Finite Group Databasebeta

← #14307485

Subgroups — #14307485

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic
2#11751313C211yesmaximal, characteristic
3C4×#6719C211yesmaximal, characteristic
4#11746347C214yesmaximal
5#10602494—31nomaximal, Hall
6C22×C3×C5×Q8C411yescharacteristic, complemented
7C23×C3×C4×C5C2211yescharacteristic
8C3×C5×C22⋊Q8—21no—
9C3×C5×C22⋊Q8—21no—
10C3×C5×C22⋊Q8—21no—
11C3×C5×C22⋊Q8—21no—
12C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
13C2×C3×C5×C4⋊C4C412yescomplemented
14C2×C3×C5×C22⋊C4C2212yes—
15C2×#6719C2212yes—
16C2×#6719C2212yes—
17C2×C4⋊Dic15—21no—
18C2×C4×Dic15—21no—
19#91115—21no—
20#91115—21no—
21#71701—51nomaximal, Hall
22C2×C5×C22⋊Q8D311yescharacteristic
23#65501—31no—
24C4×C22⋊Dic5—31no—
25#64746—34no—
26C24×C3×C5C2311yescharacteristic
27C2×C3×C5×Q8—21no—
28C2×C3×C5×Q8—21no—
29C2×C3×C5×Q8—42no—
30C22×C3×C4×C5—42no—
31C22×C3×C4×C5C2×C411yescharacteristic
32C22×C3×C4×C5C2×C412yes—
33C22×C3×C4×C5C2×C412yes—
34C22×C3×C4×C5C2×C411yescharacteristic
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5D411yescharacteristic
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5D411yescharacteristic
39C22×Dic15—22no—
40C22×Dic15—22no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—21no—
43C3×C5×C4⋊C4—22no—
44C3×C5×C4⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46C3×C5×C22⋊C4—22no—
47#6719—22no—
48#6719—24no—
49#6719Q814yescomplemented
50C4⋊Dic15—42no—
51C4×Dic15—42no—
52C2×C3×C22⋊Q8D511yescharacteristic
53#5212—51no—
54C4×#217—51no—
55#4526—54no—
56C22×C5×Q8Dic311yescharacteristic, complemented
57C23×C4×C5C2×D311yescharacteristic
58C5×C22⋊Q8—21no—
59C5×C22⋊Q8—21no—
60C5×C22⋊Q8—21no—
61C5×C22⋊Q8—21no—
62C2×C5×C4⋊C4Dic311yescharacteristic, complemented
63C2×C5×C4⋊C4Dic312yescomplemented
64C2×C5×C22⋊C4C2×D312yes—
65C2×C22⋊Dic5—32no—
66C2×C22⋊Dic5—32no—
67C2×C4⋊Dic5—61no—
68C2×C4×Dic5—61no—
69#3963—61no—
70#3963—61no—
71#1996—151noSylow
72C23×C3×C5—41no—
73C23×C3×C5—41no—
74C23×C3×C5C22×C411yescharacteristic
75C23×C3×C5C2×D411yescharacteristic
76C23×C3×C5C2×D411yescharacteristic
77C23×C3×C5C2×Q812yes—
78C23×C3×C5#4112yes—
79C3×C5×Q8—42no—
80C3×C5×Q8—42no—
81C2×C3×C4×C5—22no—
82C2×C3×C4×C5—22no—
83C2×C3×C4×C5—22no—
84C2×C3×C4×C5—22no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5—41no—
88C2×C3×C4×C5—41no—
89C2×C3×C4×C5—42no—
90C2×C3×C4×C5—42no—
91C2×C3×C4×C5C22⋊C411yescharacteristic
92C2×C3×C4×C5C22⋊C411yescharacteristic
93C2×C3×C4×C5C22⋊C411yescharacteristic
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C2×Dic15—24no—
96C2×Dic15—42no—
97C2×Dic15—42no—
98C2×Dic15—42no—
99C2×Dic15—42no—
100C2×Dic15—42no—
101C22×C3×Q8Dic511yescharacteristic, complemented
102C23×C3×C4C2×D511yescharacteristic
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8—21no—
106C3×C22⋊Q8—21no—
107C2×C3×C4⋊C4Dic511yescharacteristic, complemented
108C2×C3×C4⋊C4Dic512yescomplemented
109C2×C3×C22⋊C4C2×D512yes—
110C2×#217—52no—
111C2×#217—52no—
112C2×C4⋊Dic3—101no—
113C2×C4×Dic3—101no—
114C3⋊#95—101no—
115C3⋊#95—101no—
116C24×C5C22×D311yescharacteristic
117C2×C5×Q8—21no—
118C2×C5×Q8—21no—
119C2×C5×Q8—42no—
120C22×C4×C5—42no—
121C22×C4×C5C2×Dic311yescharacteristic
122C22×C4×C5C2×Dic312yes—
123C22×C4×C5C2×Dic312yes—
124C22×C4×C5C2×Dic311yescharacteristic
