Finite Group Databasebeta

← C3×#35294 #10653901

Subgroups — C3×#35294 #10653901

Order 768.

1722 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2868C211yesmaximal, characteristic, complemented
2C3×#2859C212yesmaximal, complemented
3C3×#2859C211yesmaximal, characteristic, complemented
4C3×#2629C212yesmaximal
5C3×#2628C212yesmaximal
6C3×#2570C214yesmaximal, complemented
7C3×#2550C214yesmaximal, complemented
8C3×#2441C214yesmaximal
9C3×#2439C214yesmaximal
10C3×#2331C212yesmaximal, complemented
11C3×#2330C212yesmaximal, complemented
12C3×#2285C212yesmaximal
13C3×C4×#117C211yesmaximal, characteristic
14#35294C311yesmaximal, characteristic, direct product factor, Sylow
15C2×C3×C4⋊Q8C2212yes—
16C2×C3×C4⋊Q8C2214yes—
17C2×C3×#125C2212yes—
18C2×C3×#125C2214yes—
19C2×C3×#117C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C2×C3×#117C2214yes—
22C22×C3×C42C2211yescharacteristic
23C3×#400C2214yes—
24C3×#399C2214yes—
25C3×#398C2218yes—
26C3×#396C2218yes—
27C3×#395C2214yes—
28C3×#393C2214yes—
29C3×#391C2218yes—
30C3×#389C2214yes—
31C3×#389C2214yes—
32C3×#389C2214yes—
33C3×#389C2214yes—
34C3×#389C2218yes—
35C3×#389C2214yes—
36C3×#389C2214yes—
37C3×#387C2218yes—
38C3×#387C2218yes—
39C3×#384C2212yes—
40C3×#384C2214yes—
41C3×#384C2212yes—
42C3×#384C2212yes—
43C3×#382C2212yes—
44C3×#382C2214yes—
45C3×#382C2214yes—
46C3×#380C2214yes—
47C3×#380C2214yes—
48C3×C4×C4⋊C4C2212yes—
49C3×C4×C4⋊C4C2212yes—
50C3×C4×C4⋊C4C2214yes—
51C3×C4×C4⋊C4C2214yes—
52C3×C4×C22⋊C4C2212yes—
53C3×C4×C22⋊C4C2212yes—
54C3×#376C2211yescharacteristic
55C3×#376C2211yescharacteristic
56C3×C43C2211yescharacteristic
57C3×C43C2211yescharacteristic
58#2868C2×C311yescharacteristic, complemented
59#2859C2×C311yescharacteristic, complemented
60#2859C2×C312yescomplemented
61#2629C2×C312yes—
62#2628C2×C312yes—
63#2570C2×C314yescomplemented
64#2550C2×C314yescomplemented
65#2441C2×C314yes—
66#2439C2×C314yes—
67#2331C2×C312yescomplemented
68#2330C2×C312yescomplemented
69#2285C2×C312yes—
70C4×#117C2×C311yescharacteristic
71C22×C3×Q8C2314yes—
72C23×C3×C4C2312yes—
73C23×C3×C4C2311yescharacteristic
74C3×C4⋊Q8—28no—
75C3×C4⋊Q8—48no—
76C3×#125—28no—
77C3×#125—48no—
78C3×#117—24no—
79C3×#117—24no—
80C3×#117—48no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2314yes—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C4⋊C4C2314yes—
90C2×C3×C4⋊C4C2314yes—
91C2×C3×C4⋊C4C2318yes—
92C2×C3×C4⋊C4C2318yes—
93C2×C3×C4⋊C4C2318yes—
94C2×C3×C4⋊C4C2318yes—
95C2×C3×C22⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4C2312yes—
97C2×C3×C22⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4C2318yes—
99C2×C3×C42Q818yescomplemented
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2312yes—
103C2×C3×C42C2312yes—
104C2×C3×C42C2312yes—
105C2×C3×C42C2312yes—
106C2×C3×C42C2318yes—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2312yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C3×#95C2312yes—
113C3×#95C2312yes—
114C3×#95C2312yes—
115C3×#95C2312yes—
116C3×#95C2318yes—
117C3×#95C2318yes—
118C3×#95C2318yes—
119C3×#95C2314yes—
120C3×#95C2318yes—
121C3×#95C2314yes—
122C2×C4⋊Q8C22×C312yes—
123C2×C4⋊Q8C22×C314yes—
124C2×#125C22×C312yes—
125C2×#125C22×C314yes—
126C2×#117C22×C311yescharacteristic
127C2×#117C22×C311yescharacteristic
128C2×#117C22×C314yes—
129C22×C42C22×C311yescharacteristic
130#400C22×C314yes—
131#399C22×C314yes—
132#398C22×C318yes—
133#396C22×C318yes—
134#395C22×C314yes—
135#393C22×C314yes—
136#391C22×C318yes—
