Finite Group Databasebeta

← C3×#35854 #10654461

Subgroups — C3×#35854 #10654461

Order 768.

2286 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×C3×#548C211yesmaximal, characteristic
3C3×#2640C213yesmaximal, complemented
4C3×#2592C213yesmaximal, complemented
5C3×#2563C213yesmaximal, complemented
6C3×#2490C216yesmaximal, complemented
7C3×#2489C216yesmaximal, complemented
8C3×#2398C213yesmaximal, complemented
9C3×#2341C212yesmaximal, complemented
10C3×Q8×C22⋊C4C212yesmaximal, complemented
11C3×#2276C211yesmaximal, characteristic, complemented
12#35854C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C3×#557—28no—
16C3×#548—28no—
17C2×C3×C4⋊Q8C2213yes—
18C2×C3×#125C2216yes—
19C2×C3×#124C2213yes—
20C2×C3×C22⋊Q8C2213yes—
21C2×C3×#121C2213yes—
22C2×C3×C4×Q8C2213yes—
23C2×C3×C4×Q8C2212yes—
24C2×C3×C4×Q8C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2213yes—
27C3×#400C2216yes—
28C3×#397C22112yes—
29C3×#397C2211yescharacteristic
30C3×#397C2213yes—
31C3×#394C2216yes—
32C3×#393C2213yes—
33C3×#393C2216yes—
34C3×#393C2213yes—
35C3×#391C2216yes—
36C3×#391C2216yes—
37C3×#390C2216yes—
38C3×#390C2216yes—
39C3×#390C2216yes—
40C3×#388C22112yes—
41C3×#382C2216yes—
42C3×#380C2216yes—
43C3×#380C2216yes—
44C3×C4×C4⋊C4C2216yes—
45C3×C4×C4⋊C4C2213yes—
46C3×C4×C22⋊C4C2216yes—
47C3×C4×C22⋊C4C2216yes—
48C3×#376C2213yes—
49C2×#557C2×C311yescharacteristic, complemented
50C2×#548C2×C311yescharacteristic
51#2640C2×C313yescomplemented
52#2592C2×C313yescomplemented
53#2563C2×C313yescomplemented
54#2490C2×C316yescomplemented
55#2489C2×C316yescomplemented
56#2398C2×C313yescomplemented
57#2341C2×C312yescomplemented
58Q8×C22⋊C4C2×C312yescomplemented
59#2276C2×C311yescharacteristic, complemented
60C2×C3×#41—24no—
61C2×C3×#41—28no—
62C22×C3×Q8—24no—
63C22×C3×Q8—28no—
64C22×C3×Q8C2313yes—
65C22×C3×Q8C2313yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4C2313yes—
68C22×C3×D4C2313yes—
69C23×C3×C4C2313yes—
70C23×C3×C4C2313yes—
71C3×C4⋊Q8—212no—
72C3×#125—224no—
73C3×#124—212no—
74C3×C22⋊Q8—212no—
75C3×#121—212no—
76C3×C4×Q8—28no—
77C3×C4×Q8—28no—
78C3×C4×Q8—212no—
79C3×C4×Q8D418yes—
80C3×C4×D4—212no—
81C2×C3×C4⋊C4C2316yes—
82C2×C3×C4⋊C4C2316yes—
83C2×C3×C4⋊C4C2313yes—
84C2×C3×C4⋊C4C2316yes—
85C2×C3×C4⋊C4C2316yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2313yes—
88C2×C3×C22⋊C4C2312yes—
89C2×C3×C22⋊C4C2316yes—
90C2×C3×C22⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4C2313yes—
92C2×C3×C22⋊C4C2316yes—
93C2×C3×C22⋊C4C2316yes—
94C2×C3×C22⋊C4C2316yes—
95C2×C3×C22⋊C4C23112yes—
96C2×C3×C42C2316yes—
97C2×C3×C42C2316yes—
98C2×C3×C42C2316yes—
99C2×C3×C42C2313yes—
100C3×#95C2311yescharacteristic
101C3×#95C2313yes—
102C3×#95C2316yes—
103C3×#95C2312yes—
104C3×#95C2316yes—
105C3×#95C2316yes—
106C3×#95C2316yes—
107C3×#95C23112yes—
108C22×#41C22×C311yescharacteristic
109C23×Q8C22×C311yescharacteristic
110#557—28no—
111#548—28no—
112C2×C4⋊Q8C22×C313yes—
113C2×#125C22×C316yes—
114C2×#124C22×C313yes—
115C2×C22⋊Q8C22×C313yes—
116C2×#121C22×C313yes—
117C2×C4×Q8C22×C313yes—
118C2×C4×Q8C22×C312yes—
119C2×C4×Q8C22×C312yes—
120C2×C4×Q8C22×C311yescharacteristic
121C2×C4×D4C22×C313yes—
122#400C22×C316yes—
123#397C22×C313yes—
124#397C22×C3112yes—
125#397C22×C311yescharacteristic
126#394C22×C316yes—
127#393C22×C313yes—
128#393C22×C316yes—
129#393C22×C313yes—
130#391C22×C316yes—
131#391C22×C316yes—
132#390C22×C316yes—
