes—
132#380C22×C314yes—
133#380C22×C318yes—
134C4×C4⋊C4C22×C312yes—
135C4×C4⋊C4C22×C314yes—
136C4×C4⋊C4C22×C311yescharacteristic
137C4×C4⋊C4C22×C312yes—
138C4×C22⋊C4C22×C312yes—
139C4×C22⋊C4C22×C314yes—
140C2×#95C22×C318yes—
141C43C22×C311yescharacteristic
142C24×C3C2412yes—
143C2×C3×D4C2×Q814yes—
144C2×C3×D4#4118yes—
145C22×C3×C4—24no—
146C22×C3×C4—24no—
147C22×C3×C4—28no—
148C22×C3×C4—216no—
149C22×C3×C4—28no—
150C22×C3×C4—44no—
151C22×C3×C4—48no—
152C22×C3×C4—48no—
153C22×C3×C4C2×Q818yes—
154C22×C3×C4C2414yes—
155C22×C3×C4C2414yes—
156C22×C3×C4C2414yes—
157C22×C3×C4C2418yes—
158C22×C3×C4C2412yes—
159C22×C3×C4C2411yescharacteristic
160C22×C3×C4C2412yes—
161C22×C3×C4C2412yes—
162C22×C3×C4C2411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C3×C4⋊C4—24no—
165C3×C4⋊C4—28no—
166C3×C4⋊C4—28no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—28no—
170C3×C4⋊C4—48no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4—44no—
173C3×C4⋊C4—48no—
174C3×C4⋊C4—48no—
175C3×C4⋊C4C2×Q814yes—
176C3×C22⋊C4—28no—
177C3×C22⋊C4—216no—
178C3×C22⋊C4C2×Q818yes—
179C3×C42—28no—
180C3×C42—24no—
181C3×C42—24no—
182C3×C42—28no—
183C3×C42—22no—
184C3×C42—42no—
185C3×C42C2×Q814yes—
186C3×C42#4114yes—
187C22×D4C23×C311yescharacteristic
188C23×C4C23×C314yes—
189C23×C4C23×C312yes—
190C23×C4C23×C318yes—
191#125—24no—
192#125—44no—
193C4×D4—28no—
194C4×D4—216no—
195C4×D4C3×Q818yes—
196C2×C4⋊C4—48no—
197C2×C4⋊C4C23×C312yes—
198C2×C4⋊C4C23×C314yes—
199C2×C4⋊C4C23×C312yes—
200C2×C4⋊C4C23×C314yes—
201C2×C4⋊C4C23×C312yes—
202C2×C4⋊C4C23×C314yes—
203C2×C4⋊C4C23×C318yes—
204C2×C4⋊C4C23×C318yes—
205C2×C4⋊C4C23×C314yes—
206C2×C4⋊C4C23×C318yes—
207C2×C4⋊C4C23×C314yes—
208C2×C4⋊C4C23×C311yescharacteristic
209C2×C22⋊C4C23×C314yes—
210C2×C22⋊C4C23×C318yes—
211C2×C22⋊C4C23×C312yes—
212C2×C42—28no—
213C2×C42C23×C312yes—
214C2×C42C23×C314yes—
215C2×C42C23×C311yescharacteristic
216C2×C42C23×C314yes—
217C2×C42C23×C312yes—
218C2×C42C23×C312yes—
219C2×C42C23×C311yescharacteristic
220C2×C42C23×C311yescharacteristic
221#95C23×C314yes—
222#95C23×C318yes—
223#95C23×C318yes—
224#95C23×C318yes—
225#95C23×C314yes—
226#95C23×C312yes—
227#95C23×C314yes—
228#95C23×C314yes—
229#95C23×C318yes—
230#95C23×C318yes—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—24no—
234C23×C3C22×Q814yes—
235C23×C3C2×#4118yes—
236C23×C3C2511yescharacteristic
237C3×D4#125116yescomplemented
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—216no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—28no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—28no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—28no—
258C2×C3×C4—24no—
259C2×C3×C4—416no—
260C2×C3×C4—816no—
261C2×C3×C4C22×Q812yes—
262C2×C3×C4C22×Q814yes—
263C2×C3×C4C22×Q812yes—
264C2×C3×C4C22×Q812yes—
265C2×C3×C4C2×#4114yes—
266C2×C3×C4C2×#4114yes—
267C2×C3×C4C2×#4112yes—
268C2×C3×C4H(5,𝔽2)12yes—
269C24C24×C312yes—
270C2×D4C2×C3×Q814yes—
271C2×D4C3×#4118yes—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—216no—
276C22×C4—28no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4C2×C3×Q818yes—
