Finite Group Databasebeta

← C3×#29805 #10648412

Subgroups — C3×#29805 #10648412

Order 768.

838 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#464C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C3×#2658C211yesmaximal, characteristic
4C3×#2320C211yesmaximal, characteristic, complemented
5C3×#2043C212yesmaximal, complemented
6C3×#2042C212yesmaximal
7C3×#2016C212yesmaximal, complemented
8C3×#2015C212yesmaximal
9C3×#1983C211yesmaximal, characteristic
10C3×#1981C211yesmaximal, characteristic, complemented
11C3×#1832C211yesmaximal, characteristic, complemented
12#29805C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C22×C3×SD16C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×#124C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C3×#464—24no—
22C3×#462—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#102C2211yescharacteristic
31C2×C3×#102C2212yes—
32C3×#399—21no—
33C3×#399—21no—
34C3×#399—22no—
35C3×#391C2212yes—
36C3×#389C2212yes—
37C3×#387—22no—
38C3×#383—21no—
39C3×#382—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42C3×#336C2212yes—
43C3×#336C2212yes—
44C2×#464C2×C311yescharacteristic
45C2×#462C2×C311yescharacteristic, complemented
46#2658C2×C311yescharacteristic
47#2320C2×C311yescharacteristic, complemented
48#2043C2×C312yescomplemented
49#2042C2×C312yes—
50#2016C2×C312yescomplemented
51#2015C2×C312yes—
52#1983C2×C311yescharacteristic
53#1981C2×C311yescharacteristic, complemented
54#1832C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×Q8C2312yes—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C2×C3×Q16—24no—
60C2×C3×Q16—22no—
61C2×C3×Q16—42no—
62C2×C3×Q16—42no—
63C2×C3×Q16—42no—
64C2×C3×SD16—22no—
65C2×C3×SD16—42no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—42no—
72C3×C4⋊Q8—42no—
73C3×#124—22no—
74C3×#124—22no—
75C3×#124—42no—
76C3×#124—42no—
77C3×C4×Q8—22no—
78C3×C4×Q8D414yes—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2312yes—
94C2×C3×C42C2311yescharacteristic
95C3×C4⋊C8—22no—
96C3×C4⋊C8—22no—
97C3×#103—24no—
98C3×#103—22no—
99C3×#103—42no—
100C3×#103—42no—
101C3×#103—42no—
102C3×#102—22no—
103C3×#102—42no—
104C3×#95—21no—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—22no—
108C3×#95D411yescharacteristic
109C3×#95D411yescharacteristic
110C3×#95D412yes—
111C22×Q16C22×C312yes—
112C22×Q16C22×C311yescharacteristic
113C22×SD16C22×C311yescharacteristic
114C2×C4⋊Q8C22×C311yescharacteristic
115C2×C4⋊Q8C22×C312yes—
116C2×#124C22×C311yescharacteristic
117C2×#124C22×C312yes—
118C2×C4×Q8C22×C311yescharacteristic
119#464—24no—
120#462—24no—
121C2×C4⋊C8C22×C311yescharacteristic
122C2×C4⋊C8C22×C311yescharacteristic
123C2×#103C22×C312yes—
124C2×#103C22×C311yescharacteristic
125C2×#103C22×C312yes—
126C2×#103C22×C312yes—
127C2×#103C22×C312yes—
128C2×#102C22×C311yescharacteristic
129C2×#102C22×C312yes—
130#399—21no—
131#399—21no—
132#399—22no—
133#391C22×C312yes—
134#389C22×C312yes—
135#387—22no—
136#383—21no—
137#382—21no—
138C4×C4⋊C4C22×C311yescharacteristic
139#376C22×C311yescharacteristic
140#336C22×C312yes—
141#336C22×C312yes—
142C24×C3—21no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—22no—
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D412yes—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C3×Q16—44no—
172C3×Q16—48no—
173C3×SD16—44no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—44no—
186C3×C4⋊C4—44no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—82no—
190C3×C4⋊C4C2×D412yes—
191C3×C22⋊C4—44no—
192C3×C22⋊C4—44no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—42no—
195C3×C42—24no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—81no—
199C3×C42C2×D412yes—
200C3×C42#4112yes—
201C22×Q8C23×C311yescharacteristic
202C22×Q8C23×C312yes—
203C22×Q8C23×C311yescharacteristic
204C22×D4C23×C311yescharacteristic
205C2×Q16—24no—
206C2×Q16—22no—
207C2×Q16—42no—
208C2×Q16—42no—
209C2×Q16—42no—
210C2×SD16—22no—
211C2×SD16—42no—
212C22×C8C23×C311yescharacteristic
213C22×C8C23×C311yescharacteristic
214C22×C8C23×C312yes—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—42no—
