Finite Group Databasebeta

← C2×C3×#3080 #10679090

Subgroups — C2×C3×#3080 #10679090

Order 768.

2142 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic, complemented
2C2×C3×C4×#41C211yesmaximal, characteristic
3C3×#3080C2116yesmaximal, direct product factor
4C2×C3×#479C214yesmaximal, complemented
5C2×C3×#478C214yesmaximal, complemented
6C2×C3×#428C212yesmaximal, complemented
7C2×C3×#423C211yesmaximal, characteristic, complemented
8C2×C3×#417C212yesmaximal
9C2×#3080C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41C2211yescharacteristic
11C22×C3×M16C2212yes—
12C2×C3×#125—21no—
13C2×C3×#125—21no—
14C2×C3×#125—24no—
15C2×C3×#125C2214yes—
16C2×C3×#125C2218yescomplemented
17C3×C4×#41—24no—
18C3×C4×#41C2218yes—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×D4—22no—
21C2×C3×C4×D4C2212yes—
22C2×C3×#117—21no—
23C2×C3×#117C2211yescharacteristic
24C22×C3×C4⋊C4—22no—
25C22×C3×C4⋊C4C2212yes—
26C22×C3×C42—21no—
27C22×C3×C42C2211yescharacteristic
28C3×#479C22132yescomplemented
29C3×#478C22132yescomplemented
30C3×#428C22116yescomplemented
31C2×C3×#107C2214yes—
32C2×C3×#106C2214yes—
33C3×#423C2218yescomplemented
34C2×C3×C4⋊C8C2214yes—
35C3×#417C22116yes—
36C2×C3×#103C2214yes—
37C2×C3×#102C2214yes—
38C22×#125C2×C311yescharacteristic, complemented
39C2×C4×#41C2×C311yescharacteristic
40#3080C2×C3116yesdirect product factor
41C2×#479C2×C314yescomplemented
42C2×#478C2×C314yescomplemented
43C2×#428C2×C312yescomplemented
44C2×#423C2×C311yescharacteristic, complemented
45C2×#417C2×C312yes—
46C2×C3×#41—22no—
47C2×C3×#41—24no—
48C2×C3×#41Q818yes—
49C2×C3×#41C2314yes—
50C22×C3×Q8C2311yescharacteristic
51C22×C3×D4—21no—
52C22×C3×D4C2311yescharacteristic
53C23×C3×C4—21no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C2×C3×M16—28no—
59C2×C3×M16C2318yes—
60C22×C3×C8C2314yes—
61C3×#125—28no—
62C3×#125—28no—
63C3×#125—28no—
64C3×#125C23116yes—
65C3×C4×Q8—24no—
66C3×C4×Q8C2318yes—
67C3×C4×D4—216no—
68C3×C4×D4—24no—
69C3×C4×D4C2318yes—
70C3×#117—22no—
71C3×#117—28no—
72C3×#117C2314yes—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—216no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4C2314yes—
80C2×C3×C4⋊C4C2318yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—21no—
85C2×C3×C42—22no—
86C2×C3×C42—28no—
87C2×C3×C42D411yescharacteristic
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D414yes—
90C2×C3×C42C2312yes—
91C2×C3×C42C2314yes—
92C3×#107C23116yes—
93C3×#106C23116yes—
94C3×C4⋊C8C23116yes—
95C3×#103C23116yes—
96C3×#102C23116yes—
97C22×#41C22×C311yescharacteristic
98C22×M16C22×C312yes—
99C2×#125—21no—
100C2×#125—24no—
101C2×#125—21no—
102C2×#125C22×C314yes—
103C2×#125C22×C318yescomplemented
104C4×#41—24no—
105C4×#41C22×C318yes—
106C2×C4×Q8C22×C312yes—
107C2×C4×D4—22no—
108C2×C4×D4C22×C312yes—
109C2×#117—21no—
110C2×#117C22×C311yescharacteristic
111C22×C4⋊C4—22no—
112C22×C4⋊C4C22×C312yes—
113C22×C42—21no—
114C22×C42C22×C311yescharacteristic
115#479C22×C3132yescomplemented
116#478C22×C3132yescomplemented
117#428C22×C3116yescomplemented
118C2×#107C22×C314yes—
119C2×#106C22×C314yes—
120#423C22×C318yescomplemented
121C2×C4⋊C8C22×C314yes—
122#417C22×C3116yes—
123C2×#103C22×C314yes—
124C2×#102C22×C314yes—
125C24×C3—21no—
126C24×C3C2×D411yescharacteristic
127C3×#41—216no—
128C3×#41—28no—
