Finite Group Databasebeta

← C3×#35910 #10654517

Subgroups — C3×#35910 #10654517

Order 768.

2228 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×C3×D4×Q8C211yesmaximal, characteristic
3C3×#2642C213yesmaximal, complemented
4C3×#2568C213yesmaximal, complemented
5C3×#2562C213yesmaximal
6C3×#2492C216yesmaximal, complemented
7C3×#2489C216yesmaximal, complemented
8C3×#2398C213yesmaximal
9C3×Q8×C22⋊C4C212yesmaximal, complemented
10C3×Q8×C22⋊C4C212yesmaximal, complemented
11C3×#2276C211yesmaximal, characteristic, complemented
12#35910C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2212yes—
14C3×#557—28no—
15C3×D4×Q8—28no—
16C2×C3×C4⋊Q8C2213yes—
17C2×C3×C4⋊Q8C2213yes—
18C2×C3×#125C2213yes—
19C2×C3×#125C2213yes—
20C2×C3×C22⋊Q8C2216yes—
21C2×C3×C4×Q8C2213yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2212yes—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×D4C2213yes—
27C3×#400C2213yes—
28C3×#400C22112yes—
29C3×#400C2216yes—
30C3×#400C2213yes—
31C3×#397C2216yes—
32C3×#394C2211yescharacteristic
33C3×#394C2213yes—
34C3×#393C2216yes—
35C3×#391C2216yes—
36C3×#391C2216yes—
37C3×#391C2216yes—
38C3×#391C2216yes—
39C3×#391C2216yes—
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×D4×Q8C2×C311yescharacteristic
51#2642C2×C313yescomplemented
52#2568C2×C313yescomplemented
53#2562C2×C313yes—
54#2492C2×C316yescomplemented
55#2489C2×C316yescomplemented
56#2398C2×C313yes—
57Q8×C22⋊C4C2×C312yescomplemented
58Q8×C22⋊C4C2×C312yescomplemented
59#2276C2×C311yescharacteristic, complemented
60C22×C3×Q8—28no—
61C22×C3×Q8—28no—
62C22×C3×Q8—28no—
63C22×C3×Q8C2316yes—
64C22×C3×Q8C2313yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4C2316yes—
68C3×C4⋊Q8—212no—
69C3×C4⋊Q8—212no—
70C3×#125—212no—
71C3×#125—212no—
72C3×C22⋊Q8—224no—
73C3×C4×Q8—24no—
74C3×C4×Q8—24no—
75C3×C4×Q8—28no—
76C3×C4×Q8—212no—
77C3×C4×Q8D418yes—
78C3×C4×D4—212no—
79C2×C3×C4⋊C4C2316yes—
80C2×C3×C4⋊C4C2313yes—
81C2×C3×C4⋊C4C2313yes—
82C2×C3×C4⋊C4C2313yes—
83C2×C3×C4⋊C4C2316yes—
84C2×C3×C4⋊C4C2316yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2313yes—
87C2×C3×C4⋊C4C2313yes—
88C2×C3×C4⋊C4C2313yes—
89C2×C3×C22⋊C4C2312yes—
90C2×C3×C22⋊C4C2312yes—
91C2×C3×C22⋊C4C2316yes—
92C2×C3×C22⋊C4C2316yes—
93C2×C3×C22⋊C4C2316yes—
94C2×C3×C42C2316yes—
95C2×C3×C42C2313yes—
96C2×C3×C42C2313yes—
97C2×C3×C42C2316yes—
98C2×C3×C42C2313yes—
99C3×#95C2311yescharacteristic
100C3×#95C2316yes—
101C3×#95C2313yes—
102C3×#95C2316yes—
103C3×#95C2316yes—
104C3×#95C2316yes—
105C3×#95C2316yes—
106C3×#95C23112yes—
107C3×#95C23112yes—
108C23×Q8C22×C312yes—
109#557—28no—
110D4×Q8—28no—
111C2×C4⋊Q8C22×C313yes—
112C2×C4⋊Q8C22×C313yes—
113C2×#125C22×C313yes—
114C2×#125C22×C313yes—
115C2×C22⋊Q8C22×C316yes—
116C2×C4×Q8C22×C313yes—
117C2×C4×Q8C22×C311yescharacteristic
118C2×C4×Q8C22×C311yescharacteristic
119C2×C4×Q8C22×C312yes—
120C2×C4×Q8C22×C311yescharacteristic
121C2×C4×D4C22×C313yes—
122#400C22×C316yes—
123#400C22×C313yes—
124#400C22×C313yes—
125#400C22×C3112yes—
126#397C22×C316yes—
127#394C22×C313yes—
128#394C22×C311yescharacteristic
129#393C22×C316yes—
130#391C22×C316yes—
