Finite Group Databasebeta

← #14289228

Subgroups — #14289228

Order 1920.

500 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×D60C211yesmaximal, characteristic, complemented
2#11751128C211yesmaximal, characteristic, complemented
3#11750944C211yesmaximal, characteristic
4#11750795C211yesmaximal, characteristic, complemented
5C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
6#11746307C211yesmaximal, characteristic, complemented
7#11746280C211yesmaximal, characteristic, complemented
8#10599059—31nomaximal, Hall
9C2×Q8×D15—21no—
10C3×C4×C5×Q8C2211yescharacteristic
11C2×#6717—21no—
12C2×D120—21no—
13#91786—21no—
14#91786—22no—
15#91763C2211yescharacteristic
16C4×D60—21no—
17C4×D60C2211yescharacteristic
18#91758—21no—
19#91758C2211yescharacteristic
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C4×C5×C8C2211yescharacteristic
22#91105—21no—
23#91105—22no—
24#91093—21no—
25C4⋊(C3⋊#168)C2211yescharacteristic
26#68266—51nomaximal, Hall
27Q8×D20—31no—
28#65316—31no—
29#65132—31no—
30#64983—31no—
31C5×Q8⋊C8D311yescharacteristic, complemented
32#64706—31no—
33#64679—31no—
34C2×C3×C5×Q8D411yescharacteristic
35Q8×D15—42no—
36Q8×D15—42no—
37C2×D60—21no—
38C2×D60—21no—
39C2×D60—21no—
40C2×D60D411yescharacteristic
41C2×C4×D15—22no—
42C2×C4×D15—21no—
43C2×Dic30—22no—
44C2×Dic30—21no—
45C2×C3×C5×C8—21no—
46C2×C3×C5×C8—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4D411yescharacteristic
49C3×C42×C5—21no—
50C3×C42×C5C2311yescharacteristic
51#6717—42no—
52#6714—22no—
53#6714—21no—
54C4⋊Dic15—22no—
55C4⋊Dic15—21no—
56C4⋊Dic15D411yescharacteristic
57C4⋊Dic15D412yes—
58C2×C3⋊#168—21no—
59D120—42no—
60Q8×D12—51no—
61#5027—51no—
62#4843—51no—
63#4694—51no—
64C3×Q8⋊C8D511yescharacteristic, complemented
65#4486—51no—
66#4459—51no—
67C2×Q8×D5—61no—
68C4×C5×Q8C2×D311yescharacteristic
69C2×#688—61no—
70C2×D40—61no—
71#4043—62no—
72#4043—61no—
73#4020—31no—
74C4×D20—31no—
75C4×D20—61no—
76#4015—31no—
77#4015—61no—
78C5×C4⋊C8C2×D311yescharacteristic
79C4×C5×C8C2×D311yescharacteristic
80#3953—61no—
81#3953—62no—
82#3941—61no—
83C4⋊#168—31no—
84#1627—151noSylow
85C22×D15—21no—
86C22×D15—41no—
87C3×C5×Q8—21no—
88C3×C5×Q8D812yescomplemented
89C2×C3×C4×C5—21no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C3×C4×C5C2×D411yescharacteristic
93C2×C3×C4×C5C2×D411yescharacteristic
94C2×Dic15—21no—
95C2×Dic15—22no—
96D60—21no—
97D60—42no—
98D60—41no—
99D60—41no—
100D60—41no—
101D60—41no—
102D60SD1612yescomplemented
103C4×D15—44no—
104C4×D15—41no—
105C4×D15—41no—
106Dic30—44no—
107Dic30—41no—
108Dic30—41no—
109C3×C5×C8—41no—
110C3×C5×C8—41no—
111C3⋊#168—41no—
112C2×D3×Q8—101no—
113C3×C4×Q8C2×D511yescharacteristic
114C2×#215—101no—
115C2×D24—101no—
116#897—102no—
117#897—101no—
118#874—51no—
119C4×D12—51no—
120C4×D12—101no—
121#869—51no—
122#869—101no—
123C3×C4⋊C8C2×D511yescharacteristic
124C3×C4×C8C2×D511yescharacteristic
125#821—102no—
126#821—101no—
127#809—101no—
128C4⋊(C3⋊C8)—51no—
129C2×C5×Q8D1211yescharacteristic
130Q8×D5—122no—
131Q8×D5—122no—
132C2×D20—31no—
133C2×D20—61no—
134C2×D20—61no—
135C2×D20—61no—
