Finite Group Databasebeta

← C3×#35957 #10654564

Subgroups — C3×#35957 #10654564

Order 768.

2000 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic
2C2×C3×#557C211yesmaximal, characteristic
3C3×#2660C213yesmaximal
4C3×#2584C213yesmaximal
5C3×#2567C213yesmaximal
6C3×#2552C216yesmaximal
7C3×#2550C216yesmaximal
8C3×#2399C213yesmaximal
9C3×#2351C212yesmaximal
10C3×Q8×C4⋊C4C212yesmaximal
11C3×#2276C211yesmaximal, characteristic
12#35957C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#557—28no—
14C3×#557—28no—
15C2×C3×C4⋊Q8C2213yes—
16C2×C3×C4⋊Q8C2213yes—
17C2×C3×#125C2213yes—
18C2×C3×#125C2216yes—
19C2×C3×#125C2213yes—
20C2×C3×C4×Q8C2213yes—
21C2×C3×C4×Q8C2213yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2212yes—
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×Q8C2212yes—
27C3×#398C2211yescharacteristic
28C3×#398C2216yes—
29C3×#398C2213yes—
30C3×#398C22112yes—
31C3×#395C2213yes—
32C3×#395C2216yes—
33C3×#395C2213yes—
34C3×#395C2216yes—
35C3×#391C2216yes—
36C3×#389C22112yes—
37C3×#389C2216yes—
38C3×#389C2216yes—
39C3×#389C2216yes—
40C3×#389C2216yes—
41C3×#384C2216yes—
42C3×#384C2216yes—
43C3×#382C2216yes—
44C3×C4×C4⋊C4C2216yes—
45C3×C4×C4⋊C4C2216yes—
46C3×C4×C4⋊C4C2216yes—
47C3×C4×C4⋊C4C2213yes—
48C3×#376C2213yes—
49C2×#557C2×C311yescharacteristic
50C2×#557C2×C311yescharacteristic
51#2660C2×C313yes—
52#2584C2×C313yes—
53#2567C2×C313yes—
54#2552C2×C316yes—
55#2550C2×C316yes—
56#2399C2×C313yes—
57#2351C2×C312yes—
58Q8×C4⋊C4C2×C312yes—
59#2276C2×C311yescharacteristic
60C22×C3×Q8C2313yes—
61C22×C3×Q8C2313yes—
62C22×C3×Q8C2311yescharacteristic
63C3×C4⋊Q8—212no—
64C3×C4⋊Q8—212no—
65C3×#125—212no—
66C3×#125—224no—
67C3×#125—212no—
68C3×C4×Q8—28no—
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×C4×Q8—212no—
72C3×C4×Q8—212no—
73C3×C4×Q8D418yes—
74C3×C4×Q8Q8116yescomplemented
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2313yes—
77C2×C3×C4⋊C4C2316yes—
78C2×C3×C4⋊C4C2313yes—
79C2×C3×C4⋊C4C2313yes—
80C2×C3×C4⋊C4C2316yes—
81C2×C3×C4⋊C4C2316yes—
82C2×C3×C4⋊C4C2316yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2313yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2316yes—
87C2×C3×C4⋊C4C2316yes—
88C2×C3×C4⋊C4C2316yes—
89C2×C3×C4⋊C4C23112yes—
90C2×C3×C4⋊C4C2313yes—
91C2×C3×C4⋊C4C2316yes—
92C2×C3×C42C2313yes—
93C2×C3×C42C2316yes—
94C2×C3×C42C2313yes—
95C2×C3×C42C2316yes—
96C2×C3×C42C2313yes—
97C2×C3×C42C2316yes—
98C3×#95C2311yescharacteristic
99C3×#95C2313yes—
100C3×#95C2316yes—
101C3×#95C2312yes—
102C3×#95C2316yes—
103C3×#95C2316yes—
104C3×#95C2316yes—
105C3×#95C23112yes—
106#557—28no—
107#557—28no—
108C2×C4⋊Q8C22×C313yes—
109C2×C4⋊Q8C22×C313yes—
110C2×#125C22×C313yes—
111C2×#125C22×C316yes—
112C2×#125C22×C313yes—
113C2×C4×Q8C22×C313yes—
114C2×C4×Q8C22×C313yes—
115C2×C4×Q8C22×C311yescharacteristic
116C2×C4×Q8C22×C312yes—
117C2×C4×Q8C22×C311yescharacteristic
118C2×C4×Q8C22×C311yescharacteristic
