Finite Group Databasebeta

← #14289021

Subgroups — #14289021

Order 1920.

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

normal subgroups only

This group has 456 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×Dic30C211yesmaximal, characteristic, complemented
2#11751128C211yesmaximal, characteristic
3#11750949C211yesmaximal, characteristic
4#11750794C211yesmaximal, characteristic
5C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
6#11746302C211yesmaximal, characteristic, complemented
7#11746277C211yesmaximal, characteristic, complemented
8#10599057—31nomaximal, Hall
9C3×C4×C5×Q8C2211yescharacteristic
10Q8×Dic15—21no—
11C2×#6717—21no—
12C2×#6706—21no—
13C2×#6706—22no—
14#91777—22no—
15#91777—21no—
16#91763C2211yescharacteristic
17#91758—21no—
18#91758C2211yescharacteristic
19C4×Dic30—21no—
20C4×Dic30C2211yescharacteristic
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C4×C5×C8C2211yescharacteristic
23#91105—21no—
24#91093—21no—
25C4⋊(C3⋊#168)C2211yescharacteristic
26#68264—51nomaximal, Hall
27Q8×Dic10—31no—
28#65316—31no—
29#65137—31no—
30#64982—31no—
31C5×Q8⋊C8D311yescharacteristic, complemented
32#64701—31no—
33#64676—31no—
34C2×C3×C5×Q8D411yescharacteristic
35C2×D60—21no—
36C2×D60—21no—
37C2×D60—21no—
38C2×Dic30—22no—
39C2×Dic30D411yescharacteristic
40C2×Dic30D412yes—
41C2×C3×C5×C8—21no—
42C2×C3×C5×C8—21no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4D411yescharacteristic
45C3×C42×C5—21no—
46C3×C42×C5C2311yescharacteristic
47#6717—42no—
48C4⋊Dic15—22no—
49C4⋊Dic15—21no—
50C4⋊Dic15—21no—
51C4⋊Dic15D411yescharacteristic
52#6712—22no—
53#6712—21no—
54C4×Dic15—22no—
55C4×Dic15—21no—
56C2×C3⋊#168—21no—
57#6706—44no—
58#6706—42no—
59Q8×Dic6—51no—
60#5027—51no—
61#4848—51no—
62#4693—51no—
63C3×Q8⋊C8D511yescharacteristic, complemented
64#4481—51no—
65#4456—51no—
66C4×C5×Q8C2×D311yescharacteristic
67Q8×Dic5—61no—
68C2×#688—61no—
69C2×#677—61no—
70C2×#677—62no—
71#4034—62no—
72#4034—61no—
73#4020—31no—
74#4015—31no—
75#4015—61no—
76C4×Dic10—31no—
77C4×Dic10—61no—
78C5×C4⋊C8C2×D311yescharacteristic
79C4×C5×C8C2×D311yescharacteristic
80#3953—61no—
81#3941—61no—
82C4⋊#168—31no—
83#1623—151noSylow
84C22×D15—41no—
85C3×C5×Q8—21no—
86C3×C5×Q8SD1612yescomplemented
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5C2×D411yescharacteristic
90C2×C3×C4×C5C2×D411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×Dic15—21no—
93C2×Dic15—22no—
94C2×Dic15—21no—
95D60—42no—
96D60—41no—
97D60—41no—
98D60—41no—
99D60—41no—
100Dic30—21no—
101Dic30—42no—
102Dic30—82no—
103Dic30SD1612yescomplemented
104C3×C5×C8—22no—
105C3×C5×C8—41no—
106C3⋊#168—41no—
107C3×C4×Q8C2×D511yescharacteristic
108Q8×Dic3—101no—
109C2×#215—101no—
110C2×#204—102no—
111C2×#204—101no—
112#888—102no—
113#888—101no—
114#874—51no—
115#869—51no—
116#869—101no—
117C4×Dic6—51no—
118C4×Dic6—101no—
119C3×C4⋊C8C2×D511yescharacteristic
120C3×C4×C8C2×D511yescharacteristic
