Finite Group Databasebeta

← C3×#29648 #10648255

Subgroups — C3×#29648 #10648255

Order 768.

948 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic
2C2×C3×#485C211yesmaximal, characteristic
3C2×C3×#483C211yesmaximal, characteristic
4C3×#2625C211yesmaximal, characteristic
5C3×#2059C212yesmaximal, complemented
6C3×#2058C212yesmaximal, complemented
7C3×#2026C212yesmaximal, complemented
8C3×#2024C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic
10C3×#1824C211yesmaximal, characteristic
11C3×#1818C211yesmaximal, characteristic
12#29648C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#544—24no—
14C3×#544—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×#125—21no—
18C2×C3×#125C2212yes—
19C2×C3×C22⋊Q8—22no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C2×C3×#117C2211yescharacteristic
23C22×C3×C4⋊C4C2211yescharacteristic
24C3×#485—24no—
25C3×#483—24no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#103C2211yescharacteristic
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#400—21no—
35C3×#398—21no—
36C3×#397—22no—
37C3×#391C2212yes—
38C3×#389C2212yes—
39C3×#380C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#544C2×C311yescharacteristic
47C2×#485C2×C311yescharacteristic
48C2×#483C2×C311yescharacteristic
49#2625C2×C311yescharacteristic
50#2059C2×C312yescomplemented
51#2058C2×C312yescomplemented
52#2026C2×C312yescomplemented
53#2024C2×C312yescomplemented
54#1825C2×C311yescharacteristic
55#1824C2×C311yescharacteristic
56#1818C2×C311yescharacteristic
57C22×C3×Q8—22no—
58C22×C3×Q8C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2312yes—
63C22×C3×C8C2311yescharacteristic
64C3×C4⋊Q8—28no—
65C3×C4⋊Q8—24no—
66C3×C4⋊Q8—24no—
67C3×#125—28no—
68C3×#125—24no—
69C3×#125—24no—
70C3×C22⋊Q8—216no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×#117—22no—
74C3×#117—22no—
75C3×#117—22no—
76C3×#117D414yes—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4D414yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4D412yes—
96C2×C3×C42—21no—
97C2×C3×C42C2312yes—
98C3×#107—22no—
99C3×#106—24no—
100C3×#106—22no—
101C3×#103—24no—
102C3×#103—22no—
103C3×#103—22no—
104C3×C22⋊C8—22no—
105C3×C22⋊C8—22no—
106C3×#95—21no—
107C3×#95—22no—
108C3×#95—22no—
109C3×#95—22no—
110#544—24no—
111#544—24no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C22×C312yes—
114C2×#125—21no—
115C2×#125C22×C312yes—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8C22×C311yescharacteristic
118C2×#117C22×C311yescharacteristic
119C2×#117C22×C311yescharacteristic
120C22×C4⋊C4C22×C311yescharacteristic
121#485—24no—
122#483—24no—
123C2×#107C22×C311yescharacteristic
124C2×#106C22×C311yescharacteristic
125C2×#106C22×C312yes—
126C2×#103C22×C312yes—
127C2×#103C22×C311yescharacteristic
128C2×#103C22×C311yescharacteristic
129C2×C22⋊C8C22×C311yescharacteristic
130C2×C22⋊C8C22×C311yescharacteristic
131#400—21no—
132#398—21no—
133#397—22no—
134#391C22×C312yes—
135#389C22×C312yes—
136#380C22×C311yescharacteristic
137#340C22×C312yes—
138#340C22×C312yes—
139#340C22×C312yes—
140#340C22×C312yes—
141#336C22×C312yes—
142#336C22×C312yes—
143C24×C3C2×D411yescharacteristic
144C2×C3×Q8—28no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—84no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—82no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2411yescharacteristic
166C2×C3×C8—22no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C3×C4⋊C4—28no—
174C3×C4⋊C4—28no—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—82no—
188C3×C4⋊C4C2×D414yes—
189C3×C4⋊C4C2×D412yes—
190C3×C4⋊C4C2×D412yes—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—81no—
195C3×C42—24no—
196C3×C42—22no—
197C3×C42—22no—
198C22×Q8—22no—
199C22×Q8C23×C311yescharacteristic
200C23×C4—21no—
201C23×C4C23×C311yescharacteristic
202C22×C8C23×C311yescharacteristic
203C22×C8C23×C312yes—