129C3×#41C2×Q8116yes—
130C2×C3×Q8—21no—
131C2×C3×Q8—22no—
132C2×C3×Q8C2×Q814yes—
133C2×C3×Q8C2412yes—
134C2×C3×D4—28no—
135C2×C3×D4—24no—
136C2×C3×D4—21no—
137C2×C3×D4—22no—
138C2×C3×D4C2×Q814yes—
139C2×C3×D4C2412yes—
140C22×C3×C4—21no—
141C22×C3×C4—28no—
142C22×C3×C4—24no—
143C22×C3×C4—28no—
144C22×C3×C4—22no—
145C22×C3×C4—24no—
146C22×C3×C4—28no—
147C22×C3×C4—44no—
148C22×C3×C4—48no—
149C22×C3×C4—48no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D418yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×Q814yes—
156C22×C3×C4#4114yes—
157C22×C3×C4C2412yes—
158C22×C3×C4C2411yescharacteristic
159C3×M16—216no—
160C2×C3×C8—28no—
161C2×C3×C8C2418yes—
162C3×C4⋊C4—216no—
163C3×C4⋊C4—216no—
164C3×C4⋊C4—28no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—28no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4C2418yes—
169C3×C4⋊C4C2414yes—
170C3×C22⋊C4—24no—
171C3×C22⋊C4—28no—
172C3×C42—24no—
173C3×C42—28no—
174C3×C42C2×D414yes—
175C3×C42C2×D414yes—
176C3×C42C2×D414yes—
177C3×C42C2414yes—
178C2×#41—22no—
179C2×#41—24no—
180C2×#41C3×Q818yes—
181C2×#41C23×C314yes—
182C22×Q8C23×C311yescharacteristic
183C22×D4—21no—
184C22×D4C23×C311yescharacteristic
185C23×C4—21no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4C3×D412yes—
189C23×C4C23×C311yescharacteristic
190C2×M16—28no—
191C2×M16C23×C318yes—
192C22×C8C23×C314yes—
193#125—28no—
194#125—28no—
195#125—28no—
196#125C23×C3116yes—
197C4×Q8—24no—
198C4×Q8C23×C318yes—
199C4×D4—216no—
200C4×D4—24no—
201C4×D4C23×C318yes—
202#117—22no—
203#117—28no—
204#117C23×C314yes—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—216no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C23×C314yes—
212C2×C4⋊C4C23×C318yes—
213C2×C4⋊C4C23×C312yes—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—28no—
219C2×C42C3×D411yescharacteristic
220C2×C42C3×D414yes—
221C2×C42C3×D411yescharacteristic
222C2×C42C23×C312yes—
223C2×C42C23×C314yes—
224#107C23×C3116yes—
225#106C23×C3116yes—
226C4⋊C8C23×C3116yes—
227#103C23×C3116yes—
228#102C23×C3116yes—
229C23×C3—24no—
230C23×C3—24no—
231C23×C3—22no—
232C23×C3—44no—
233C23×C3C22⋊Q814yes—
234C23×C3C22×D412yes—
235C23×C3C22×D411yescharacteristic
236C3×Q8—24no—
237C3×Q8—22no—
238C3×Q8C22×Q814yes—
239C3×D4—216no—
240C3×D4—24no—
241C3×D4—22no—
242C3×D4C22×Q814yes—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—216no—
248C2×C3×C4—24no—
249C2×C3×C4—28no—
250C2×C3×C4—48no—
251C2×C3×C4—416no—
252C2×C3×C4—416no—
253C2×C3×C4—44no—
254C2×C3×C4—48no—
255C2×C3×C4C22⋊Q814yes—
256C2×C3×C4C22⋊Q818yes—
257C2×C3×C4C22×D414yes—
258C2×C3×C4C22×D414yes—
259C2×C3×C4C22×D412yes—
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C22×Q812yes—
262C2×C3×C4C22×Q814yes—
263C2×C3×C4C2×#4112yes—
264C2×C3×C4C2×#4114yes—
265C2×C3×C4C2511yescharacteristic
266C3×C8—28no—
267C24—21no—
268C24C2×C3×D411yescharacteristic
269#41—216no—
270#41—28no—
271#41C2×C3×Q8116yes—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8C2×C3×Q814yes—
275C2×Q8C24×C312yes—
276C2×D4—28no—
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4C2×C3×Q814yes—
281C2×D4C24×C312yes—
282C22×C4—21no—
283C22×C4—28no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—44no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4C2×C3×D412yes—
293C22×C4C2×C3×D412yes—