136C2×C4×D5—62no—
137C2×C4×D5—61no—
138C2×Dic10—62no—
139C2×Dic10—61no—
140C2×C5×C8—21no—
141C2×C5×C8—21no—
142C5×C4⋊C4—21no—
143C5×C4⋊C4D1211yescharacteristic
144C42×C5—21no—
145C42×C5C22×D311yescharacteristic
146#688—122no—
147#685—62no—
148#685—61no—
149C4⋊Dic5—32no—
150C4⋊Dic5—31no—
151C4⋊Dic5—62no—
152C4⋊Dic5—61no—
153C2×#168—61no—
154D40—122no—
155D4×Q8—151no—
156#486—151no—
157#460—151no—
158#459—151no—
159#332—151no—
160Q8⋊C8D1511yescharacteristic, complemented
161D4⋊C8—151no—
162C22×C3×C5#12011yescharacteristic
163C2×D15—22no—
164C2×D15—41no—
165C2×D15—41no—
166C2×D15—81no—
167C2×D15—81no—
168C2×D15—81no—
169C3×C4×C5—22no—
170C3×C4×C5—21no—
171C3×C4×C5—41no—
172C3×C4×C5C2×D811yescharacteristic
173C3×C4×C5C2×SD1611yescharacteristic
174C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
175C3×C4×C5#13711yescharacteristic
176Dic15—41no—
177Dic15—42no—
178C2×C3×Q8D2011yescharacteristic
179D3×Q8—202no—
180D3×Q8—202no—
181C2×D12—51no—
182C2×D12—101no—
183C2×D12—101no—
184C2×D12—101no—
185C2×C4×D3—102no—
186C2×C4×D3—101no—
187C2×Dic6—102no—
188C2×Dic6—101no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4D2011yescharacteristic
193C3×C42—21no—
194C3×C42C22×D511yescharacteristic
195#215—202no—
196(C2×D3)⋊C4—102no—
197(C2×D3)⋊C4—101no—
198C4⋊Dic3—52no—
199C4⋊Dic3—51no—
200C4⋊Dic3—102no—
201C4⋊Dic3—101no—
202C2×C3⋊C8—101no—
203D24—202no—
204C22×D5—61no—
205C22×D5—121no—
206C5×Q8—21no—
207C5×Q8D2412yescomplemented
208C2×C4×C5—21no—
209C2×C4×C5—21no—
210C2×C4×C5C2×D1211yescharacteristic
211C2×C4×C5D3×D411yescharacteristic
212C2×C4×C5D3×D411yescharacteristic
213C2×Dic5—62no—
214C2×Dic5—61no—
215D20—32no—
216D20—61no—
217D20—122no—
218D20—121no—
219D20—121no—
220D20—121no—
221D20—121no—
222C4×D5—124no—
223C4×D5—121no—
224C4×D5—121no—
225Dic10—124no—
226Dic10—121no—
227Dic10—121no—
228C5×C8—41no—
229C5×C8—41no—
230#168—121no—
231C22×Q8—301no—
232C2×SD16—301no—
233C2×D8—301no—
234C4⋊Q8—151no—
235C4⋊Q8—301no—
236#127—151no—
237C22⋊Q8—302no—
238C22⋊Q8—301no—
239C4×Q8C2×D1511yescharacteristic
240C4×D4—151no—
241C4×D4—301no—
242C4⋊C8—151no—
243C4⋊C8C2×D1511yescharacteristic
244#102—301no—
245#102—302no—
246#102—301no—
247C4×C8C2×D1511yescharacteristic
248C2×C3×C5#44711yescharacteristic
249C2×C3×C5#44811yescharacteristic
250C2×C3×C5#45411yescharacteristic
251D15—42no—
252D15—161no—
253C22×D3—101no—
254C22×D3—201no—
255C3×Q8—21no—
256C3×Q8D4012yescomplemented
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4C2×D2011yescharacteristic
260C2×C3×C4D4×D511yescharacteristic
261C2×C3×C4D4×D511yescharacteristic
262C2×Dic3—102no—
263C2×Dic3—101no—
264D12—52no—
265D12—101no—
266D12—202no—
267D12—201no—
268D12—201no—
269D12—201no—
270D12—201no—
271C4×D3—204no—
272C4×D3—201no—
273C4×D3—201no—
274Dic6—204no—
275Dic6—201no—
276Dic6—201no—
277C3×C8—41no—
278C3×C8—41no—
279C3⋊C8—201no—
280C22×C5#88211yescharacteristic
281C2×D5—62no—
282C2×D5—121no—
283C2×D5—121no—
284C2×D5—241no—
285C2×D5—241no—
286C2×D5—241no—
287C4×C5—22no—
288C4×C5—21no—
289C4×C5—41no—
290C4×C5C2×D2411yescharacteristic
291C4×C5#90811yescharacteristic