204C22×C8C23×C311yescharacteristic
205C4⋊Q8—28no—
206C4⋊Q8—24no—
207C4⋊Q8—24no—
208#125—28no—
209#125—24no—
210#125—24no—
211C22⋊Q8—216no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117C3×D414yes—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4C3×D411yescharacteristic
226C2×C4⋊C4C3×D411yescharacteristic
227C2×C4⋊C4C3×D414yes—
228C2×C4⋊C4C23×C311yescharacteristic
229C2×C4⋊C4C23×C312yes—
230C2×C4⋊C4C23×C312yes—
231C2×C4⋊C4C23×C311yescharacteristic
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4C3×D412yes—
237C2×C42—21no—
238C2×C42C23×C312yes—
239#107—22no—
240#106—24no—
241#106—22no—
242#103—24no—
243#103—22no—
244#103—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251C23×C3—22no—
252C23×C3—21no—
253C23×C3—22no—
254C23×C3—21no—
255C23×C3#12012yes—
256C23×C3C22×D411yescharacteristic
257C3×Q8—42no—
258C3×Q8—41no—
259C3×Q8—41no—
260C3×Q8—88no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—41no—
274C2×C3×C4—44no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—84no—
286C2×C3×C4#12012yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C42−1+412yes—
290C2×C3×C42−1+412yes—
291C3×C8—42no—
292C3×C8—41no—
293C3×C8—41no—
294C24C2×C3×D411yescharacteristic
295C2×Q8—28no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—84no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—82no—
311C22×C4C2×C3×D412yes—
312C22×C4C2×C3×D412yes—
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D412yes—
316C22×C4C24×C311yescharacteristic
317C2×C8—22no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C4⋊C4—28no—
325C4⋊C4—28no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—42no—
338C4⋊C4—82no—
339C4⋊C4C2×C3×D414yes—
340C4⋊C4C2×C3×D412yes—
341C4⋊C4C2×C3×D412yes—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C22⋊C4—81no—
346C42—24no—
347C42—22no—
348C42—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3C2×#12011yescharacteristic
355C22×C3#53612yes—
356C22×C3#57712yes—
357C22×C3#57711yescharacteristic
358C22×C3#57811yescharacteristic
359C3×C4—22no—
360C3×C4—42no—
361C3×C4—44no—
362C3×C4—42no—
363C3×C4—42no—
364C3×C4—42no—
365C3×C4—81no—
366C3×C4—82no—
367C3×C4—81no—
368C3×C4#53611yescharacteristic
369C3×C4#53612yes—
370C3×C4#53611yescharacteristic
371C23—21no—
372C23—22no—
373C23—21no—
374C23—22no—
375C23C3×#12012yes—
376C23C22×C3×D411yescharacteristic
377Q8—42no—
378Q8—41no—
379Q8—41no—
380Q8—88no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—84no—
406C2×C4C3×#12012yes—
407C2×C4C22×C3×D411yescharacteristic
408C2×C4C22×C3×D411yescharacteristic
409C2×C4C3×2−1+412yes—
410C2×C4C3×2−1+412yes—
411C8—42no—
412C8—41no—
413C8—41no—
414C2×C3—42no—
415C2×C3#261411yescharacteristic
416C2×C3#300911yescharacteristic
417C2×C3#301311yescharacteristic
418C2×C3#324212yes—
419C2×C3#324412yes—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—42no—
424C22—42no—
425C22C2×C3×#12011yescharacteristic
426C22C3×#53612yes—
427C22C3×#57712yes—
428C22C3×#57711yescharacteristic
429C22C3×#57811yescharacteristic
430C4—22no—
431C4—42no—
432C4—44no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—82no—
437C4—81no—
438C4—81no—
439C4C3×#53611yescharacteristic
440C4C3×#53612yes—
441C4C3×#53611yescharacteristic
442C3#2964811yescharacteristic, direct product factor, Sylow
443C2—42no—
444C2C3×#261411yescharacteristic
445C2C3×#300911yescharacteristic
446C2C3×#301311yescharacteristic
447C2C3×#324212yes—
448C2C3×#324412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
5. C3×#2059C2semidirect
6. C3×#2058C2semidirect
7. C3×#2026C2semidirect
8. C3×#2024C2semidirect
12. #29648C3direct (N × H)
50. #2059C2×C3semidirect
51. #2058C2×C3semidirect
52. #2026C2×C3semidirect
53. #2024C2×C3semidirect
442. C3#29648direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×C3×#4852
C2×C3×#4832
C3×#26252
C3×#20592
C3×#20582
C3×#20262
C3×#20242
C3×#18252
C3×#18242
C3×#18182
#296483