125C22×C4×C5#6711yescharacteristic
126C22×C4×C5#6711yescharacteristic
127C22×C4×C5#6711yescharacteristic
128C22×C4×C5#6711yescharacteristic
129C22×Dic5—62no—
130C22×Dic5—62no—
131C5×C4⋊C4—21no—
132C5×C4⋊C4—21no—
133C5×C4⋊C4—22no—
134C5×C4⋊C4—22no—
135C5×C22⋊C4—22no—
136C5×C22⋊C4—22no—
137C22⋊Dic5—34no—
138C22⋊Dic5—62no—
139C22⋊Dic5—64no—
140C4⋊Dic5—122no—
141C4×Dic5—122no—
142C2×C22⋊Q8D1511yescharacteristic
143#380—151no—
144C4×C22⋊C4—151no—
145#342—154no—
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5—21no—
152C22×C3×C5—42no—
153C22×C3×C5—42no—
154C22×C3×C5C2×C22⋊C411yescharacteristic
155C22×C3×C5C4×Q812yes—
156C22×C3×C5C22⋊Q812yes—
157C22×C3×C5#12411yescharacteristic
158C22×C3×C5C4⋊Q811yescharacteristic
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161C3×C4×C5—21no—
162C3×C4×C5—42no—
163C3×C4×C5—42no—
164C3×C4×C5#9912yes—
165Dic15—44no—
166Dic15—82no—
167C24×C3C22×D511yescharacteristic
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—42no—
171C22×C3×C4—42no—
172C22×C3×C4C2×Dic511yescharacteristic
173C22×C3×C4C2×Dic512yes—
174C22×C3×C4C2×Dic512yes—
175C22×C3×C4C2×Dic511yescharacteristic
176C22×C3×C4C22⋊D511yescharacteristic
177C22×C3×C4C22⋊D511yescharacteristic
178C22×C3×C4C22⋊D511yescharacteristic
179C22×C3×C4C22⋊D511yescharacteristic
180C22×Dic3—102no—
181C22×Dic3—102no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—22no—
188#217—54no—
189#217—102no—
190#217—104no—
191C4⋊Dic3—202no—
192C4×Dic3—202no—
193C23×C5—41no—
194C23×C5—41no—
195C23×C5D3×Q812yes—
196C23×C5#23912yes—
197C23×C5C22×Dic311yescharacteristic
198C23×C5C2×#6711yescharacteristic
199C23×C5C2×#6711yescharacteristic
200C5×Q8—42no—
201C5×Q8—42no—
202C2×C4×C5—22no—
203C2×C4×C5—22no—
204C2×C4×C5—22no—
205C2×C4×C5—22no—
206C2×C4×C5—22no—
207C2×C4×C5—41no—
208C2×C4×C5—41no—
209C2×C4×C5—41no—
210C2×C4×C5—42no—
211C2×C4×C5—42no—
212C2×C4×C5#21711yescharacteristic
213C2×C4×C5#21711yescharacteristic
214C2×C4×C5#21711yescharacteristic
215C2×C4×C5#21711yescharacteristic
216C2×Dic5—64no—
217C2×Dic5—122no—
218C2×Dic5—122no—
219C2×Dic5—122no—
220C2×Dic5—122no—
221C2×Dic5—122no—
222C22×Q8Dic1511yescharacteristic, complemented
223C23×C4C2×D1511yescharacteristic
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C2×C4⋊C4—301no—
229C2×C4⋊C4Dic1511yescharacteristic, complemented
230C2×C4⋊C4Dic1512yescomplemented
231C2×C22⋊C4—152no—
232C2×C22⋊C4—152no—
233C2×C22⋊C4C2×D1512yes—
234C2×C42—301no—
235#95—301no—
236#95—301no—
237C2×C3×C5—22no—
238C2×C3×C5—21no—
239C2×C3×C5—22no—
240C2×C3×C5—21no—
241C2×C3×C5#39111yescharacteristic
242C2×C3×C5C2×#9911yescharacteristic
243C2×C3×C5#41011yescharacteristic
244C23×C3—41no—
245C23×C3—41no—
246C23×C3Q8×D512yes—
247C23×C3#71312yes—
248C23×C3C22×Dic511yescharacteristic
249C23×C3C2×C22⋊D511yescharacteristic
250C23×C3C2×C22⋊D511yescharacteristic
251C3×Q8—42no—
252C3×Q8—42no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4C22⋊Dic511yescharacteristic
264C2×C3×C4C22⋊Dic511yescharacteristic
265C2×C3×C4C22⋊Dic511yescharacteristic
266C2×C3×C4C22⋊Dic511yescharacteristic
267C2×Dic3—104no—
268C2×Dic3—202no—
269C2×Dic3—202no—
270C2×Dic3—202no—
271C2×Dic3—202no—
272C2×Dic3—202no—
273C22×C5—21no—
274C22×C5—21no—
275C22×C5—21no—
276C22×C5—21no—
277C22×C5—21no—
278C22×C5—21no—
279C22×C5—42no—
280C22×C5—42no—
