Finite Group Databasebeta

← C3×#10602724 #14289284

Subgroups — C3×#10602724 #14289284

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#556C211yesmaximal, characteristic, complemented
2C3×#65377C211yesmaximal, characteristic, complemented
3C3×#65316C211yesmaximal, characteristic, complemented
4C3×#65304C211yesmaximal, characteristic
5C3×#64767C211yesmaximal, characteristic, complemented
6C3×Q8⋊#168C211yesmaximal, characteristic, complemented
7C3×D4⋊#168C211yesmaximal, characteristic, complemented
8#10602724C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×#126—22no—
11C3×C5×#124—21no—
12C3×C5×C22⋊Q8—21no—
13C3×C4×C5×Q8—21no—
14C3×C4×C5×Q8C2211yescharacteristic
15C3×C4×C5×D4C2211yescharacteristic
16C3×C5×#117—21no—
17C2×C3×#688—21no—
18C2×C3×#686—21no—
19C3×#4020C2211yescharacteristic
20C3×#3941—21no—
21C3×#3941—21no—
22C3×#3941—22no—
23C3×C4⋊#168C2211yescharacteristic
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1628—51nomaximal, Hall
27C5×#556C2×C311yescharacteristic, complemented
28#65377C2×C311yescharacteristic, complemented
29#65316C2×C311yescharacteristic, complemented
30#65304C2×C311yescharacteristic
31#64767C2×C311yescharacteristic, complemented
32Q8⋊#168C2×C311yescharacteristic, complemented
33D4⋊#168C2×C311yescharacteristic, complemented
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C3×C5×D4D411yescharacteristic
37C22×C3×C4×C5—21no—
38C2×C3×D20—21no—
39C2×C3×D20—21no—
40C2×C3×D20—21no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4D411yescharacteristic
45C3×C5×C4⋊C4D411yescharacteristic
46C3×C5×C4⋊C4D412yes—
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—22no—
49C3×C5×C22⋊C4—21no—
50C3×C42×C5—22no—
51C3×C42×C5—21no—
52C3×C42×C5C2311yescharacteristic
53C3×#688—42no—
54C3×#686—42no—
55C2×C3×#168—21no—
56C2×C3×#168—21no—
57C2×C3×#168—21no—
58C3×#556D511yescharacteristic, complemented
59C3×#486—51no—
60C3×#460—51no—
61C3×#459—51no—
62C3×#332—51no—
63C3×Q8⋊C8—51no—
64C3×D4⋊C8—51no—
65C5×C4⋊Q8C22×C311yescharacteristic
66C5×#126—22no—
67C5×#124—21no—
68C5×C22⋊Q8—21no—
69C4×C5×Q8—21no—
70C4×C5×Q8C22×C311yescharacteristic
71C4×C5×D4C22×C311yescharacteristic
72C5×#117—21no—
73C2×#688—21no—
74C2×#686—21no—
75#4020C22×C311yescharacteristic
76#3941—21no—
77#3941—21no—
78#3941—22no—
79C4⋊#168C22×C311yescharacteristic
80C4⋊#168C22×C311yescharacteristic
81C4×#168C22×C311yescharacteristic
82#1628—51noSylow
83C23×C3×C5—21no—
84C22×C3×D5—41no—
85C3×C5×Q8—21no—
86C3×C5×Q8—21no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×D4—21no—
90C3×C5×D4—21no—
91C2×C3×C4×C5—21no—
92C2×C3×C4×C5—21no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—22no—
95C2×C3×C4×C5—41no—
96C2×C3×C4×C5—41no—
97C2×C3×C4×C5C2×D411yescharacteristic
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C3×D20—42no—
101C3×D20—41no—
102C3×D20—41no—
103C3×D20—41no—
104C3×D20—41no—
105C3×#168—41no—
106C3×#168—41no—
107C3×#168—41no—
108C2×C3×SD16—101no—
109C2×C3×D8—101no—
110C3×C4⋊Q8C2×D511yescharacteristic
111C3×#127—51no—
112C3×#126—22no—
113C3×#124—21no—
114C3×C22⋊Q8—21no—
115C3×C4×Q8—21no—
116C3×C4×Q8C2×D511yescharacteristic
117C3×C4×D4C2×D511yescharacteristic
118C3×#117—21no—
119C3×C4⋊C8—51no—
120C3×C4⋊C8—51no—
121C3×#102—101no—
