Finite Group Databasebeta

← C3×#10602712 #14289425

Subgroups — C3×#10602712 #14289425

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#550C211yesmaximal, characteristic
2C3×#65365C211yesmaximal, characteristic, complemented
3C3×#65310C211yesmaximal, characteristic, complemented
4C3×#65297C211yesmaximal, characteristic, complemented
5C3×#64764C211yesmaximal, characteristic, complemented
6C3×Q8⋊#168C211yesmaximal, characteristic, complemented
7C3×D4⋊#168C211yesmaximal, characteristic, complemented
8#10602712C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41—21no—
10C3×C5×#127—21no—
11C3×C5×#127C2211yescharacteristic
12C3×C5×#121—22no—
13C3×C5×#121—21no—
14C3×C4×C5×Q8C2211yescharacteristic
15C3×C4×C5×D4—21no—
16C3×C4×C5×D4C2211yescharacteristic
17C2×C3×#687—22no—
18C3×#4015C2211yescharacteristic
19C3×#3967—21no—
20C3×#3964—21no—
21C3×#3940—21no—
22C3×#3939—21no—
23C3×C4⋊#168C2211yescharacteristic
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1631—51nomaximal, Hall
27C5×#550C2×C311yescharacteristic
28#65365C2×C311yescharacteristic, complemented
29#65310C2×C311yescharacteristic, complemented
30#65297C2×C311yescharacteristic, complemented
31#64764C2×C311yescharacteristic, complemented
32Q8⋊#168C2×C311yescharacteristic, complemented
33D4⋊#168C2×C311yescharacteristic, complemented
34C3×C5×#41—42no—
35C3×C5×#41—42no—
36C2×C3×C5×Q8D411yescharacteristic
37C2×C3×C5×D4—22no—
38C2×C3×C5×D4—22no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4D411yescharacteristic
42C2×C3×C5×D4D412yes—
43C22×C3×C4×C5—22no—
44C22×C3×C4×C5—21no—
45C2×C3×Dic10—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4D411yescharacteristic
48C3×C5×C4⋊C4D411yescharacteristic
49C3×C5×C22⋊C4—22no—
50C3×C5×C22⋊C4—21no—
51C3×C42×C5—21no—
52C3×C42×C5C2311yescharacteristic
53C3×#687—44no—
54C3×C4⋊Dic5—21no—
55C3×C4⋊Dic5—21no—
56C2×C3×#168—21no—
57C2×C3×#168—21no—
58C2×C3×#168—21no—
59C3×#550D511yescharacteristic
60C3×#492—51no—
61C3×#477—51no—
62C3×#476—51no—
63C3×#331—51no—
64C3×Q8⋊C8—51no—
65C3×D4⋊C8—51no—
66C2×C5×#41—21no—
67C5×#127—21no—
68C5×#127C22×C311yescharacteristic
69C5×#121—22no—
70C5×#121—21no—
71C4×C5×Q8C22×C311yescharacteristic
72C4×C5×D4—21no—
73C4×C5×D4C22×C311yescharacteristic
74C2×#687—22no—
75#4015C22×C311yescharacteristic
76#3967—21no—
77#3964—21no—
78#3940—21no—
79#3939—21no—
80C4⋊#168C22×C311yescharacteristic
81C4⋊#168C22×C311yescharacteristic
82C4×#168C22×C311yescharacteristic
83#1631—51noSylow
84C23×C3×C5—21no—
85C23×C3×C5—22no—
86C3×C5×Q8—21no—
87C3×C5×Q8SD1612yescomplemented
88C3×C5×D4—21no—
89C3×C5×D4—21no—
90C3×C5×D4—44no—
91C3×C5×D4—42no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C3×C5×D4—41no—
95C3×C5×D4—41no—
96C3×C5×D4—82no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—44no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×Dic5—41no—
107C3×Dic10—42no—
108C3×#168—22no—
109C3×#168—41no—
110C3×#168—41no—
111C2×C3×#41—21no—
112C2×C3×SD16—102no—
113C3×C4⋊Q8—51no—
114C3×#127—21no—
115C3×#127C2×D511yescharacteristic
116C3×#121—22no—
117C3×#121—21no—
118C3×C4×Q8C2×D511yescharacteristic
119C3×C4×D4—21no—
120C3×C4×D4C2×D511yescharacteristic
121C3×#107—101no—
122C3×#106—101no—
123C3×C4⋊C8—51no—
124C3×C4⋊C8—51no—
125C3×#103—101no—
126C3×#102—101no—