137#389C22×C314yes—
138#389C22×C314yes—
139#389C22×C314yes—
140#389C22×C318yes—
141#389C22×C314yes—
142#389C22×C314yes—
143#389C22×C314yes—
144#387C22×C318yes—
145#387C22×C318yes—
146#384C22×C314yes—
147#384C22×C312yes—
148#384C22×C312yes—
149#384C22×C312yes—
150#382C22×C314yes—
151#382C22×C314yes—
152#382C22×C312yes—
153#380C22×C314yes—
154#380C22×C314yes—
155C4×C4⋊C4C22×C312yes—
156C4×C4⋊C4C22×C314yes—
157C4×C4⋊C4C22×C314yes—
158C4×C4⋊C4C22×C312yes—
159C4×C22⋊C4C22×C312yes—
160C4×C22⋊C4C22×C312yes—
161#376C22×C311yescharacteristic
162#376C22×C311yescharacteristic
163C43C22×C311yescharacteristic
164C43C22×C311yescharacteristic
165C24×C3C2411yescharacteristic
166C2×C3×Q8—28no—
167C2×C3×Q8—48no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4C2×Q814yes—
173C22×C3×C4C2×Q818yes—
174C22×C3×C4C2411yescharacteristic
175C22×C3×C4C2412yes—
176C22×C3×C4C2418yes—
177C22×C3×C4C2418yes—
178C22×C3×C4C2411yescharacteristic
179C22×C3×C4C2412yes—
180C22×C3×C4C2411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C22×C3×C4C2412yes—
184C22×C3×C4C2411yescharacteristic
185C22×C3×C4C2412yes—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—216no—
190C3×C4⋊C4—28no—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—28no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—48no—
198C3×C4⋊C4—48no—
199C3×C4⋊C4—48no—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—48no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—24no—
207C3×C42—24no—
208C3×C42—24no—
209C3×C42—24no—
210C3×C42—48no—
211C3×C42C2×Q814yes—
212C3×C42C2×Q818yes—
213C3×C42C2×Q814yes—
214C3×C42#4114yes—
215C3×C42#4114yes—
216C22×Q8C23×C314yes—
217C23×C4C23×C312yes—
218C23×C4C23×C311yescharacteristic
219C4⋊Q8—28no—
220C4⋊Q8—48no—
221#125—28no—
222#125—48no—
223#117—24no—
224#117—24no—
225#117—48no—
226C2×C4⋊C4C23×C314yes—
227C2×C4⋊C4C23×C314yes—
228C2×C4⋊C4C23×C311yescharacteristic
229C2×C4⋊C4C23×C312yes—
230C2×C4⋊C4C23×C311yescharacteristic
231C2×C4⋊C4C23×C314yes—
232C2×C4⋊C4C23×C314yes—
233C2×C4⋊C4C23×C318yes—
234C2×C4⋊C4C23×C314yes—
235C2×C4⋊C4C23×C314yes—
236C2×C4⋊C4C23×C314yes—
237C2×C4⋊C4C23×C318yes—
238C2×C4⋊C4C23×C318yes—
239C2×C4⋊C4C23×C318yes—
240C2×C22⋊C4C23×C311yescharacteristic
241C2×C22⋊C4C23×C312yes—
242C2×C22⋊C4C23×C311yescharacteristic
243C2×C22⋊C4C23×C318yes—
244C2×C42C3×Q818yescomplemented
245C2×C42C23×C311yescharacteristic
246C2×C42C23×C311yescharacteristic
247C2×C42C23×C312yes—
248C2×C42C23×C312yes—
249C2×C42C23×C312yes—
250C2×C42C23×C312yes—
251C2×C42C23×C318yes—
252C2×C42C23×C311yescharacteristic
253C2×C42C23×C311yescharacteristic
254C2×C42C23×C312yes—
255C2×C42C23×C311yescharacteristic
256C2×C42C23×C311yescharacteristic
257#95C23×C312yes—
258#95C23×C312yes—
259#95C23×C312yes—
260#95C23×C312yes—
261#95C23×C318yes—
262#95C23×C318yes—
263#95C23×C318yes—
264#95C23×C314yes—
265#95C23×C318yes—
266#95C23×C314yes—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3C22×Q812yes—
272C23×C3C2511yescharacteristic
273C3×Q8—88no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—28no—
283C2×C3×C4—216no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—22no—
287C2×C3×C4—28no—
288C2×C3×C4—28no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—28no—
292C2×C3×C4—28no—
293C2×C3×C4—28no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—28no—
297C2×C3×C4C22×Q812yes—
298C2×C3×C4C22×Q814yes—
299C2×C3×C4C22×Q812yes—
300C2×C3×C4C22×Q814yes—
301C2×C3×C4C2×#4112yes—
302C2×C3×C4C2×#4114yes—
303C2×C3×C4C2×#4112yes—