133#390C22×C316yes—
134#390C22×C316yes—
135#388C22×C3112yes—
136#382C22×C316yes—
137#380C22×C316yes—
138#380C22×C316yes—
139C4×C4⋊C4C22×C313yes—
140C4×C4⋊C4C22×C316yes—
141C4×C22⋊C4C22×C316yes—
142C4×C22⋊C4C22×C316yes—
143#376C22×C313yes—
144C24×C3C2413yes—
145C24×C3C2411yescharacteristic
146C3×#41—416no—
147C2×C3×Q8—26no—
148C2×C3×Q8—26no—
149C2×C3×Q8—212no—
150C2×C3×Q8—46no—
151C2×C3×Q8—416no—
152C2×C3×Q8C2×D414yes—
153C2×C3×Q8#4118yes—
154C2×C3×D4—26no—
155C2×C3×D4—26no—
156C2×C3×D4—26no—
157C2×C3×D4—212no—
158C2×C3×D4—43no—
159C22×C3×C4—26no—
160C22×C3×C4—26no—
161C22×C3×C4—212no—
162C22×C3×C4—212no—
163C22×C3×C4C2412yes—
164C22×C3×C4C2413yes—
165C22×C3×C4C2411yescharacteristic
166C22×C3×C4C2416yes—
167C22×C3×C4C2412yes—
168C22×C3×C4C2416yes—
169C22×C3×C4C2411yescharacteristic
170C22×C3×C4C2413yes—
171C22×C3×C4C2413yes—
172C3×C4⋊C4—26no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—212no—
175C3×C4⋊C4—212no—
176C3×C4⋊C4—212no—
177C3×C4⋊C4—212no—
178C3×C4⋊C4C2×D4112yes—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—26no—
182C3×C22⋊C4—212no—
183C3×C22⋊C4—46no—
184C3×C22⋊C4—46no—
185C3×C22⋊C4—46no—
186C3×C22⋊C4—412no—
187C3×C42—212no—
188C3×C42—212no—
189C3×C42—46no—
190C3×C42C2×D4112yes—
191C2×#41—24no—
192C2×#41—28no—
193C22×Q8—24no—
194C22×Q8—28no—
195C22×Q8C23×C313yes—
196C22×Q8C23×C313yes—
197C22×Q8C23×C311yescharacteristic
198C22×D4C23×C313yes—
199C22×D4C23×C313yes—
200C23×C4C23×C313yes—
201C23×C4C23×C313yes—
202C4⋊Q8—212no—
203#125—224no—
204#124—212no—
205C22⋊Q8—212no—
206#121—212no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—212no—
210C4×Q8C3×D418yes—
211C4×D4—212no—
212C2×C4⋊C4C23×C313yes—
213C2×C4⋊C4C23×C311yescharacteristic
214C2×C4⋊C4C23×C316yes—
215C2×C4⋊C4C23×C316yes—
216C2×C4⋊C4C23×C316yes—
217C2×C4⋊C4C23×C316yes—
218C2×C4⋊C4C23×C313yes—
219C2×C22⋊C4C23×C311yescharacteristic
220C2×C22⋊C4C23×C313yes—
221C2×C22⋊C4C23×C316yes—
222C2×C22⋊C4C23×C312yes—
223C2×C22⋊C4C23×C316yes—
224C2×C22⋊C4C23×C316yes—
225C2×C22⋊C4C23×C316yes—
226C2×C22⋊C4C23×C3112yes—
227C2×C42C23×C316yes—
228C2×C42C23×C316yes—
229C2×C42C23×C316yes—
230C2×C42C23×C313yes—
231#95C23×C311yescharacteristic
232#95C23×C313yes—
233#95C23×C316yes—
234#95C23×C312yes—
235#95C23×C316yes—
236#95C23×C316yes—
237#95C23×C316yes—
238#95C23×C3112yes—
239C23×C3—21no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—21no—
243C23×C3—23no—
244C23×C3—26no—
245C23×C3—23no—
246C23×C3—26no—
247C23×C3—23no—
248C23×C32−1+412yes—
249C23×C3C2511yescharacteristic
250C3×Q8—412no—
251C3×Q8—412no—
252C3×Q8#124116yes—
253C3×D4—412no—
254C3×D4—412no—
255C2×C3×C4—23no—
256C2×C3×C4—23no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—26no—
260C2×C3×C4—23no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—26no—
264C2×C3×C4—22no—
265C2×C3×C4—26no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—26no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—26no—
272C2×C3×C4—26no—
273C2×C3×C4—26no—
274C2×C3×C4—412no—
275C2×C3×C4—412no—
276C2×C3×C4C22×D412yes—
277C2×C3×C4C22×D416yes—
278C2×C3×C4C22×D416yes—
279C2×C3×C4C2×#41112yes—
280C24C24×C313yes—
281C24C24×C311yescharacteristic
282#41—416no—
283C2×Q8—26no—
284C2×Q8—26no—
285C2×Q8—212no—
286C2×Q8—46no—
287C2×Q8—416no—
288C2×Q8C2×C3×D414yes—
289C2×Q8C3×#4118yes—
290C2×D4—26no—
291C2×D4—26no—
292C2×D4—26no—
293C2×D4—212no—
294C2×D4—43no—
295C22×C4—26no—
