Finite Group Databasebeta

← C3×#10602716 #14288954

Subgroups — C3×#10602716 #14288954

Order 1920.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#554C211yesmaximal, characteristic, complemented
2C3×#65374C211yesmaximal, characteristic, complemented
3C3×#65316C211yesmaximal, characteristic, complemented
4C3×#65304C211yesmaximal, characteristic
5C3×#64766C211yesmaximal, characteristic, complemented
6C3×Q8⋊#168C211yesmaximal, characteristic, complemented
7C3×D4⋊#168C211yesmaximal, characteristic, complemented
8#10602716C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×#125—21no—
11C3×C5×C22⋊Q8—21no—
12C3×C5×C22⋊Q8—22no—
13C3×C4×C5×Q8C2211yescharacteristic
14C3×C4×C5×D4—21no—
15C3×C4×C5×D4C2211yescharacteristic
16C2×C3×C5×C4⋊C4—21no—
17C2×C3×#688—21no—
18C2×C3×#688—22no—
19C2×C3×#686—21no—
20C3×#4020C2211yescharacteristic
21C3×#3941—21no—
22C3×#3941—21no—
23C3×C4⋊#168C2211yescharacteristic
24C3×C4⋊#168C2211yescharacteristic
25C3×C4×#168C2211yescharacteristic
26C3×#1622—51nomaximal, Hall
27C5×#554C2×C311yescharacteristic, complemented
28#65374C2×C311yescharacteristic, complemented
29#65316C2×C311yescharacteristic, complemented
30#65304C2×C311yescharacteristic
31#64766C2×C311yescharacteristic, complemented
32Q8⋊#168C2×C311yescharacteristic, complemented
33D4⋊#168C2×C311yescharacteristic, complemented
34C2×C3×C5×Q8D411yescharacteristic
35C2×C3×C5×Q8D412yes—
36C2×C3×C5×D4D411yescharacteristic
37C22×C3×C4×C5—22no—
38C22×C3×C4×C5—21no—
39C2×C3×D20—21no—
40C2×C3×D20—21no—
41C2×C3×D20—21no—
42C3×C5×C4⋊C4—22no—
43C3×C5×C4⋊C4—22no—
44C3×C5×C4⋊C4—21no—
45C3×C5×C4⋊C4—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×#688—44no—
54C3×#688—42no—
55C3×#686—42no—
56C2×C3×#168—21no—
57C2×C3×#168—21no—
58C2×C3×#168—21no—
59C3×#554D511yescharacteristic, complemented
60C3×#492—51no—
61C3×#460—51no—
62C3×#459—51no—
63C3×#333—51no—
64C3×Q8⋊C8—51no—
65C3×D4⋊C8—51no—
66C5×C4⋊Q8C22×C311yescharacteristic
67C5×#125—21no—
68C5×C22⋊Q8—21no—
69C5×C22⋊Q8—22no—
70C4×C5×Q8C22×C311yescharacteristic
71C4×C5×D4—21no—
72C4×C5×D4C22×C311yescharacteristic
73C2×C5×C4⋊C4—21no—
74C2×#688—21no—
75C2×#688—22no—
76C2×#686—21no—
77#4020C22×C311yescharacteristic
78#3941—21no—
79#3941—21no—
80C4⋊#168C22×C311yescharacteristic
81C4⋊#168C22×C311yescharacteristic
82C4×#168C22×C311yescharacteristic
83#1622—51noSylow
84C23×C3×C5—21no—
85C22×C3×D5—41no—
86C3×C5×Q8—21no—
87C3×C5×Q8—21no—
88C3×C5×Q8—42no—
89C3×C5×D4—21no—
90C3×C5×D4SD1612yescomplemented
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×C5—82no—
98C2×C3×C4×C5C2×D411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C3×D20—42no—
102C3×D20—41no—
103C3×D20—41no—
104C3×D20—41no—
105C3×D20—41no—
106C3×#168—22no—
107C3×#168—41no—
108C3×#168—41no—
109C2×C3×SD16—101no—
110C2×C3×SD16—102no—
111C2×C3×D8—101no—
112C3×C4⋊Q8C2×D511yescharacteristic
113C3×#127—51no—
114C3×#125—21no—
115C3×C22⋊Q8—21no—
116C3×C22⋊Q8—22no—
117C3×C4×Q8C2×D511yescharacteristic
118C3×C4×D4—21no—
119C3×C4×D4C2×D511yescharacteristic
120C2×C3×C4⋊C4—21no—
121C3×C4⋊C8—51no—
122C3×C4⋊C8—51no—
123C3×#102—101no—
124C3×#102—101no—
125C3×C4×C8—51no—
126C2×C5×Q8C3×D411yescharacteristic
127C2×C5×Q8C3×D412yes—
128C2×C5×D4C3×D411yescharacteristic