122C3×#102—101no—
123C3×#102—102no—
124C3×C4×C8—51no—
125C2×C5×Q8—21no—
126C2×C5×Q8C3×D411yescharacteristic
127C2×C5×D4C3×D411yescharacteristic
128C22×C4×C5—21no—
129C2×D20—21no—
130C2×D20—21no—
131C2×D20—21no—
132C5×C4⋊C4—22no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4—21no—
135C5×C4⋊C4C3×D411yescharacteristic
136C5×C4⋊C4C3×D411yescharacteristic
137C5×C4⋊C4C3×D412yes—
138C5×C22⋊C4—22no—
139C5×C22⋊C4—22no—
140C5×C22⋊C4—21no—
141C42×C5—22no—
142C42×C5—21no—
143C42×C5C23×C311yescharacteristic
144#688—42no—
145#686—42no—
146C2×#168—21no—
147C2×#168—21no—
148C2×#168—21no—
149#556C3×D511yescharacteristic, complemented
150#486—51no—
151#460—51no—
152#459—51no—
153#332—51no—
154Q8⋊C8—51no—
155D4⋊C8—51no—
156C22×C3×C5—41no—
157C22×C3×C5—41no—
158C22×C3×C5—41no—
159C22×C3×C5#12011yescharacteristic
160C2×C3×D5—81no—
161C2×C3×D5—81no—
162C2×C3×D5—81no—
163C3×C4×C5—21no—
164C3×C4×C5—41no—
165C3×C4×C5—41no—
166C3×C4×C5—42no—
167C3×C4×C5—41no—
168C3×C4×C5#13511yescharacteristic
169C3×C4×C5#13511yescharacteristic
170C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
171C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
172C2×C3×Q8—21no—
173C2×C3×Q8C22⋊D511yescharacteristic
174C2×C3×D4—101no—
175C2×C3×D4—101no—
176C2×C3×D4—101no—
177C2×C3×D4C22⋊D511yescharacteristic
178C22×C3×C4—21no—
179C3×SD16—202no—
180C3×D8—202no—
181C2×C3×C8—101no—
182C2×C3×C8—101no—
183C2×C3×C8—101no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4C22⋊D511yescharacteristic
188C3×C4⋊C4C22⋊D511yescharacteristic
189C3×C4⋊C4C22⋊D512yes—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—21no—
193C3×C42—22no—
194C3×C42—21no—
195C3×C42C22×D511yescharacteristic
196C23×C5—21no—
197C22×D5—41no—
198C5×Q8—21no—
199C5×Q8—21no—
200C5×Q8—41no—
201C5×Q8—41no—
202C5×D4—21no—
203C5×D4—21no—
204C2×C4×C5—21no—
205C2×C4×C5—21no—
206C2×C4×C5—21no—
207C2×C4×C5—22no—
208C2×C4×C5—41no—
209C2×C4×C5—41no—
210C2×C4×C5C2×C3×D411yescharacteristic
211C2×C4×C5C2×C3×D411yescharacteristic
212C2×C4×C5C2×C3×D411yescharacteristic
213D20—42no—
214D20—41no—
215D20—41no—
216D20—41no—
217D20—41no—
218#168—41no—
219#168—41no—
220#168—41no—
221C2×SD16—101no—
222C2×D8—101no—
223C4⋊Q8C2×C3×D511yescharacteristic
224#127—51no—
225#126—22no—
226#124—21no—
227C22⋊Q8—21no—
228C4×Q8—21no—
229C4×Q8C2×C3×D511yescharacteristic
230C4×D4C2×C3×D511yescharacteristic
231#117—21no—
232C4⋊C8—51no—
233C4⋊C8—51no—
234#102—101no—
235#102—101no—
236#102—102no—
237C4×C8—51no—
238C2×C3×C5—81no—
239C2×C3×C5#44911yescharacteristic
240C2×C3×C5#44911yescharacteristic
241C2×C3×C5#45411yescharacteristic
242C3×D5—161no—
243C23×C3—21no—
244C23×C3—201no—
245C3×Q8—21no—
246C3×Q8—21no—
247C3×Q8—41no—
248C3×Q8—41no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—202no—
252C3×D4—201no—
253C3×D4—201no—
254C3×D4—201no—
255C3×D4—201no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—22no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4C2×C22⋊D511yescharacteristic
263C2×C3×C4C2×C22⋊D511yescharacteristic
264C2×C3×C4C2×C22⋊D511yescharacteristic
265C3×C8—201no—
266C3×C8—201no—
267C3×C8—201no—
268C22×C5—41no—
269C22×C5—41no—
270C22×C5—41no—
271C22×C5C3×#12011yescharacteristic
272C2×D5—81no—
273C2×D5—81no—
274C2×D5—81no—