131#391C22×C316yes—
132#391C22×C316yes—
133#391C22×C316yes—
134#391C22×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×C3C2412yes—
145C2×C3×Q8—26no—
146C2×C3×Q8—212no—
147C2×C3×Q8—212no—
148C2×C3×Q8—212no—
149C2×C3×Q8—26no—
150C2×C3×Q8—43no—
151C2×C3×Q8—432no—
152C2×C3×Q8C2×D414yes—
153C2×C3×Q8#4118yes—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C22×C3×C4—212no—
157C22×C3×C4—212no—
158C22×C3×C4—212no—
159C22×C3×C4C2412yes—
160C22×C3×C4C2416yes—
161C22×C3×C4C2416yes—
162C22×C3×C4C2411yescharacteristic
163C22×C3×C4C2413yes—
164C22×C3×C4C2413yes—
165C22×C3×C4C2411yescharacteristic
166C22×C3×C4C2413yes—
167C22×C3×C4C2411yescharacteristic
168C22×C3×C4C2413yes—
169C3×C4⋊C4—26no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—26no—
172C3×C4⋊C4—26no—
173C3×C4⋊C4—26no—
174C3×C4⋊C4—26no—
175C3×C4⋊C4—212no—
176C3×C4⋊C4—212no—
177C3×C4⋊C4—212no—
178C3×C4⋊C4C2×D4112yes—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—212no—
181C3×C22⋊C4—212no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—46no—
184C3×C42—26no—
185C3×C42—26no—
186C3×C42—212no—
187C3×C42—46no—
188C3×C42C2×D4112yes—
189C22×Q8—28no—
190C22×Q8—28no—
191C22×Q8—28no—
192C22×Q8C23×C316yes—
193C22×Q8C23×C313yes—
194C22×Q8C23×C311yescharacteristic
195C22×D4C23×C311yescharacteristic
196C23×C4C23×C316yes—
197C4⋊Q8—212no—
198C4⋊Q8—212no—
199#125—212no—
200#125—212no—
201C22⋊Q8—224no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—28no—
205C4×Q8—212no—
206C4×Q8C3×D418yes—
207C4×D4—212no—
208C2×C4⋊C4C23×C313yes—
209C2×C4⋊C4C23×C311yescharacteristic
210C2×C4⋊C4C23×C313yes—
211C2×C4⋊C4C23×C313yes—
212C2×C4⋊C4C23×C313yes—
213C2×C4⋊C4C23×C313yes—
214C2×C4⋊C4C23×C316yes—
215C2×C4⋊C4C23×C316yes—
216C2×C4⋊C4C23×C316yes—
217C2×C4⋊C4C23×C313yes—
218C2×C22⋊C4C23×C316yes—
219C2×C22⋊C4C23×C312yes—
220C2×C22⋊C4C23×C312yes—
221C2×C22⋊C4C23×C316yes—
222C2×C22⋊C4C23×C316yes—
223C2×C42C23×C313yes—
224C2×C42C23×C313yes—
225C2×C42C23×C316yes—
226C2×C42C23×C316yes—
227C2×C42C23×C313yes—
228#95C23×C311yescharacteristic
229#95C23×C316yes—
230#95C23×C313yes—
231#95C23×C316yes—
232#95C23×C316yes—
233#95C23×C316yes—
234#95C23×C316yes—
235#95C23×C3112yes—
236#95C23×C3112yes—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C32−1+414yes—
244C23×C3C2511yescharacteristic
245C3×Q8—412no—
246C3×Q8—424no—
247C3×Q8#124116yes—
248C3×D4—28no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—23no—
252C2×C3×C4—23no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—23no—
256C2×C3×C4—23no—
257C2×C3×C4—26no—
258C2×C3×C4—22no—
259C2×C3×C4—26no—
260C2×C3×C4—26no—
261C2×C3×C4—26no—
262C2×C3×C4—22no—
263C2×C3×C4—26no—
264C2×C3×C4—26no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—23no—
268C2×C3×C4—26no—
269C2×C3×C4—23no—
270C2×C3×C4—424no—
271C2×C3×C4C22×D412yes—
272C2×C3×C4C22×D416yes—
273C2×C3×C4C22×D416yes—
274C2×C3×C4C2×#41112yes—
275C2×C3×C42−1+412yes—
276C24C24×C312yes—
277C2×Q8—26no—
278C2×Q8—212no—
279C2×Q8—212no—
280C2×Q8—212no—
281C2×Q8—26no—
282C2×Q8—43no—
283C2×Q8—432no—
284C2×Q8C2×C3×D414yes—
285C2×Q8C3×#4118yes—
286C2×D4—24no—
287C2×D4—22no—
288C22×C4—212no—