119C2×C4×Q8C22×C312yes—
120#398C22×C313yes—
121#398C22×C311yescharacteristic
122#398C22×C316yes—
123#398C22×C3112yes—
124#395C22×C313yes—
125#395C22×C316yes—
126#395C22×C313yes—
127#395C22×C316yes—
128#391C22×C316yes—
129#389C22×C316yes—
130#389C22×C316yes—
131#389C22×C316yes—
132#389C22×C316yes—
133#389C22×C3112yes—
134#384C22×C316yes—
135#384C22×C316yes—
136#382C22×C316yes—
137C4×C4⋊C4C22×C313yes—
138C4×C4⋊C4C22×C316yes—
139C4×C4⋊C4C22×C316yes—
140C4×C4⋊C4C22×C316yes—
141#376C22×C313yes—
142C2×C3×Q8—26no—
143C2×C3×Q8—26no—
144C2×C3×Q8—26no—
145C2×C3×Q8—43no—
146C2×C3×Q8—46no—
147C2×C3×Q8C2×D414yes—
148C2×C3×Q8C2×Q818yes—
149C22×C3×C4C2413yes—
150C22×C3×C4C2412yes—
151C22×C3×C4C2416yes—
152C22×C3×C4C2411yescharacteristic
153C22×C3×C4C2413yes—
154C22×C3×C4C2411yescharacteristic
155C22×C3×C4C2413yes—
156C22×C3×C4C2412yes—
157C22×C3×C4C2416yes—
158C22×C3×C4C2411yescharacteristic
159C22×C3×C4C2413yes—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—26no—
163C3×C4⋊C4—212no—
164C3×C4⋊C4—26no—
165C3×C4⋊C4—26no—
166C3×C4⋊C4—26no—
167C3×C4⋊C4—212no—
168C3×C4⋊C4—212no—
169C3×C4⋊C4—212no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—46no—
172C3×C4⋊C4—46no—
173C3×C4⋊C4—46no—
174C3×C4⋊C4—412no—
175C3×C4⋊C4C2×D4112yes—
176C3×C4⋊C4C2×Q8124yes—
177C3×C42—26no—
178C3×C42—212no—
179C3×C42—26no—
180C3×C42—46no—
181C3×C42C2×D4112yes—
182C3×C42C2×Q8124yes—
183C22×Q8C23×C313yes—
184C22×Q8C23×C313yes—
185C22×Q8C23×C311yescharacteristic
186C4⋊Q8—212no—
187C4⋊Q8—212no—
188#125—212no—
189#125—224no—
190#125—212no—
191C4×Q8—28no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—212no—
195C4×Q8—212no—
196C4×Q8C3×D418yes—
197C4×Q8C3×Q8116yescomplemented
198C2×C4⋊C4C23×C311yescharacteristic
199C2×C4⋊C4C23×C311yescharacteristic
200C2×C4⋊C4C23×C313yes—
201C2×C4⋊C4C23×C313yes—
202C2×C4⋊C4C23×C316yes—
203C2×C4⋊C4C23×C313yes—
204C2×C4⋊C4C23×C313yes—
205C2×C4⋊C4C23×C316yes—
206C2×C4⋊C4C23×C316yes—
207C2×C4⋊C4C23×C316yes—
208C2×C4⋊C4C23×C312yes—
209C2×C4⋊C4C23×C316yes—
210C2×C4⋊C4C23×C316yes—
211C2×C4⋊C4C23×C316yes—
212C2×C4⋊C4C23×C3112yes—
213C2×C4⋊C4C23×C313yes—
214C2×C4⋊C4C23×C316yes—
215C2×C42C23×C313yes—
216C2×C42C23×C316yes—
217C2×C42C23×C313yes—
218C2×C42C23×C316yes—
219C2×C42C23×C313yes—
220C2×C42C23×C316yes—
221#95C23×C311yescharacteristic
222#95C23×C313yes—
223#95C23×C316yes—
224#95C23×C312yes—
225#95C23×C316yes—
226#95C23×C316yes—
227#95C23×C316yes—
228#95C23×C3112yes—
229C23×C3C2511yescharacteristic
230C3×Q8—412no—
231C3×Q8—412no—
232C3×Q8C4⋊Q8116yes—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—23no—
236C2×C3×C4—26no—
237C2×C3×C4—23no—
238C2×C3×C4—22no—
239C2×C3×C4—26no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—23no—
245C2×C3×C4—26no—
246C2×C3×C4—23no—
247C2×C3×C4—22no—
248C2×C3×C4—23no—
249C2×C3×C4—22no—
250C2×C3×C4—26no—
251C2×C3×C4—22no—
252C2×C3×C4—26no—
253C2×C3×C4—26no—
254C2×C3×C4—23no—
255C2×C3×C4C22×D412yes—