294C22×C4C2×C3×D418yes—
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C2×C3×D412yes—
297C22×C4C2×C3×Q814yes—
298C22×C4C3×#4114yes—
299C22×C4C24×C311yescharacteristic
300C22×C4C24×C312yes—
301M16—216no—
302C2×C8—28no—
303C2×C8C24×C318yes—
304C4⋊C4—216no—
305C4⋊C4—216no—
306C4⋊C4—28no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—22no—
310C4⋊C4C24×C318yes—
311C4⋊C4C24×C314yes—
312C22⋊C4—24no—
313C22⋊C4—28no—
314C42—24no—
315C42—28no—
316C42C2×C3×D414yes—
317C42C2×C3×D414yes—
318C42C2×C3×D414yes—
319C42C24×C314yes—
320C22×C3—28no—
321C22×C3—24no—
322C22×C3—24no—
323C22×C3—48no—
324C22×C3—44no—
325C22×C3C2×C22⋊Q814yes—
326C22×C3C2×C22⋊Q812yes—
327C22×C3#57514yes—
328C22×C3C23×D411yescharacteristic
329C3×C4—24no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—44no—
333C3×C4—48no—
334C3×C4C2×C22⋊Q814yes—
335C3×C4C2×C22⋊Q812yes—
336C3×C4C23×Q811yescharacteristic
337C3×C4C22×#4111yescharacteristic
338C23—24no—
339C23—22no—
340C23—24no—
341C23—44no—
342C23C3×C22⋊Q814yes—
343C23C22×C3×D412yes—
344C23C22×C3×D411yescharacteristic
345Q8—22no—
346Q8—24no—
347Q8C22×C3×Q814yes—
348D4—216no—
349D4—22no—
350D4—24no—
351D4C22×C3×Q814yes—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—216no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—48no—
360C2×C4—416no—
361C2×C4—416no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4C3×C22⋊Q814yes—
365C2×C4C3×C22⋊Q818yes—
366C2×C4C22×C3×D414yes—
367C2×C4C22×C3×D414yes—
368C2×C4C22×C3×D412yes—
369C2×C4C22×C3×D411yescharacteristic
370C2×C4C22×C3×Q812yes—
371C2×C4C22×C3×Q814yes—
372C2×C4C2×C3×#4112yes—
373C2×C4C2×C3×#4114yes—
374C2×C4C25×C311yescharacteristic
375C8—28no—
376C2×C3—24no—
377C2×C3—44no—
378C2×C3#308014yesdirect product factor
379C2×C3C22×C22⋊Q811yescharacteristic
380C2×C3C2×#57512yes—
381C22—24no—
382C22—28no—
383C22—24no—
384C22—44no—
385C22—48no—
386C22C2×C3×C22⋊Q814yes—
387C22C2×C3×C22⋊Q812yes—
388C22C3×#57514yes—
389C22C23×C3×D411yescharacteristic
390C4—24no—
391C4—24no—
392C4—24no—
393C4—44no—
394C4—48no—
395C4C2×C3×C22⋊Q814yes—
396C4C2×C3×C22⋊Q812yes—
397C4C23×C3×Q811yescharacteristic
398C4C22×C3×#4111yescharacteristic
399C3C2×#308011yescharacteristic, direct product factor, Sylow
400C2—24no—
401C2—44no—
402C2C3×#308014yesdirect product factor
403C2C22×C3×C22⋊Q811yescharacteristic
404C2C2×C3×#57512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC22×C3×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#125C2semidirect
3. C3×#3080C2direct (N × H)
4. C2×C3×#479C2semidirect
5. C2×C3×#478C2semidirect
6. C2×C3×#428C2semidirect
7. C2×C3×#423C2semidirect
9. C2×#3080C3direct (N × H)
16. C2×C3×#125C22semidirect
28. C3×#479C22semidirect
29. C3×#478C22semidirect
30. C3×#428C22semidirect
33. C3×#423C22semidirect
38. C22×#125C2×C3semidirect
40. #3080C2×C3direct (N × H)
41. C2×#479C2×C3semidirect
42. C2×#478C2×C3semidirect
43. C2×#428C2×C3semidirect
44. C2×#423C2×C3semidirect
103. C2×#125C22×C3semidirect
115. #479C22×C3semidirect
116. #478C22×C3semidirect
117. #428C22×C3semidirect
120. #423C22×C3semidirect
378. C2×C3#3080direct (N × H)
399. C3C2×#3080direct (N × H)
402. C2C3×#3080direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×C3×C4×#412
C3×#30802
C2×C3×#4792
C2×C3×#4782
C2×C3×#4282
C2×C3×#4232
C2×C3×#4172
C2×#30803