218C4⋊Q8—42no—
219#124—22no—
220#124—22no—
221#124—42no—
222#124—42no—
223C4×Q8—22no—
224C4×Q8C3×D414yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4C23×C312yes—
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C23×C312yes—
240C2×C42C23×C311yescharacteristic
241C4⋊C8—22no—
242C4⋊C8—22no—
243#103—24no—
244#103—22no—
245#103—42no—
246#103—42no—
247#103—42no—
248#102—22no—
249#102—42no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95C3×D411yescharacteristic
255#95C3×D411yescharacteristic
256#95C3×D412yes—
257C23×C3—41no—
258C23×C3—42no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—42no—
262C23×C3C22×D411yescharacteristic
263C3×Q8—22no—
264C3×Q8—22no—
265C3×Q8—42no—
266C3×Q8—41no—
267C3×Q8—42no—
268C3×Q8—41no—
269C3×Q8—42no—
270C3×Q8—42no—
271C3×Q8—42no—
272C3×Q8—42no—
273C3×D4—41no—
274C3×D4—41no—
275C3×D4—42no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4#12112yes—
299C2×C3×C4#12112yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4H(5,𝔽2)12yes—
303C2×C3×C4H(5,𝔽2)12yes—
304C3×C8—41no—
305C3×C8—41no—
306C3×C8—42no—
307C24—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8C2×C3×D412yes—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—22no—
325C2×D4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4C2×C3×D412yes—
332C22×C4C2×C3×D412yes—
333C22×C4C2×C3×D411yescharacteristic
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C24×C311yescharacteristic
336Q16—44no—
337Q16—48no—
338SD16—44no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C4⋊C4—24no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—82no—
355C4⋊C4C2×C3×D412yes—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C42—24no—
361C42—22no—
362C42—22no—
363C42—81no—
364C42C2×C3×D412yes—
365C42C3×#4112yes—
366C22×C3—81no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—81no—
370C22×C3—81no—
371C22×C3C2×#12111yescharacteristic
372C22×C3#53512yes—
373C22×C3#57711yescharacteristic
374C22×C3#57811yescharacteristic
375C22×C3#57812yes—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—81no—
383C3×C4—82no—
384C3×C4—82no—
385C3×C4#53511yescharacteristic
386C3×C4#53511yescharacteristic
387C3×C4#54012yes—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—41no—
392C23—42no—
393C23C22×C3×D411yescharacteristic
394Q8—22no—
395Q8—22no—
396Q8—42no—
397Q8—41no—
398Q8—42no—
399Q8—41no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404D4—41no—
405D4—41no—
406D4—42no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4C3×#12112yes—
430C2×C4C3×#12112yes—
431C2×C4C22×C3×D411yescharacteristic
432C2×C4C2×C3×#4111yescharacteristic
433C2×C4C3×H(5,𝔽2)12yes—
434C2×C4C3×H(5,𝔽2)12yes—
435C8—41no—
436C8—41no—
437C8—42no—
438C2×C3—161no—
439C2×C3#261911yescharacteristic
440C2×C3#304411yescharacteristic
441C2×C3#304611yescharacteristic
442C2×C3#322312yes—
443C2×C3#322512yes—
444C22—81no—
445C22—82no—
446C22—82no—
447C22—81no—
448C22—81no—
449C22C2×C3×#12111yescharacteristic
450C22C3×#53512yes—
451C22C3×#57711yescharacteristic
452C22C3×#57811yescharacteristic
453C22C3×#57812yes—
454C4—22no—
455C4—22no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—81no—
461C4—82no—
462C4—82no—
463C4C3×#53511yescharacteristic
464C4C3×#53511yescharacteristic
465C4C3×#54012yes—
466C3#2980511yescharacteristic, direct product factor, Sylow
467C2—161no—
468C2C3×#261911yescharacteristic
469C2C3×#304411yescharacteristic
470C2C3×#304611yescharacteristic
471C2C3×#322312yes—
472C2C3×#322512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#462C2semidirect
4. C3×#2320C2semidirect
5. C3×#2043C2semidirect
7. C3×#2016C2semidirect
10. C3×#1981C2semidirect
11. C3×#1832C2semidirect
12. #29805C3direct (N × H)
45. C2×#462C2×C3semidirect
47. #2320C2×C3semidirect
48. #2043C2×C3semidirect
50. #2016C2×C3semidirect
53. #1981C2×C3semidirect
54. #1832C2×C3semidirect
466. C3#29805direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4642
C2×C3×#4622
C3×#26582
C3×#23202
C3×#20432
C3×#20422
C3×#20162
C3×#20152
C3×#19832
C3×#19812
C3×#18322
#298053