292C4×C5D3×SD1611yescharacteristic
293C4×C5#91811yescharacteristic
294Dic5—121no—
295Dic5—122no—
296C2×Q8—302no—
297C2×Q8—301no—
298C2×Q8—602no—
299C2×Q8—602no—
300C2×Q8D6011yescharacteristic
301C2×D4—151no—
302C2×D4—301no—
303C2×D4—301no—
304C2×D4—301no—
305C22×C4—302no—
306C22×C4—301no—
307SD16—602no—
308D8—602no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—301no—
312C4⋊C4—21no—
313C4⋊C4—151no—
314C4⋊C4—152no—
315C4⋊C4—302no—
316C4⋊C4—301no—
317C4⋊C4D6011yescharacteristic
318C22⋊C4—302no—
319C22⋊C4—301no—
320C42—21no—
321C42C22×D1511yescharacteristic
322C3×C5#162711yescharacteristic, complemented, Hall
323C22×C3#402811yescharacteristic
324C2×D3—102no—
325C2×D3—201no—
326C2×D3—201no—
327C2×D3—401no—
328C2×D3—401no—
329C2×D3—401no—
330C3×C4—21no—
331C3×C4—22no—
332C3×C4—41no—
333C3×C4C2×D4011yescharacteristic
334C3×C4#405411yescharacteristic
335C3×C4D5×SD1611yescharacteristic
336C3×C4#406411yescharacteristic
337Dic3—201no—
338Dic3—202no—
339C2×C5#473211yescharacteristic
340C2×C5#480611yescharacteristic
341C2×C5#482711yescharacteristic
342D5—122no—
343D5—481no—
344C23—301no—
345C23—601no—
346Q8—21no—
347Q8—604no—
348Q8—601no—
349Q8—601no—
350Q8D12012yescomplemented
351D4—152no—
352D4—301no—
353D4—602no—
354D4—601no—
355D4—601no—
356D4—601no—
357D4—601no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—302no—
361C2×C4—301no—
362C2×C4—604no—
363C2×C4—601no—
364C2×C4—601no—
365C2×C4C2×D6011yescharacteristic
366C2×C4D4×D1511yescharacteristic
367C2×C4D4×D1511yescharacteristic
368C8—41no—
369C8—41no—
370C8—601no—
371C2×C3#6502111yescharacteristic
372C2×C3#6509511yescharacteristic
373C2×C3#6511611yescharacteristic
374D3—202no—
375D3—801no—
376C5#6826611yescharacteristic, complemented, Sylow
377C22—302no—
378C22—601no—
379C22—601no—
380C22—1201no—
381C22—1201no—
382C22—1201no—
383C22#9177111yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—602no—
388C4—601no—
389C4C2×D12011yescharacteristic
390C4#9179711yescharacteristic
391C4SD16×D1511yescharacteristic
392C4#9180711yescharacteristic
393C3#1059905911yescharacteristic, complemented, Sylow
394C2—602no—
395C2—2401no—
396C2#1175083311yescharacteristic
397C2#1175090711yescharacteristic
398C2#1175092811yescharacteristic
subgroupG/H
centerC22#91771
commutator, cosocleC3×C42×C5C23
FrattiniC42C22×D15
FittingC3×C5×Q8⋊C8C2
socleC22×C3×C5#120
prime radical#11750944C2
hypercenterQ8⋊C8D15
hypocenterC3×C5#1627
LCSG ▷ C3×C42×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. Q8×D60C2semidirect
2. #11751128C2semidirect
4. #11750795C2semidirect
5. C3×C5×Q8⋊C8C2semidirect
6. #11746307C2semidirect
7. #11746280C2semidirect
31. C5×Q8⋊C8D3semidirect
64. C3×Q8⋊C8D5semidirect
88. C3×C5×Q8D8semidirect
102. D60SD16semidirect
160. Q8⋊C8D15semidirect
207. C5×Q8D24semidirect
256. C3×Q8D40semidirect
322. C3×C5#1627semidirect
350. Q8D120semidirect
376. C5#68266semidirect
393. C3#10599059semidirect

Maximal subgroups

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

subgroupindex
Q8×D602
#117511282
#117509442
#117507952
C3×C5×Q8⋊C82
#117463072
#117462802
#105990593
#682665