129C22×C4×C5—22no—
130C22×C4×C5—21no—
131C2×D20—21no—
132C2×D20—21no—
133C2×D20—21no—
134C5×C4⋊C4—22no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—21no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4C3×D411yescharacteristic
140C5×C4⋊C4C3×D411yescharacteristic
141C5×C22⋊C4—22no—
142C5×C22⋊C4—21no—
143C42×C5—21no—
144C42×C5C23×C311yescharacteristic
145#688—44no—
146#688—42no—
147#686—42no—
148C2×#168—21no—
149C2×#168—21no—
150C2×#168—21no—
151#554C3×D511yescharacteristic, complemented
152#492—51no—
153#460—51no—
154#459—51no—
155#333—51no—
156Q8⋊C8—51no—
157D4⋊C8—51no—
158C22×C3×C5—22no—
159C22×C3×C5—41no—
160C22×C3×C5—41no—
161C22×C3×C5#12011yescharacteristic
162C2×C3×D5—81no—
163C2×C3×D5—81no—
164C2×C3×D5—81no—
165C3×C4×C5—21no—
166C3×C4×C5—41no—
167C3×C4×C5—41no—
168C3×C4×C5—41no—
169C3×C4×C5—42no—
170C3×C4×C5C2×SD1611yescharacteristic
171C3×C4×C5#13511yescharacteristic
172C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
173C3×C4×C5#13711yescharacteristic
174C2×C3×Q8C22⋊D511yescharacteristic
175C2×C3×Q8C22⋊D512yes—
176C2×C3×D4—101no—
177C2×C3×D4—101no—
178C2×C3×D4—101no—
179C2×C3×D4C22⋊D511yescharacteristic
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C3×SD16—204no—
183C3×SD16—202no—
184C3×D8—202no—
185C2×C3×C8—101no—
186C2×C3×C8—101no—
187C2×C3×C8—101no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4C22⋊D511yescharacteristic
194C3×C4⋊C4C22⋊D511yescharacteristic
195C3×C22⋊C4—22no—
196C3×C22⋊C4—21no—
197C3×C42—21no—
198C3×C42C22×D511yescharacteristic
199C23×C5—21no—
200C22×D5—41no—
201C5×Q8—21no—
202C5×Q8—21no—
203C5×Q8—42no—
204C5×D4—21no—
205C5×D4C3×SD1612yescomplemented
206C2×C4×C5—21no—
207C2×C4×C5—21no—
208C2×C4×C5—21no—
209C2×C4×C5—22no—
210C2×C4×C5—41no—
211C2×C4×C5—41no—
212C2×C4×C5—82no—
213C2×C4×C5C2×C3×D411yescharacteristic
214C2×C4×C5C2×C3×D411yescharacteristic
215C2×C4×C5C2×C3×D411yescharacteristic
216D20—42no—
217D20—41no—
218D20—41no—
219D20—41no—
220D20—41no—
221#168—22no—
222#168—41no—
223#168—41no—
224C2×SD16—101no—
225C2×SD16—102no—
226C2×D8—101no—
227C4⋊Q8C2×C3×D511yescharacteristic
228#127—51no—
229#125—21no—
230C22⋊Q8—21no—
231C22⋊Q8—22no—
232C4×Q8C2×C3×D511yescharacteristic
233C4×D4—21no—
234C4×D4C2×C3×D511yescharacteristic
235C2×C4⋊C4—21no—
236C4⋊C8—51no—
237C4⋊C8—51no—
238#102—101no—
239#102—101no—
240C4×C8—51no—
241C2×C3×C5—42no—
242C2×C3×C5#44811yescharacteristic
243C2×C3×C5#44911yescharacteristic
244C2×C3×C5#45311yescharacteristic
245C3×D5—161no—
246C23×C3—21no—
247C23×C3—201no—
248C3×Q8—21no—
249C3×Q8—21no—
250C3×Q8—42no—
251C3×D4—21no—
252C3×D4—202no—
253C3×D4—201no—
254C3×D4—201no—
255C3×D4—201no—
256C3×D4—201no—
257C3×D4#68812yescomplemented
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—82no—
265C2×C3×C4C2×C22⋊D511yescharacteristic
266C2×C3×C4C2×C22⋊D511yescharacteristic
267C2×C3×C4C2×C22⋊D511yescharacteristic
268C3×C8—102no—
269C3×C8—201no—
270C3×C8—201no—
271C22×C5—22no—
272C22×C5—41no—
273C22×C5—41no—
274C22×C5C3×#12011yescharacteristic
275C2×D5—81no—
276C2×D5—81no—
277C2×D5—81no—
278C4×C5—21no—
279C4×C5—41no—
280C4×C5—41no—
281C4×C5—41no—
282C4×C5—42no—
283C4×C5C2×C3×SD1611yescharacteristic
284C4×C5C3×#13511yescharacteristic
