Finite Group Databasebeta

← #29596958

Subgroups — #29596958

Order 15616.

574 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840000C211yesmaximal, characteristic
2C2×#107×D61C211yesmaximal, characteristic
3C2×#16839566C211yesmaximal, characteristic, complemented
4C2×#16839541C211yesmaximal, characteristic
5#21443911C211yesmaximal, characteristic, complemented
6#21443910C211yesmaximal, characteristic, complemented
7C2×C61×#358C211yesmaximal, characteristic, complemented
8C22×C8×D61C2211yescharacteristic
9C2×#14519753—21no—
10C2×C4⋊C4×D61—21no—
11C22×#11758185C2211yescharacteristic
12#16840000—22no—
13#16840000C414yescomplemented
14C2×#14519711C2211yescharacteristic
15C2×#14519711C2211yescharacteristic
16C2×#14519710—21no—
17#107×D61—22no—
18#107×D61—22no—
19C2×#14519701—21no—
20#16839566—22no—
21#16839541—22no—
22C22×Q16×C61C2211yescharacteristic
23C22×C16×C61C2211yescharacteristic
24C2×#107×C61C2211yescharacteristic
25C61×#358—22no—
26C22×C4×D61—41no—
27C22×C4×D61D411yescharacteristic
28C22×#92418D411yescharacteristic
29C2×C8×D61—21no—
30C2×C8×D61—21no—
31C2×C8×D61—21no—
32C2×C8×D61—21no—
33C2×C8×D61C2×C412yes—
34C2×C8×D61D412yes—
35#14519753—24no—
36#14519753—42no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×#11758196—41no—
40C2×#11758196—41no—
41C4⋊C4×D61—24no—
42C4⋊C4×D61—42no—
43C2×#11758198—41no—
44C2×#11758185—21no—
45C2×#11758185—21no—
46C2×#11758185—21no—
47#14519711—21no—
48#14519711—21no—
49#14519711C2×C412yescomplemented
50#14519711D412yes—
51#14519710—24no—
52#14519701—24no—
53C22×Q8×C61—21no—
54C2×Q16×C61—21no—
55C2×Q16×C61—21no—
56C2×Q16×C61—21no—
57C2×Q16×C61—21no—
58C2×Q16×C61—21no—
59C2×Q16×C61C2×C412yescomplemented
60C22×C8×C61C2311yescharacteristic
61C2×C4⋊C4×C61—21no—
62C2×C16×C61—21no—
63C2×C16×C61—21no—
64C2×C16×C61—21no—
65#107×C61—21no—
66#107×C61D412yes—
67C23×D61D811yescharacteristic
68C2×Q8×C61—22no—
69C2×Q8×C61—41no—
70C2×Q8×C61—41no—
71C2×Q8×C61—41no—
72C2×Q8×C61—41no—
73C2×Q8×C61—41no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61C2×D411yescharacteristic
77C22×Dic61—41no—
78C22×Dic61—41no—
79C22×Dic61D811yescharacteristic
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—44no—
85C2×C4×D61—82no—
86C2×C4×D61—82no—
87C2×C4×D61C22⋊C412yes—
88C2×C4×D61SD1612yes—
89Q16×C61—22no—
90Q16×C61—22no—
91Q16×C61—41no—
92Q16×C61—41no—
93C2×C8×C61C22×C411yescharacteristic
94C2×C8×C61C2×D411yescharacteristic
95C2×C8×C61C2×D411yescharacteristic
96C2×C8×C61C2×D411yescharacteristic
97C2×C8×C61#4111yescharacteristic
98C2×C8×C61#4111yescharacteristic
99C4⋊C4×C61—22no—
100C4⋊C4×C61—41no—
101#11758198—44no—
102C4⋊Dic61—22no—
103C4⋊Dic61—22no—
104C4⋊Dic61—41no—
105C4⋊Dic61—41no—
106#11758196—44no—
107#11758196—44no—
108C2×#92418—21no—
109C2×#92418—21no—
110C2×#92418C22⋊C412yes—
111C8×D61—22no—
112C8×D61—22no—
113C8×D61—22no—
114C8×D61—22no—
115C16×C61—41no—
116#11758185—41no—
117C23×C61C2×D811yescharacteristic
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—21no—
122C22×D61#10212yes—
123C22×D61SD3212yes—
124C22×D61Q3212yes—
125Q8×C61—42no—
126Q8×C61—42no—