121#821—101no—
122#809—101no—
123C4⋊(C3⋊C8)—51no—
124C2×C5×Q8D1211yescharacteristic
125C2×D20—61no—
126C2×D20—61no—
127C2×D20—61no—
128C2×Dic10—31no—
129C2×Dic10—32no—
130C2×Dic10—62no—
131C2×C5×C8—21no—
132C2×C5×C8—21no—
133C5×C4⋊C4—21no—
134C5×C4⋊C4D1211yescharacteristic
135C42×C5—21no—
136C42×C5C22×D311yescharacteristic
137#688—122no—
138C4⋊Dic5—31no—
139C4⋊Dic5—62no—
140C4⋊Dic5—61no—
141C4⋊Dic5—61no—
142#683—62no—
143#683—61no—
144C4×Dic5—62no—
145C4×Dic5—61no—
146C2×#168—61no—
147#677—124no—
148#677—122no—
149Q82—151no—
150#492—151no—
151#460—151no—
152#460—151no—
153#333—151no—
154Q8⋊C8—151no—
155Q8⋊C8D1511yescharacteristic, complemented
156C22×C3×C5#12011yescharacteristic
157C2×D15—81no—
158C2×D15—81no—
159C2×D15—81no—
160C3×C4×C5—22no—
161C3×C4×C5—21no—
162C3×C4×C5—41no—
163C3×C4×C5C2×SD1611yescharacteristic
164C3×C4×C5C2×SD1611yescharacteristic
165C3×C4×C5#13711yescharacteristic
166C3×C4×C5#13711yescharacteristic
167Dic15—22no—
168Dic15—42no—
169Dic15—41no—
170C2×C3×Q8D2011yescharacteristic
171C2×D12—101no—
172C2×D12—101no—
173C2×D12—101no—
174C2×Dic6—51no—
175C2×Dic6—52no—
176C2×Dic6—102no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4D2011yescharacteristic
181C3×C42—21no—
182C3×C42C22×D511yescharacteristic
183#215—202no—
184C4⋊Dic3—51no—
185C4⋊Dic3—102no—
186C4⋊Dic3—101no—
187C4⋊Dic3—101no—
188Dic3⋊C4—102no—
189Dic3⋊C4—101no—
190C4×Dic3—102no—
191C4×Dic3—101no—
192C2×C3⋊C8—101no—
193#204—204no—
194#204—202no—
195C22×D5—121no—
196C5×Q8—21no—
197C5×Q8#20412yescomplemented
198C2×C4×C5—21no—
199C2×C4×C5—21no—
200C2×C4×C5C2×D1211yescharacteristic
201C2×C4×C5D3×D411yescharacteristic
202C2×C4×C5D3×D411yescharacteristic
203C2×Dic5—61no—
204C2×Dic5—61no—
205C2×Dic5—62no—
206D20—122no—
207D20—121no—
208D20—121no—
209D20—121no—
210D20—121no—
211Dic10—32no—
212Dic10—61no—
213Dic10—122no—
214Dic10—242no—
215C5×C8—22no—
216C5×C8—41no—
217#168—121no—
218C2×SD16—301no—
219C2×SD16—302no—
220C2×SD16—301no—
221C4⋊Q8—151no—
222C4⋊Q8—301no—
223C4⋊Q8—302no—
224C4⋊Q8—301no—
225#127—151no—
226C4×Q8—151no—
227C4×Q8—301no—
228C4×Q8—301no—
229C4×Q8C2×D1511yescharacteristic
230C4⋊C8—151no—
231C4⋊C8C2×D1511yescharacteristic
232#102—301no—
233#102—301no—
234C4×C8C2×D1511yescharacteristic
235C2×C3×C5#44811yescharacteristic
236C2×C3×C5#44811yescharacteristic
237C2×C3×C5#45311yescharacteristic
238D15—161no—
239C22×D3—201no—
240C3×Q8—21no—
241C3×Q8#67712yescomplemented
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4C2×D2011yescharacteristic
245C2×C3×C4D4×D511yescharacteristic
246C2×C3×C4D4×D511yescharacteristic
247C2×Dic3—101no—
248C2×Dic3—101no—
249C2×Dic3—102no—
250D12—202no—
251D12—201no—
252D12—201no—
253D12—201no—
254D12—201no—
255Dic6—52no—
256Dic6—101no—
257Dic6—202no—
258Dic6—402no—
259C3×C8—22no—
260C3×C8—41no—
261C3⋊C8—201no—
262C22×C5#88211yescharacteristic
263C2×D5—241no—
264C2×D5—241no—
265C2×D5—241no—
266C4×C5—22no—