285C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
286C4×C5C3×#13711yescharacteristic
287C2×Q8C3×C22⋊D511yescharacteristic
288C2×Q8C3×C22⋊D512yes—
289C2×D4—101no—
290C2×D4—101no—
291C2×D4—101no—
292C2×D4C3×C22⋊D511yescharacteristic
293C22×C4—22no—
294C22×C4—21no—
295SD16—204no—
296SD16—202no—
297D8—202no—
298C2×C8—101no—
299C2×C8—101no—
300C2×C8—101no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4C3×C22⋊D511yescharacteristic
307C4⋊C4C3×C22⋊D511yescharacteristic
308C22⋊C4—22no—
309C22⋊C4—21no—
310C42—21no—
311C42C22×C3×D511yescharacteristic
312C3×C5#162211yescharacteristic, complemented, Hall
313C22×C3—22no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—401no—
317C22×C3—401no—
318C22×C3—401no—
319C22×C3#409911yescharacteristic
320C3×C4—21no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—42no—
325C3×C4C2×#68811yescharacteristic
326C3×C4#408811yescharacteristic
327C3×C4#409511yescharacteristic
328C3×C4#409611yescharacteristic
329C2×C5—42no—
330C2×C5C3×#44811yescharacteristic
331C2×C5C3×#44911yescharacteristic
332C2×C5C3×#45311yescharacteristic
333D5—161no—
334C23—21no—
335C23—201no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339D4—21no—
340D4—202no—
341D4—201no—
342D4—201no—
343D4—201no—
344D4—201no—
345D4C3×#68812yescomplemented
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—82no—
353C2×C4C2×C3×C22⋊D511yescharacteristic
354C2×C4C2×C3×C22⋊D511yescharacteristic
355C2×C4C2×C3×C22⋊D511yescharacteristic
356C8—102no—
357C8—201no—
358C8—201no—
359C2×C3—42no—
360C2×C3—801no—
361C2×C3#6551611yescharacteristic
362C2×C3#6552711yescharacteristic
363C2×C3#6553011yescharacteristic
364C5C3×#162211yescharacteristic, complemented, Sylow
365C22—22no—
366C22—41no—
367C22—41no—
368C22—401no—
369C22—401no—
370C22—401no—
371C22C3×#409911yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—42no—
377C4C2×C3×#68811yescharacteristic
378C4C3×#408811yescharacteristic
379C4C3×#409511yescharacteristic
380C4C3×#409611yescharacteristic
381C3#1060271611yescharacteristic, direct product factor, Sylow
382C2—42no—
383C2—801no—
384C2C3×#6551611yescharacteristic
385C2C3×#6552711yescharacteristic
386C2C3×#6553011yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC42×C5C23×C3
FrattiniC42C22×C3×D5
FittingC3×C5×#554C2
socleC22×C3×C5#120
prime radicalC3×#65304C2
hypercenterC3×#554D5
hypocenterC5C3×#1622
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#554

Complements

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

NH ≅ G/Ntype
1. C3×C5×#554C2semidirect
2. C3×#65374C2semidirect
3. C3×#65316C2semidirect
5. C3×#64766C2semidirect
6. C3×Q8⋊#168C2semidirect
7. C3×D4⋊#168C2semidirect
8. #10602716C3direct (N × H)
27. C5×#554C2×C3semidirect
28. #65374C2×C3semidirect
29. #65316C2×C3semidirect
31. #64766C2×C3semidirect
32. Q8⋊#168C2×C3semidirect
33. D4⋊#168C2×C3semidirect
59. C3×#554D5semidirect
90. C3×C5×D4SD16semidirect
151. #554C3×D5semidirect
205. C5×D4C3×SD16semidirect
257. C3×D4#688semidirect
312. C3×C5#1622semidirect
345. D4C3×#688semidirect
364. C5C3×#1622semidirect
381. C3#10602716direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5542
C3×#653742
C3×#653162
C3×#653042
C3×#647662
C3×Q8⋊#1682
C3×D4⋊#1682
#106027163
C3×#16225