281C22×C5#94411yescharacteristic
282C22×C5Q8×Dic312yes—
283C22×C5#94612yes—
284C22×C5#94711yescharacteristic
285C22×C5C2×#21711yescharacteristic
286C4×C5—21no—
287C4×C5—21no—
288C4×C5—21no—
289C4×C5—42no—
290C4×C5—42no—
291C4×C5#83412yes—
292Dic5—124no—
293Dic5—242no—
294C24C22×D1511yescharacteristic
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—42no—
298C22×C4—42no—
299C22×C4—302no—
300C22×C4—302no—
301C22×C4C2×Dic1511yescharacteristic
302C22×C4C2×Dic1512yes—
303C22×C4C2×Dic1512yes—
304C22×C4C2×Dic1511yescharacteristic
305C22×C4#122011yescharacteristic
306C22×C4#122011yescharacteristic
307C22×C4#122011yescharacteristic
308C22×C4#122011yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—602no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—154no—
317C22⋊C4—302no—
318C22⋊C4—304no—
319C42—602no—
320C3×C5#199611yescharacteristic, complemented, Hall
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3#409011yescharacteristic
330C22×C3Q8×Dic512yes—
331C22×C3#409212yes—
332C22×C3#409311yescharacteristic
333C22×C3C2×C22⋊Dic511yescharacteristic
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4#396612yes—
340Dic3—204no—
341Dic3—402no—
342C2×C5—22no—
343C2×C5—21no—
344C2×C5—21no—
345C2×C5—22no—
346C2×C5C2×#83411yescharacteristic
347C2×C5#522911yescharacteristic
348C2×C5#524011yescharacteristic
349C23—41no—
350C23—41no—
351C23Q8×D1512yes—
352C23#682112yes—
353C23C22×Dic1511yescharacteristic
354C23C2×#122011yescharacteristic
355C23C2×#122011yescharacteristic
356Q8—42no—
357Q8—42no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—304no—
369C2×C4—602no—
370C2×C4—602no—
371C2×C4—602no—
372C2×C4—602no—
373C2×C4—602no—
374C2×C4#671911yescharacteristic
375C2×C4#671911yescharacteristic
376C2×C4#671911yescharacteristic
377C2×C4#671911yescharacteristic
378C2×C3—22no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—22no—
382C2×C3C2×#396611yescharacteristic
383C2×C3#6551911yescharacteristic
384C2×C3#6552911yescharacteristic
385C5#7170111yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22#9183311yescharacteristic
395C22Q8×Dic1512yes—
396C22#9183512yes—
397C22#9183611yescharacteristic
398C22C2×#671911yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—604no—
405C4—1202no—
406C4#9111812yes—
407C3#1060249411yescharacteristic, complemented, Sylow
408C2—22no—
409C2—21no—
410C2—21no—
411C2—22no—
412C2C2×#9111811yescharacteristic
413C2#1175133011yescharacteristic
414C2#1175134111yescharacteristic
subgroupG/H
centerC22C2×#6719
commutatorC23×C3×C5C22×C4
FrattiniC24C22×D15
FittingC2×C3×C5×C22⋊Q8C2
socleC22×C3×C5C2×C22⋊C4
prime radical, cosocleC24×C3×C5C23
hypercenterC2×C22⋊Q8D15
hypocenterC3×C5#1996
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
6. C22×C3×C5×Q8C4semidirect
12. C2×C3×C5×C4⋊C4C4semidirect
13. C2×C3×C5×C4⋊C4C4semidirect
49. #6719Q8semidirect
56. C22×C5×Q8Dic3semidirect
62. C2×C5×C4⋊C4Dic3semidirect
63. C2×C5×C4⋊C4Dic3semidirect
101. C22×C3×Q8Dic5semidirect
107. C2×C3×C4⋊C4Dic5semidirect
108. C2×C3×C4⋊C4Dic5semidirect
222. C22×Q8Dic15semidirect
229. C2×C4⋊C4Dic15semidirect
230. C2×C4⋊C4Dic15semidirect
320. C3×C5#1996semidirect
385. C5#71701semidirect
407. C3#10602494semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×C22⋊Q82
#117513132
C4×#67192
#117463472
#106024943
#717015