Finite Group Databasebeta

← #29733627

Subgroups — #29733627

Order 15616.

1280 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1280 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C2×#16840368C211yesmaximal, characteristic, complemented
2C2×#16840138C211yesmaximal, characteristic, complemented
3#21451570C211yesmaximal, characteristic, complemented
4#21446816C214yesmaximal, complemented
5#21446785C211yesmaximal, characteristic, complemented
6#21446643C211yesmaximal, characteristic, complemented
7#21446164C214yesmaximal, complemented
8#21445588C211yesmaximal, characteristic, complemented
9C2×C61×#490C211yesmaximal, characteristic, complemented
10C22×#11758199C2214yes—
11#16840368—24no—
12#16840368—24no—
13C2×#14519774C2211yescharacteristic
14C2×#14519762—22no—
15C2×#14519752—22no—
16C2×#14519752C2212yes—
17C2×#14519748C2212yes—
18C2×#14519739C2212yes—
19C2×#14519734C2211yescharacteristic
20#16840138—24no—
21C2×#14519699C2214yes—
22C2×#14519689C2212yes—
23C2×C4×#92418C2211yescharacteristic
24#16839676—22no—
25#16839670—22no—
26#16839665C2212yes—
27#16839628C2211yescharacteristic
28#16839524C2214yes—
29#16839516C2212yes—
30C2×#127×C61C2211yescharacteristic
31C2×#125×C61C2211yescharacteristic
32C61×#490—24no—
33C2×#102×C61C2214yes—
34C2×#97×C61C2211yescharacteristic
35C22×D244—22no—
36C22×D244—22no—
37C22×D244C2312yes—
38C22×C22⋊D61—22no—
39C22×C4×D61—22no—
40C22×C4×D61C2311yescharacteristic
41C2×#11758199—28no—
42C2×#11758199—28no—
43C2×#11758199—44no—
44C22×#92418C2312yes—
45#14519774—22no—
46#14519774D414yescomplemented
47#14519762—216no—
48#14519752—216no—
49#14519752—24no—
50#14519752—24no—
51#14519748—44no—
52#14519739—24no—
53#14519739—24no—
54#14519734—22no—
55#14519734—22no—
56C2×C4⋊Dic61C2312yes—
57C2×#11758196D412yes—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—24no—
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×#11758198D412yes—
63C2×C4×Dic61C2311yescharacteristic
64#14519699—44no—
65#14519689—24no—
66C4×#92418—22no—
67#14519684—22no—
68C22×D4×C61—22no—
69C22×D4×C61C2312yes—
70C22×C8×C61C2312yes—
71#127×C61—22no—
72#127×C61—22no—
73#125×C61—22no—
74#125×C61D414yescomplemented
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61C2312yes—
77C2×C42×C61C2311yescharacteristic
78#102×C61—28no—
79#97×C61—22no—
80C24×C61—22no—
81C23×D61—22no—
82C23×D61C2×D411yescharacteristic
83C2×D4×C61—22no—
84C2×D4×C61—22no—
85C2×D4×C61—24no—
86C2×D4×C61—24no—
87C2×D4×C61—44no—
88C2×D4×C61—84no—
89C22×C4×C61—22no—
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2411yescharacteristic
92C2×C22⋊D61—44no—
93C2×C22⋊D61—84no—
94C22×Dic61—22no—
95C22×Dic61C2×D412yes—
96C22×Dic61C2×D411yescharacteristic
97C2×D244—24no—
98C2×D244—24no—
99C2×D244—22no—
100C2×D244—28no—
101C2×D244—28no—
102C2×D244—84no—
103C2×D244—84no—
104C2×D244C2×D414yes—
105C2×C4×D61—21no—
106C2×C4×D61—21no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—28no—
110C2×C4×D61—84no—
111C2×C8×C61—22no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C4⋊C4×C61—28no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61C2×D414yes—
117C42×C61C2×D412yes—
118C42×C61C2×D412yes—
119C22⋊Dic61—44no—
120C22⋊Dic61—84no—
121#11758199—416no—
122#11758198—28no—
123#11758198—28no—
124#11758198—24no—
125C4⋊Dic61—24no—
126#11758196—24no—
127C4×Dic61—21no—
128C4×Dic61—21no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418#4114yes—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61—42no—
135C23×C61—44no—
136C23×C61—44no—
137C23×C61C22×D411yescharacteristic
138C22×D61—21no—
139C22×D61—24no—
140C22×D61—21no—
141C22×D61—21no—
142C22×D61—24no—
143C22×D61—42no—
144C22×D61—42no—
145C22×D61—44no—
146C22×D61—44no—
147D4×C61—44no—
148D4×C61—44no—
149D4×C61—88no—
150C2×C4×C61—24no—
151C2×C4×C61—24no—
152C2×C4×C61—44no—
153C2×C4×C61#12014yes—
154C2×C4×C61C22×D411yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C2×#4112yes—
157C2×C4×C61H(5,𝔽2)12yes—
158C22⋊D61—88no—
159C2×Dic61—22no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—24no—
164C2×Dic61—42no—