296C22×C4—26no—
297C22×C4—212no—
298C22×C4—212no—
299C22×C4C24×C312yes—
300C22×C4C24×C313yes—
301C22×C4C24×C311yescharacteristic
302C22×C4C24×C316yes—
303C22×C4C24×C312yes—
304C22×C4C24×C316yes—
305C22×C4C24×C311yescharacteristic
306C22×C4C24×C313yes—
307C22×C4C24×C313yes—
308C4⋊C4—26no—
309C4⋊C4—22no—
310C4⋊C4—212no—
311C4⋊C4—212no—
312C4⋊C4—212no—
313C4⋊C4—212no—
314C4⋊C4C2×C3×D4112yes—
315C22⋊C4—24no—
316C22⋊C4—22no—
317C22⋊C4—26no—
318C22⋊C4—212no—
319C22⋊C4—46no—
320C22⋊C4—46no—
321C22⋊C4—46no—
322C22⋊C4—412no—
323C42—212no—
324C42—212no—
325C42—46no—
326C42C2×C3×D4112yes—
327C22×C3—22no—
328C22×C3—24no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—42no—
332C22×C3—43no—
333C22×C3—43no—
334C22×C3—43no—
335C22×C3—46no—
336C22×C3—46no—
337C22×C3C23×D411yescharacteristic
338C22×C3C22×#4112yes—
339C22×C3C2×2−1+411yescharacteristic
340C22×C3#58512yes—
341C22×C3#58511yescharacteristic
342C3×C4—22no—
343C3×C4—26no—
344C3×C4—41no—
345C3×C4—43no—
346C3×C4—46no—
347C3×C4—42no—
348C3×C4—42no—
349C3×C4—46no—
350C3×C4C2×#124112yes—
351C23—21no—
352C23—22no—
353C23—22no—
354C23—21no—
355C23—23no—
356C23—26no—
357C23—23no—
358C23—26no—
359C23—23no—
360C23C3×2−1+412yes—
361C23C25×C311yescharacteristic
362Q8—412no—
363Q8—412no—
364Q8C3×#124116yes—
365D4—412no—
366D4—412no—
367C2×C4—23no—
368C2×C4—23no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—26no—
372C2×C4—23no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—26no—
376C2×C4—22no—
377C2×C4—26no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—26no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—26no—
384C2×C4—26no—
385C2×C4—26no—
386C2×C4—412no—
387C2×C4—412no—
388C2×C4C22×C3×D412yes—
389C2×C4C22×C3×D416yes—
390C2×C4C22×C3×D416yes—
391C2×C4C2×C3×#41112yes—
392C2×C3—42no—
393C2×C3—83no—
394C2×C3C22×#12411yescharacteristic
395C2×C3#348711yescharacteristic
396C2×C3#348911yescharacteristic
397C2×C3#351612yes—
398C2×C3#352412yes—
399C22—22no—
400C22—24no—
401C22—41no—
402C22—41no—
403C22—42no—
404C22—43no—
405C22—43no—
406C22—43no—
407C22—46no—
408C22—46no—
409C22C23×C3×D411yescharacteristic
410C22C22×C3×#4112yes—
411C22C2×C3×2−1+411yescharacteristic
412C22C3×#58512yes—
413C22C3×#58511yescharacteristic
414C4—22no—
415C4—26no—
416C4—41no—
417C4—43no—
418C4—46no—
419C4—42no—
420C4—42no—
421C4—46no—
422C4C2×C3×#124112yes—
423C3#3585411yescharacteristic, direct product factor, Sylow
424C2—42no—
425C2—83no—
426C2C22×C3×#12411yescharacteristic
427C2C3×#348711yescharacteristic
428C2C3×#348911yescharacteristic
429C2C3×#351612yes—
430C2C3×#352412yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC2×C3×#548C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
3. C3×#2640C2semidirect
4. C3×#2592C2semidirect
5. C3×#2563C2semidirect
6. C3×#2490C2semidirect
7. C3×#2489C2semidirect
8. C3×#2398C2semidirect
9. C3×#2341C2semidirect
10. C3×Q8×C22⋊C4C2semidirect
11. C3×#2276C2semidirect
12. #35854C3direct (N × H)
49. C2×#557C2×C3semidirect
51. #2640C2×C3semidirect
52. #2592C2×C3semidirect
53. #2563C2×C3semidirect
54. #2490C2×C3semidirect
55. #2489C2×C3semidirect
56. #2398C2×C3semidirect
57. #2341C2×C3semidirect
58. Q8×C22⋊C4C2×C3semidirect
59. #2276C2×C3semidirect
423. C3#35854direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×C3×#5482
C3×#26402
C3×#25922
C3×#25632
C3×#24902
C3×#24892
C3×#23982
C3×#23412
C3×Q8×C22⋊C42
C3×#22762
#358543