281C22×C4C24×C314yes—
282C22×C4C24×C314yes—
283C22×C4C24×C314yes—
284C22×C4C24×C318yes—
285C22×C4C24×C312yes—
286C22×C4C24×C311yescharacteristic
287C22×C4C24×C312yes—
288C22×C4C24×C312yes—
289C22×C4C24×C311yescharacteristic
290C22×C4C24×C311yescharacteristic
291C4⋊C4—24no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—28no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—48no—
302C4⋊C4C2×C3×Q814yes—
303C22⋊C4—28no—
304C22⋊C4—216no—
305C22⋊C4C2×C3×Q818yes—
306C42—28no—
307C42—24no—
308C42—24no—
309C42—28no—
310C42—22no—
311C42—42no—
312C42C2×C3×Q814yes—
313C42C3×#4114yes—
314C22×C3—24no—
315C22×C3—28no—
316C22×C3—24no—
317C22×C3—28no—
318C22×C3C2×#12518yes—
319C22×C3C23×Q811yescharacteristic
320C22×C3C22×#4112yes—
321C22×C3C22×#4111yescharacteristic
322C22×C3C2×H(5,𝔽2)11yescharacteristic
323C22×C3#58512yes—
324C3×C4—24no—
325C3×C4—24no—
326C3×C4—22no—
327C3×C4—44no—
328C3×C4—44no—
329C3×C4—42no—
330C3×C4—48no—
331C3×C4—44no—
332C3×C4C2×#12514yes—
333C3×C4#55414yes—
334C23—22no—
335C23—22no—
336C23—24no—
337C23C22×C3×Q814yes—
338C23C2×C3×#4118yes—
339C23C25×C311yescharacteristic
340D4C3×#125116yescomplemented
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—216no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—28no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—24no—
362C2×C4—416no—
363C2×C4—816no—
364C2×C4C22×C3×Q812yes—
365C2×C4C22×C3×Q814yes—
366C2×C4C22×C3×Q812yes—
367C2×C4C22×C3×Q812yes—
368C2×C4C2×C3×#4114yes—
369C2×C4C2×C3×#4114yes—
370C2×C4C2×C3×#4112yes—
371C2×C4C3×H(5,𝔽2)12yes—
372C2×C3—28no—
373C2×C3C22×#12511yescharacteristic
374C2×C3C2×#55411yescharacteristic
375C2×C3#347512yes—
376C2×C3#350411yescharacteristic
377C2×C3#351412yes—
378C22—24no—
379C22—28no—
380C22—24no—
381C22—28no—
382C22C2×C3×#12518yes—
383C22C23×C3×Q811yescharacteristic
384C22C22×C3×#4112yes—
385C22C22×C3×#4111yescharacteristic
386C22C2×C3×H(5,𝔽2)11yescharacteristic
387C22C3×#58512yes—
388C4—24no—
389C4—24no—
390C4—22no—
391C4—44no—
392C4—44no—
393C4—44no—
394C4—48no—
395C4—42no—
396C4C2×C3×#12514yes—
397C4C3×#55414yes—
398C3#3616511yescharacteristic, direct product factor, Sylow
399C2—28no—
400C2C22×C3×#12511yescharacteristic
401C2C2×C3×#55411yescharacteristic
402C2C3×#347512yes—
403C2C3×#350411yescharacteristic
404C2C3×#351412yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC22×C3×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2859C2semidirect
2. C3×#2851C2semidirect
3. C3×#2571C2semidirect
4. C3×#2508C2semidirect
5. C3×#2505C2semidirect
6. C3×#2442C2semidirect
9. C3×C4×#125C2semidirect
10. C2×C3×#382C2semidirect
12. #36165C3direct (N × H)
50. #2859C2×C3semidirect
51. #2851C2×C3semidirect
52. #2571C2×C3semidirect
53. #2508C2×C3semidirect
54. #2505C2×C3semidirect
55. #2442C2×C3semidirect
58. C4×#125C2×C3semidirect
59. C2×#382C2×C3semidirect
237. C3×D4#125semidirect
340. D4C3×#125semidirect
398. C3#36165direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28592
C3×#28512
C3×#25712
C3×#25082
C3×#25052
C3×#24422
C3×#23492
C3×D4×C4⋊C42
C3×C4×#1252
C2×C3×#3822
C3×C42×D42
#361653