275C4×C5—21no—
276C4×C5—41no—
277C4×C5—41no—
278C4×C5—42no—
279C4×C5—41no—
280C4×C5C3×#13511yescharacteristic
281C4×C5C3×#13511yescharacteristic
282C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
283C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
284C2×Q8—21no—
285C2×Q8C3×C22⋊D511yescharacteristic
286C2×D4—101no—
287C2×D4—101no—
288C2×D4—101no—
289C2×D4C3×C22⋊D511yescharacteristic
290C22×C4—21no—
291SD16—202no—
292D8—202no—
293C2×C8—101no—
294C2×C8—101no—
295C2×C8—101no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4C3×C22⋊D511yescharacteristic
300C4⋊C4C3×C22⋊D511yescharacteristic
301C4⋊C4C3×C22⋊D512yes—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—21no—
305C42—22no—
306C42—21no—
307C42C22×C3×D511yescharacteristic
308C3×C5#162811yescharacteristic, complemented, Hall
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—401no—
313C22×C3—401no—
314C22×C3—401no—
315C22×C3#409911yescharacteristic
316C3×C4—21no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—42no—
320C3×C4—41no—
321C3×C4#409511yescharacteristic
322C3×C4#409511yescharacteristic
323C3×C4#409611yescharacteristic
324C3×C4#409611yescharacteristic
325C2×C5—81no—
326C2×C5C3×#44911yescharacteristic
327C2×C5C3×#44911yescharacteristic
328C2×C5C3×#45411yescharacteristic
329D5—161no—
330C23—21no—
331C23—201no—
332Q8—21no—
333Q8—21no—
334Q8—41no—
335Q8—41no—
336D4—21no—
337D4—21no—
338D4—202no—
339D4—201no—
340D4—201no—
341D4—201no—
342D4—201no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4C2×C3×C22⋊D511yescharacteristic
350C2×C4C2×C3×C22⋊D511yescharacteristic
351C2×C4C2×C3×C22⋊D511yescharacteristic
352C8—201no—
353C8—201no—
354C8—201no—
355C2×C3—81no—
356C2×C3—801no—
357C2×C3#6552711yescharacteristic
358C2×C3#6552711yescharacteristic
359C2×C3#6553411yescharacteristic
360C5C3×#162811yescharacteristic, complemented, Sylow
361C22—41no—
362C22—41no—
363C22—41no—
364C22—401no—
365C22—401no—
366C22—401no—
367C22C3×#409911yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—41no—
373C4C3×#409511yescharacteristic
374C4C3×#409511yescharacteristic
375C4C3×#409611yescharacteristic
376C4C3×#409611yescharacteristic
377C3#1060272411yescharacteristic, direct product factor, Sylow
378C2—81no—
379C2—801no—
380C2C3×#6552711yescharacteristic
381C2C3×#6552711yescharacteristic
382C2C3×#6553411yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#556C2
socleC22×C3×C5#120
prime radicalC3×#65304C2
hypercenterC3×#556D5
hypocenterC5C3×#1628
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#556

Complements

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

NH ≅ G/Ntype
1. C3×C5×#556C2semidirect
2. C3×#65377C2semidirect
3. C3×#65316C2semidirect
5. C3×#64767C2semidirect
6. C3×Q8⋊#168C2semidirect
7. C3×D4⋊#168C2semidirect
8. #10602724C3direct (N × H)
27. C5×#556C2×C3semidirect
28. #65377C2×C3semidirect
29. #65316C2×C3semidirect
31. #64767C2×C3semidirect
32. Q8⋊#168C2×C3semidirect
33. D4⋊#168C2×C3semidirect
58. C3×#556D5semidirect
149. #556C3×D5semidirect
308. C3×C5#1628semidirect
360. C5C3×#1628semidirect
377. C3#10602724direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5562
C3×#653772
C3×#653162
C3×#653042
C3×#647672
C3×Q8⋊#1682
C3×D4⋊#1682
#106027243
C3×#16285