304C2×C3×C4C2×#4112yes—
305C2×C3×C4C2×#4112yes—
306C2×C3×C4H(5,𝔽2)14yes—
307C2×C3×C42−1+414yes—
308C24C24×C311yescharacteristic
309C2×Q8—28no—
310C2×Q8—48no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4C2×C3×Q814yes—
316C22×C4C2×C3×Q818yes—
317C22×C4C24×C311yescharacteristic
318C22×C4C24×C312yes—
319C22×C4C24×C318yes—
320C22×C4C24×C318yes—
321C22×C4C24×C311yescharacteristic
322C22×C4C24×C312yes—
323C22×C4C24×C311yescharacteristic
324C22×C4C24×C311yescharacteristic
325C22×C4C24×C311yescharacteristic
326C22×C4C24×C312yes—
327C22×C4C24×C311yescharacteristic
328C22×C4C24×C312yes—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—216no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—48no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C22⋊C4—24no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—48no—
347C42—22no—
348C42—22no—
349C42—24no—
350C42—24no—
351C42—24no—
352C42—24no—
353C42—48no—
354C42C2×C3×Q814yes—
355C42C2×C3×Q818yes—
356C42C2×C3×Q814yes—
357C42C3×#4114yes—
358C42C3×#4114yes—
359C22×C3—22no—
360C22×C3—24no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—42no—
364C22×C3C23×Q811yescharacteristic
365C22×C3C22×#4111yescharacteristic
366C22×C3C22×#4111yescharacteristic
367C22×C3C2×H(5,𝔽2)12yes—
368C22×C3C2×2−1+412yes—
369C3×C4—22no—
370C3×C4—24no—
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—24no—
374C3×C4—41no—
375C3×C4—48no—
376C3×C4—41no—
377C3×C4—42no—
378C3×C4—48no—
379C3×C4#53314yes—
380C3×C4#54418yes—
381C23—21no—
382C23—21no—
383C23—22no—
384C23—22no—
385C23C22×C3×Q812yes—
386C23C25×C311yescharacteristic
387Q8—88no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—216no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4C22×C3×Q812yes—
412C2×C4C22×C3×Q814yes—
413C2×C4C22×C3×Q812yes—
414C2×C4C22×C3×Q814yes—
415C2×C4C2×C3×#4112yes—
416C2×C4C2×C3×#4114yes—
417C2×C4C2×C3×#4112yes—
418C2×C4C2×C3×#4112yes—
419C2×C4C2×C3×#4112yes—
420C2×C4C3×H(5,𝔽2)14yes—
421C2×C4C3×2−1+414yes—
422C2×C3—42no—
423C2×C3C2×#53311yescharacteristic
424C2×C3C2×#54412yes—
425C2×C3#350811yescharacteristic
426C2×C3#350911yescharacteristic
427C2×C3#351112yes—
428C22—22no—
429C22—24no—
430C22—41no—
431C22—41no—
432C22—42no—
433C22C23×C3×Q811yescharacteristic
434C22C22×C3×#4111yescharacteristic
435C22C22×C3×#4111yescharacteristic
436C22C2×C3×H(5,𝔽2)12yes—
437C22C2×C3×2−1+412yes—
438C4—22no—
439C4—24no—
440C4—22no—
441C4—22no—
442C4—24no—
443C4—48no—
444C4—41no—
445C4—41no—
446C4—42no—
447C4—48no—
448C4C3×#53314yes—
449C4C3×#54418yes—
450C3#3529411yescharacteristic, direct product factor, Sylow
451C2—42no—
452C2C2×C3×#53311yescharacteristic
453C2C2×C3×#54412yes—
454C2C3×#350811yescharacteristic
455C2C3×#350911yescharacteristic
456C2C3×#351112yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC24×C3C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2868C2semidirect
2. C3×#2859C2semidirect
3. C3×#2859C2semidirect
6. C3×#2570C2semidirect
7. C3×#2550C2semidirect
10. C3×#2331C2semidirect
11. C3×#2330C2semidirect
14. #35294C3direct (N × H)
58. #2868C2×C3semidirect
59. #2859C2×C3semidirect
60. #2859C2×C3semidirect
63. #2570C2×C3semidirect
64. #2550C2×C3semidirect
67. #2331C2×C3semidirect
68. #2330C2×C3semidirect
99. C2×C3×C42Q8semidirect
244. C2×C42C3×Q8semidirect
450. C3#35294direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28682
C3×#28592
C3×#26292
C3×#26282
C3×#25702
C3×#25502
C3×#24412
C3×#24392
C3×#23312
C3×#23302
C3×#22852
C3×C4×#1172
#352943