256C2×C3×C4C22×D416yes—
257C2×C3×C4C22×D416yes—
258C2×C3×C4C22×Q814yes—
259C2×C3×C4C22×Q8112yes—
260C2×C3×C4C22×Q8112yes—
261C2×C3×C42−1+412yes—
262C2×Q8—26no—
263C2×Q8—26no—
264C2×Q8—26no—
265C2×Q8—43no—
266C2×Q8—46no—
267C2×Q8C2×C3×D414yes—
268C2×Q8C2×C3×Q818yes—
269C22×C4C24×C313yes—
270C22×C4C24×C312yes—
271C22×C4C24×C316yes—
272C22×C4C24×C311yescharacteristic
273C22×C4C24×C313yes—
274C22×C4C24×C311yescharacteristic
275C22×C4C24×C313yes—
276C22×C4C24×C312yes—
277C22×C4C24×C316yes—
278C22×C4C24×C311yescharacteristic
279C22×C4C24×C313yes—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—26no—
283C4⋊C4—212no—
284C4⋊C4—26no—
285C4⋊C4—26no—
286C4⋊C4—26no—
287C4⋊C4—212no—
288C4⋊C4—212no—
289C4⋊C4—212no—
290C4⋊C4—24no—
291C4⋊C4—46no—
292C4⋊C4—46no—
293C4⋊C4—46no—
294C4⋊C4—412no—
295C4⋊C4C2×C3×D4112yes—
296C4⋊C4C2×C3×Q8124yes—
297C42—26no—
298C42—212no—
299C42—26no—
300C42—46no—
301C42C2×C3×D4112yes—
302C42C2×C3×Q8124yes—
303C22×C3C23×D411yescharacteristic
304C22×C3C23×Q812yes—
305C22×C3C2×2−1+411yescharacteristic
306C22×C3#58512yes—
307C22×C3#58511yescharacteristic
308C3×C4—22no—
309C3×C4—26no—
310C3×C4—22no—
311C3×C4—24no—
312C3×C4—212no—
313C3×C4—41no—
314C3×C4—43no—
315C3×C4—46no—
316C3×C4—42no—
317C3×C4—43no—
318C3×C4C2×C4⋊Q8112yes—
319C23C25×C311yescharacteristic
320Q8—412no—
321Q8—412no—
322Q8C3×C4⋊Q8116yes—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—23no—
326C2×C4—26no—
327C2×C4—23no—
328C2×C4—22no—
329C2×C4—26no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—23no—
335C2×C4—26no—
336C2×C4—23no—
337C2×C4—22no—
338C2×C4—23no—
339C2×C4—22no—
340C2×C4—26no—
341C2×C4—22no—
342C2×C4—26no—
343C2×C4—26no—
344C2×C4—23no—
345C2×C4C22×C3×D412yes—
346C2×C4C22×C3×D416yes—
347C2×C4C22×C3×D416yes—
348C2×C4C22×C3×Q814yes—
349C2×C4C22×C3×Q8112yes—
350C2×C4C22×C3×Q8112yes—
351C2×C4C3×2−1+412yes—
352C2×C3C22×C4⋊Q811yescharacteristic
353C2×C3#348711yescharacteristic
354C2×C3#348911yescharacteristic
355C2×C3#350212yes—
356C2×C3#350412yes—
357C22C23×C3×D411yescharacteristic
358C22C23×C3×Q812yes—
359C22C2×C3×2−1+411yescharacteristic
360C22C3×#58512yes—
361C22C3×#58511yescharacteristic
362C4—22no—
363C4—26no—
364C4—22no—
365C4—24no—
366C4—212no—
367C4—46no—
368C4—41no—
369C4—43no—
370C4—43no—
371C4—42no—
372C4C2×C3×C4⋊Q8112yes—
373C3#3595711yescharacteristic, direct product factor, Sylow
374C2C22×C3×C4⋊Q811yescharacteristic
375C2C3×#348711yescharacteristic
376C2C3×#348911yescharacteristic
377C2C3×#350212yes—
378C2C3×#350412yes—
subgroupG/H
center, socle, prime radicalC23×C3C25
commutator, Frattini, cosocleC23C25×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
12. #35957C3direct (N × H)
74. C3×C4×Q8Q8semidirect
197. C4×Q8C3×Q8semidirect
373. C3#35957direct (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
C3×#26602
C3×#25842
C3×#25672
C3×#25522
C3×#25502
C3×#23992
C3×#23512
C3×Q8×C4⋊C42
C3×#22762
#359573