289C22×C4—212no—
290C22×C4—212no—
291C22×C4C24×C312yes—
292C22×C4C24×C316yes—
293C22×C4C24×C316yes—
294C22×C4C24×C311yescharacteristic
295C22×C4C24×C313yes—
296C22×C4C24×C313yes—
297C22×C4C24×C311yescharacteristic
298C22×C4C24×C313yes—
299C22×C4C24×C311yescharacteristic
300C22×C4C24×C313yes—
301C4⋊C4—26no—
302C4⋊C4—22no—
303C4⋊C4—26no—
304C4⋊C4—26no—
305C4⋊C4—26no—
306C4⋊C4—26no—
307C4⋊C4—212no—
308C4⋊C4—212no—
309C4⋊C4—212no—
310C4⋊C4C2×C3×D4112yes—
311C22⋊C4—24no—
312C22⋊C4—212no—
313C22⋊C4—212no—
314C22⋊C4—24no—
315C22⋊C4—46no—
316C42—26no—
317C42—26no—
318C42—212no—
319C42—46no—
320C42C2×C3×D4112yes—
321C22×C3—24no—
322C22×C3—24no—
323C22×C3—24no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3C23×D411yescharacteristic
329C22×C3C22×#4112yes—
330C22×C3C2×2−1+412yes—
331C22×C3C2×2−1+411yescharacteristic
332C22×C3#58511yescharacteristic
333C3×C4—22no—
334C3×C4—26no—
335C3×C4—22no—
336C3×C4—46no—
337C3×C4—41no—
338C3×C4—43no—
339C3×C4—43no—
340C3×C4—46no—
341C3×C4—42no—
342C3×C4C2×#124112yes—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23C3×2−1+414yes—
350C23C25×C311yescharacteristic
351Q8—412no—
352Q8—424no—
353Q8C3×#124116yes—
354D4—28no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—23no—
358C2×C4—23no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—23no—
362C2×C4—23no—
363C2×C4—26no—
364C2×C4—22no—
365C2×C4—26no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—22no—
369C2×C4—26no—
370C2×C4—26no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—23no—
374C2×C4—26no—
375C2×C4—23no—
376C2×C4—424no—
377C2×C4C22×C3×D412yes—
378C2×C4C22×C3×D416yes—
379C2×C4C22×C3×D416yes—
380C2×C4C2×C3×#41112yes—
381C2×C4C3×2−1+412yes—
382C2×C3—44no—
383C2×C3C22×#12411yescharacteristic
384C2×C3#348611yescharacteristic
385C2×C3#348911yescharacteristic
386C2×C3#351612yes—
387C2×C3#352312yes—
388C22—24no—
389C22—24no—
390C22—24no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22C23×C3×D411yescharacteristic
396C22C22×C3×#4112yes—
397C22C2×C3×2−1+412yes—
398C22C2×C3×2−1+411yescharacteristic
399C22C3×#58511yescharacteristic
400C4—22no—
401C4—26no—
402C4—22no—
403C4—46no—
404C4—41no—
405C4—43no—
406C4—43no—
407C4—46no—
408C4—42no—
409C4C2×C3×#124112yes—
410C3#3591011yescharacteristic, direct product factor, Sylow
411C2—44no—
412C2C22×C3×#12411yescharacteristic
413C2C3×#348611yescharacteristic
414C2C3×#348911yescharacteristic
415C2C3×#351612yes—
416C2C3×#352312yes—
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. C2×C3×#557C2semidirect
3. C3×#2642C2semidirect
4. C3×#2568C2semidirect
6. C3×#2492C2semidirect
7. C3×#2489C2semidirect
9. C3×Q8×C22⋊C4C2semidirect
10. C3×Q8×C22⋊C4C2semidirect
11. C3×#2276C2semidirect
12. #35910C3direct (N × H)
49. C2×#557C2×C3semidirect
51. #2642C2×C3semidirect
52. #2568C2×C3semidirect
54. #2492C2×C3semidirect
55. #2489C2×C3semidirect
57. Q8×C22⋊C4C2×C3semidirect
58. Q8×C22⋊C4C2×C3semidirect
59. #2276C2×C3semidirect
410. C3#35910direct (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×D4×Q82
C3×#26422
C3×#25682
C3×#25622
C3×#24922
C3×#24892
C3×#23982
C3×Q8×C22⋊C42
C3×#22762
#359103