127C3×C4×C8—51no—
128C5×#41—42no—
129C5×#41—42no—
130C2×C5×Q8C3×D411yescharacteristic
131C2×C5×D4—22no—
132C2×C5×D4—22no—
133C2×C5×D4—21no—
134C2×C5×D4—21no—
135C2×C5×D4C3×D411yescharacteristic
136C2×C5×D4C3×D412yes—
137C22×C4×C5—22no—
138C22×C4×C5—21no—
139C2×Dic10—21no—
140C5×C4⋊C4—21no—
141C5×C4⋊C4C3×D411yescharacteristic
142C5×C4⋊C4C3×D411yescharacteristic
143C5×C22⋊C4—22no—
144C5×C22⋊C4—21no—
145C42×C5—21no—
146C42×C5C23×C311yescharacteristic
147#687—44no—
148C4⋊Dic5—21no—
149C4⋊Dic5—21no—
150C2×#168—21no—
151C2×#168—21no—
152C2×#168—21no—
153#550C3×D511yescharacteristic
154#492—51no—
155#477—51no—
156#476—51no—
157#331—51no—
158Q8⋊C8—51no—
159D4⋊C8—51no—
160C22×C3×C5—42no—
161C22×C3×C5—41no—
162C22×C3×C5—41no—
163C22×C3×C5—42no—
164C22×C3×C5—42no—
165C22×C3×C5—41no—
166C22×C3×C5#12011yescharacteristic
167C3×C4×C5—22no—
168C3×C4×C5—21no—
169C3×C4×C5—41no—
170C3×C4×C5—41no—
171C3×C4×C5C2×SD1611yescharacteristic
172C3×C4×C5#13511yescharacteristic
173C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
174C3×C4×C5#13711yescharacteristic
175C3×Dic5—81no—
176C3×#41—42no—
177C3×#41—42no—
178C2×C3×Q8—101no—
179C2×C3×Q8C22⋊D511yescharacteristic
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4C22⋊D511yescharacteristic
185C2×C3×D4C22⋊D512yes—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C3×SD16—204no—
189C2×C3×C8—101no—
190C2×C3×C8—101no—
191C2×C3×C8—101no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—101no—
194C3×C4⋊C4—101no—
195C3×C4⋊C4C22⋊D511yescharacteristic
196C3×C4⋊C4C22⋊D511yescharacteristic
197C3×C22⋊C4—22no—
198C3×C22⋊C4—21no—
199C3×C42—21no—
200C3×C42C22×D511yescharacteristic
201C23×C5—21no—
202C23×C5—22no—
203C5×Q8—21no—
204C5×Q8C3×SD1612yescomplemented
205C5×D4—21no—
206C5×D4—21no—
207C5×D4—42no—
208C5×D4—44no—
209C5×D4—41no—
210C5×D4—41no—
211C5×D4—41no—
212C5×D4—41no—
213C5×D4—82no—
214C2×C4×C5—21no—
215C2×C4×C5—21no—
216C2×C4×C5—21no—
217C2×C4×C5—44no—
218C2×C4×C5—41no—
219C2×C4×C5—41no—
220C2×C4×C5C2×C3×D411yescharacteristic
221C2×C4×C5C2×C3×D411yescharacteristic
222C2×C4×C5C2×C3×D411yescharacteristic
223C2×Dic5—41no—
224Dic10—42no—
225#168—22no—
226#168—41no—
227#168—41no—
228C2×#41—21no—
229C2×SD16—102no—
230C4⋊Q8—51no—
231#127—21no—
232#127C2×C3×D511yescharacteristic
233#121—22no—
234#121—21no—
235C4×Q8C2×C3×D511yescharacteristic
236C4×D4—21no—
237C4×D4C2×C3×D511yescharacteristic
238#107—101no—
239#106—101no—
240C4⋊C8—51no—
241C4⋊C8—51no—
242#103—101no—
243#102—101no—
244C4×C8—51no—
245C2×C3×C5—81no—
246C2×C3×C5—82no—
247C2×C3×C5#45011yescharacteristic
248C2×C3×C5#45211yescharacteristic
249C2×C3×C5#45311yescharacteristic
250C23×C3—21no—
251C23×C3—22no—
252C3×Q8—21no—
253C3×Q8—202no—
254C3×Q8#68712yescomplemented
255C3×D4—21no—
256C3×D4—21no—
257C3×D4—42no—
258C3×D4—44no—
259C3×D4—41no—
260C3×D4—41no—
261C3×D4—41no—
262C3×D4—41no—
263C3×D4—82no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—44no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—201no—
271C2×C3×C4C2×C22⋊D511yescharacteristic
272C2×C3×C4C2×C22⋊D511yescharacteristic
273C2×C3×C4C2×C22⋊D511yescharacteristic
274C3×C8—102no—
275C3×C8—201no—
276C3×C8—201no—
277C22×C5—42no—
278C22×C5—41no—
279C22×C5—41no—
280C22×C5—42no—
281C22×C5—42no—
282C22×C5—41no—