267C4×C5—21no—
268C4×C5—41no—
269C4×C5C2×#20411yescharacteristic
270C4×C5#90911yescharacteristic
271C4×C5D3×SD1611yescharacteristic
272C4×C5#91811yescharacteristic
273Dic5—62no—
274Dic5—122no—
275Dic5—121no—
276C2×Q8—151no—
277C2×Q8—152no—
278C2×Q8—302no—
279C2×Q8D6011yescharacteristic
280C2×D4—301no—
281C2×D4—301no—
282C2×D4—301no—
283SD16—604no—
284SD16—602no—
285SD16—602no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—301no—
289C4⋊C4—21no—
290C4⋊C4—151no—
291C4⋊C4—302no—
292C4⋊C4—302no—
293C4⋊C4—301no—
294C4⋊C4—301no—
295C4⋊C4—301no—
296C4⋊C4D6011yescharacteristic
297C42—21no—
298C42—302no—
299C42—301no—
300C42C22×D1511yescharacteristic
301C3×C5#162311yescharacteristic, complemented, Hall
302C22×C3#402811yescharacteristic
303C2×D3—401no—
304C2×D3—401no—
305C2×D3—401no—
306C3×C4—22no—
307C3×C4—21no—
308C3×C4—41no—
309C3×C4C2×#67711yescharacteristic
310C3×C4#405511yescharacteristic
311C3×C4D5×SD1611yescharacteristic
312C3×C4#406411yescharacteristic
313Dic3—102no—
314Dic3—202no—
315Dic3—201no—
316C2×C5#473811yescharacteristic
317C2×C5#480611yescharacteristic
318C2×C5#482211yescharacteristic
319D5—481no—
320C23—601no—
321Q8—21no—
322Q8—152no—
323Q8—301no—
324Q8—602no—
325Q8—1202no—
326Q8#670612yescomplemented
327D4—602no—
328D4—601no—
329D4—601no—
330D4—601no—
331D4—601no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—301no—
335C2×C4—301no—
336C2×C4—302no—
337C2×C4C2×D6011yescharacteristic
338C2×C4D4×D1511yescharacteristic
339C2×C4D4×D1511yescharacteristic
340C8—22no—
341C8—41no—
342C8—601no—
343C2×C3#6502711yescharacteristic
344C2×C3#6509511yescharacteristic
345C2×C3#6511111yescharacteristic
346D3—801no—
347C5#6826411yescharacteristic, complemented, Sylow
348C22—1201no—
349C22—1201no—
350C22—1201no—
351C22#9177111yescharacteristic
352C4—21no—
353C4—22no—
354C4—41no—
355C4—302no—
356C4—602no—
357C4—601no—
358C4C2×#670611yescharacteristic
359C4#9179811yescharacteristic
360C4SD16×D1511yescharacteristic
361C4#9180711yescharacteristic
362C3#1059905711yescharacteristic, complemented, Sylow
363C2—2401no—
364C2#1175083911yescharacteristic
365C2#1175090711yescharacteristic
366C2#1175092311yescharacteristic
subgroupG/H
centerC22#91771
commutator, cosocleC3×C42×C5C23
FrattiniC42C22×D15
FittingC3×C5×Q8⋊C8C2
socleC22×C3×C5#120
prime radical#91763C22
hypercenterQ8⋊C8D15
hypocenterC3×C5#1623
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×Dic30C2semidirect
5. C3×C5×Q8⋊C8C2semidirect
6. #11746302C2semidirect
7. #11746277C2semidirect
31. C5×Q8⋊C8D3semidirect
63. C3×Q8⋊C8D5semidirect
86. C3×C5×Q8SD16semidirect
103. Dic30SD16semidirect
155. Q8⋊C8D15semidirect
197. C5×Q8#204semidirect
241. C3×Q8#677semidirect
301. C3×C5#1623semidirect
326. Q8#6706semidirect
347. C5#68264semidirect
362. C3#10599057semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic302
#117511282
#117509492
#117507942
C3×C5×Q8⋊C82
#117463022
#117462772
#105990573
#682645