127Q8×C61—81no—
128Q8×C61—81no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—81no—
132C2×C4×C61—81no—
133C2×C4×C61—81no—
134C2×C4×C61—81no—
135C2×C4×C61C2×C22⋊C411yescharacteristic
136C2×C4×C61#12011yescharacteristic
137C2×C4×C61#12311yescharacteristic
138C2×C4×C61C2×SD1611yescharacteristic
139C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C61#13711yescharacteristic
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—81no—
146C2×Dic61—81no—
147C2×Dic61—81no—
148C2×Dic61—81no—
149C2×Dic61#10212yes—
150C4×D61—22no—
151C4×D61—22no—
152C4×D61—22no—
153C4×D61—22no—
154C4×D61—88no—
155C8×C61C4×D411yescharacteristic
156C8×C61C4×D411yescharacteristic
157C8×C61#12111yescharacteristic
158C8×C61#12111yescharacteristic
159#92418—22no—
160#12993—611nomaximal, Sylow
161C22×C61C2×#10211yescharacteristic
162C22×C61#44711yescharacteristic
163C22×C61#48011yescharacteristic
164C22×C61C2×SD3211yescharacteristic
165C22×C61C2×Q3211yescharacteristic
166C22×C61#50911yescharacteristic
167C22×C61#51011yescharacteristic
168C2×D61—22no—
169C2×D61—22no—
170C2×D61—22no—
171C2×D61—22no—
172C2×D61—22no—
173C2×D61—22no—
174C2×D61#35814yescomplemented
175C4×C61—82no—
176C4×C61—82no—
177C4×C61#38611yescharacteristic
178C4×C61#41711yescharacteristic
179C4×C61#44811yescharacteristic
180C4×C61#48111yescharacteristic
181Dic61—22no—
182Dic61—82no—
183Dic61—82no—
184C2×#467—611no—
185C22×#107—611no—
186C2×#358—611no—
187C2×#358D6111yescharacteristic, complemented
188C2×C22⋊C16—611no—
189#1380—611no—
190#1380—611no—
191C2×C61#187411yescharacteristic
192C2×C61C2×#35811yescharacteristic
193C2×C61#213911yescharacteristic
194C2×C61#218711yescharacteristic
195C2×C61#218811yescharacteristic
196C2×C61#223511yescharacteristic
197C2×C61#223611yescharacteristic
198D61—24no—
199C22×Q16C2×D6111yescharacteristic
200C23×C8—611no—
201C2×C22⋊Q8—1221no—
202C22×C4⋊C4—1221no—
203C22×C16—611no—
204C22×C16C2×D6111yescharacteristic
205#467—614no—
206#467—1222no—
207C2×#107—611no—
208C2×#107—611no—
209C2×#107—1222no—
210C2×#107—1221no—
211C2×#107—1222no—
212C2×#107C2×D6111yescharacteristic
213C2×#103—1221no—
214#358—22no—
215#358—1222no—
216C22⋊C16—1222no—
217C61#1299311yescharacteristic, complemented, Sylow
218C22×Q8—21no—
219C23×C4—611no—
220C23×C4—2441no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16—21no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16C4×D6112yescomplemented
227C22×C8—611no—
228C22×C8—612no—
229C22×C8—612no—
230C22×C8—1221no—
231C22×C8—1221no—
232C22×C8—1221no—
233C22×C8—1221no—
234C22×C8C22×D6111yescharacteristic
235C22⋊Q8—1224no—
236C22⋊Q8—2442no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—1224no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—2442no—
242C2×C4⋊C4—2441no—
243C2×C4⋊C4—2441no—
244C2×C22⋊C4—2441no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—1221no—
249C2×C16—1221no—
250C2×C16—1221no—
251#107—21no—
252#107—612no—
253#107—612no—
254#107—1224no—
255#107—1221no—
256#107—1221no—
257#107C22⋊D6112yes—
258#103—1224no—
259C24—611no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—611no—