283C22×C5C3×#12011yescharacteristic
284C4×C5—22no—
285C4×C5—21no—
286C4×C5—41no—
287C4×C5—41no—
288C4×C5C2×C3×SD1611yescharacteristic
289C4×C5C3×#13511yescharacteristic
290C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
291C4×C5C3×#13711yescharacteristic
292Dic5—81no—
293#41—42no—
294#41—42no—
295C2×Q8—101no—
296C2×Q8C3×C22⋊D511yescharacteristic
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4C3×C22⋊D511yescharacteristic
302C2×D4C3×C22⋊D512yes—
303C22×C4—22no—
304C22×C4—21no—
305SD16—204no—
306C2×C8—101no—
307C2×C8—101no—
308C2×C8—101no—
309C4⋊C4—21no—
310C4⋊C4—101no—
311C4⋊C4—101no—
312C4⋊C4C3×C22⋊D511yescharacteristic
313C4⋊C4C3×C22⋊D511yescharacteristic
314C22⋊C4—22no—
315C22⋊C4—21no—
316C42—21no—
317C42C22×C3×D511yescharacteristic
318C3×C5#163111yescharacteristic, complemented, Hall
319C22×C3—42no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—41no—
325C22×C3#409911yescharacteristic
326C3×C4—22no—
327C3×C4—21no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—401no—
331C3×C4#407811yescharacteristic
332C3×C4C2×#68711yescharacteristic
333C3×C4#409611yescharacteristic
334C3×C4#409711yescharacteristic
335C2×C5—81no—
336C2×C5—82no—
337C2×C5C3×#45011yescharacteristic
338C2×C5C3×#45211yescharacteristic
339C2×C5C3×#45311yescharacteristic
340C23—21no—
341C23—22no—
342Q8—21no—
343Q8—202no—
344Q8C3×#68712yescomplemented
345D4—21no—
346D4—21no—
347D4—42no—
348D4—44no—
349D4—41no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—201no—
361C2×C4C2×C3×C22⋊D511yescharacteristic
362C2×C4C2×C3×C22⋊D511yescharacteristic
363C2×C4C2×C3×C22⋊D511yescharacteristic
364C8—102no—
365C8—201no—
366C8—201no—
367C2×C3—81no—
368C2×C3—82no—
369C2×C3#6550711yescharacteristic
370C2×C3#6552811yescharacteristic
371C2×C3#6553011yescharacteristic
372C5C3×#163111yescharacteristic, complemented, Sylow
373C22—42no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22C3×#409911yescharacteristic
380C4—22no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—401no—
385C4C3×#407811yescharacteristic
386C4C2×C3×#68711yescharacteristic
387C4C3×#409611yescharacteristic
388C4C3×#409711yescharacteristic
389C3#1060271211yescharacteristic, direct product factor, Sylow
390C2—81no—
391C2—82no—
392C2C3×#6550711yescharacteristic
393C2C3×#6552811yescharacteristic
394C2C3×#6553011yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
Fitting, prime radicalC3×C5×#550C2
socleC22×C3×C5#120
hypercenterC3×#550D5
hypocenterC5C3×#1631
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#550

Complements

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

NH ≅ G/Ntype
2. C3×#65365C2semidirect
3. C3×#65310C2semidirect
4. C3×#65297C2semidirect
5. C3×#64764C2semidirect
6. C3×Q8⋊#168C2semidirect
7. C3×D4⋊#168C2semidirect
8. #10602712C3direct (N × H)
28. #65365C2×C3semidirect
29. #65310C2×C3semidirect
30. #65297C2×C3semidirect
31. #64764C2×C3semidirect
32. Q8⋊#168C2×C3semidirect
33. D4⋊#168C2×C3semidirect
87. C3×C5×Q8SD16semidirect
204. C5×Q8C3×SD16semidirect
254. C3×Q8#687semidirect
318. C3×C5#1631semidirect
344. Q8C3×#687semidirect
372. C5C3×#1631semidirect
389. C3#10602712direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5502
C3×#653652
C3×#653102
C3×#652972
C3×#647642
C3×Q8⋊#1682
C3×D4⋊#1682
#106027123
C3×#16315