269C22×C4—612no—
270C22×C4—612no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—4882no—
276C22×C4—4882no—
277C22×C4—2444no—
278C22×C4—2441no—
279C22×C4—2441no—
280C22×C4D4×D6111yescharacteristic
281Q16—22no—
282Q16—22no—
283Q16—41no—
284Q16—41no—
285C2×C8—612no—
286C2×C8—1221no—
287C2×C8—1221no—
288C2×C8—1222no—
289C2×C8—1222no—
290C2×C8—1222no—
291C2×C8—1222no—
292C2×C8C2×C4×D6111yescharacteristic
293C2×C8#1175822211yescharacteristic
294C2×C8D4×D6111yescharacteristic
295C2×C8D4×D6111yescharacteristic
296C2×C8#1175822411yescharacteristic
297C2×C8C2×C22⋊D6111yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—1222no—
301C4⋊C4—1222no—
302C4⋊C4—2444no—
303C4⋊C4—2444no—
304C4⋊C4—2441no—
305C4⋊C4—2441no—
306C22⋊C4—2444no—
307C16—41no—
308C16—2441no—
309C23—612no—
310C23—612no—
311C23—612no—
312C23—1221no—
313C23—1221no—
314C23—1221no—
315C23—1221no—
316C23D8×D6111yescharacteristic
317Q8—42no—
318Q8—42no—
319Q8—81no—
320Q8—81no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—612no—
328C2×C4—1221no—
329C2×C4—1221no—
330C2×C4—1222no—
331C2×C4—1222no—
332C2×C4—1222no—
333C2×C4—1222no—
334C2×C4—4888no—
335C2×C4—4881no—
336C2×C4—4881no—
337C2×C4—4881no—
338C2×C4—4881no—
339C2×C4—2442no—
340C2×C4—2442no—
341C2×C4C22⋊C4×D6111yescharacteristic
342C2×C4#1451974511yescharacteristic
343C2×C4#1451975911yescharacteristic
344C2×C4SD16×D6111yescharacteristic
345C2×C4#1451980111yescharacteristic
346C2×C4#1451980611yescharacteristic
347C8—1222no—
348C8#1451973711yescharacteristic
349C8C4×C22⋊D6111yescharacteristic
350C8#1451974811yescharacteristic
351C8#1451975011yescharacteristic
352C22—614no—
353C22—1222no—
354C22—1222no—
355C22—1222no—
356C22—1222no—
357C22—1222no—
358C22—1222no—
359C22#102×D6111yescharacteristic
360C22#1683989411yescharacteristic
361C22#1684007411yescharacteristic
362C22SD32×D6111yescharacteristic
363C22Q32×D6111yescharacteristic
364C22#1684019911yescharacteristic
365C22#1684020411yescharacteristic
366C4—82no—
367C4—82no—
368C4—1222no—
369C4—4882no—
370C4—4882no—
371C4#1683963811yescharacteristic
372C4#1683976011yescharacteristic
373C4#1683989911yescharacteristic
374C4#1684007911yescharacteristic
375C2—1224no—
376C2#2144613311yescharacteristic
377C2#358×D6111yescharacteristic
378C2#2144753011yescharacteristic
379C2#2144774611yescharacteristic
380C2#2144775111yescharacteristic
381C2#2144801511yescharacteristic
382C2#2144802011yescharacteristic
subgroupG/H
centerC23D8×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#358C2
socleC23×C61C2×D8
prime radicalC23×D61D8
cosocleC22×C8×C61C23
hypercenterC2×#358D61
hypocenterC61#12993
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
3. C2×#16839566C2semidirect
5. #21443911C2semidirect
6. #21443910C2semidirect
7. C2×C61×#358C2semidirect
13. #16840000C4semidirect
49. #14519711C2×C4semidirect
59. C2×Q16×C61C2×C4semidirect
174. C2×D61#358semidirect
187. C2×#358D61semidirect
217. C61#12993semidirect
226. C2×Q16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400002
C2×#107×D612
C2×#168395662
C2×#168395412
#214439112
#214439102
C2×C61×#3582
#1299361