165C2×Dic61—44no—
166D244—28no—
167D244—44no—
168D244—88no—
169D244—88no—
170C4×D61—42no—
171C4×D61—42no—
172C4×D61—88no—
173C8×C61—42no—
174#92418—24no—
175#25608—611nomaximal, Sylow
176C22×C61—84no—
177C22×C61—82no—
178C22×C61—82no—
179C22×C61—82no—
180C22×C61—84no—
181C22×C61C2×#12011yescharacteristic
182C22×C61#53211yescharacteristic
183C22×C61#53511yescharacteristic
184C22×C61#57614yes—
185C2×D61—42no—
186C2×D61—41no—
187C2×D61—44no—
188C2×D61—42no—
189C2×D61—44no—
190C2×D61—42no—
191C2×D61—44no—
192C2×D61—84no—
193C2×D61—84no—
194C4×C61—24no—
195C4×C61—44no—
196C4×C61#54612yes—
197C4×C61#55012yes—
198Dic61—41no—
199Dic61—42no—
200Dic61—82no—
201C2×#540—611no—
202C2×#493—611no—
203C2×#490D6111yescharacteristic, complemented
204#2619—611no—
205#1999—614no—
206#1991—611no—
207#1961—611no—
208#1879—614no—
209#1750—611no—
210C2×C61—162no—
211C2×C61#243111yescharacteristic
212C2×C61#300812yes—
213C2×C61#317612yes—
214C2×C61#322012yes—
215D61—81no—
216D61—84no—
217C22×D8—614no—
218#540—1224no—
219#540—1224no—
220C2×#127—611no—
221C2×#127C2×D6111yescharacteristic
222C2×#125C2×D6111yescharacteristic
223C2×#123—1222no—
224C2×#121—612no—
225C2×#121—612no—
226C2×#121—1222no—
227C2×C4×D4—612no—
228C2×#117—611no—
229#493—1224no—
230#490—24no—
231C2×#102—614no—
232C2×#102C2×D6114yes—
233C2×C22⋊C8—612no—
234C2×#97C2×D6111yescharacteristic
235C2×C4×C8—611no—
236#394—1222no—
237#392—1222no—
238#390—612no—
239#382—611no—
240#340—614no—
241#336—612no—
242C61#2560811yescharacteristic, complemented, Sylow
243C22×D4—22no—
244C22×D4—612no—
245C22×D4—1222no—
246C22×D4—1222no—
247C22×D4—1222no—
248C22×D4C22×D6112yes—
249C23×C4—611no—
250C23×C4—1222no—
251C2×D8—1228no—
252C2×D8—1228no—
253C2×D8—2444no—
254C22×C8—612no—
255C22×C8C22×D6112yes—
256#127—22no—
257#127—22no—
258#127—614no—
259#127—1222no—
260#125—22no—
261#125C22⋊D6114yescomplemented
262#123—12216no—
263#121—1224no—
264#121—12216no—
265#121—1224no—
266#121—2444no—
267C4×D4—1224no—
268C4×D4—1224no—
269#117—1222no—
270#117—1222no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—612no—
273C2×C4⋊C4—612no—
274C2×C4⋊C4C22×D6112yes—
275C2×C22⋊C4—612no—
276C2×C22⋊C4—1222no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1224no—
279C2×C22⋊C4—1222no—
280C2×C42—611no—
281C2×C42C22×D6111yescharacteristic
282#102—28no—
283#102—2444no—
284C22⋊C8—1224no—
285#97—22no—
286C4×C8—1222no—
287#95—1222no—
288C24—22no—
289C24—611no—
290C24—1222no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—44no—
296C2×D4—84no—
297C2×D4—614no—
298C2×D4—1228no—
299C2×D4—1222no—
300C2×D4—1224no—
301C2×D4—1224no—
302C2×D4—1228no—
303C2×D4—4884no—
304C2×D4—4884no—
305C2×D4—4884no—
306C2×D4—2444no—
307C22×C4—22no—
308C22×C4—612no—
309C22×C4—611no—
310C22×C4—1228no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—1222no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4—4884no—
317C22×C4D4×D6112yes—
318C22×C4C23×D6111yescharacteristic
319D8—24416no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—614no—
324C2×C8—1222no—
325C2×C8—1222no—
326C4⋊C4—22no—
327C4⋊C4—28no—
328C4⋊C4—1224no—
329C4⋊C4—1224no—
330C4⋊C4C2×C22⋊D6114yes—
331C22⋊C4—1228no—
332C22⋊C4—1228no—
333C22⋊C4—1224no—
334C22⋊C4—4884no—
335C22⋊C4—2444no—
336C42—1221no—
337C42—1221no—
338C42D4×D6112yes—
339C42C2×C22⋊D6112yes—
340C23—42no—
341C23—44no—
342C23—44no—
343C23—42no—
344C23—42no—
345C23—1224no—
346C23—1221no—
347C23—1224no—
348C23—1221no—
349C23—1221no—
350C23—2444no—
351C23—2444no—
352C23—2442no—
353C23—2442no—
354C23C2×D4×D6111yescharacteristic
355D4—44no—
356D4—44no—
357D4—88no—
358D4—1228no—
359D4—4888no—
360D4—4888no—
361D4—4888no—
362D4—2444no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—44no—
366C2×C4—1222no—
367C2×C4—1221no—
368C2×C4—1224no—
369C2×C4—1221no—
370C2×C4—1221no—
371C2×C4—4888no—
372C2×C4—2444no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4#1451974514yes—
377C2×C4C2×D4×D6111yescharacteristic
378C2×C4C22×C22⋊D6111yescharacteristic
379C2×C4#41×D6112yes—
380C2×C4#1451982312yes—
381C8—42no—
382C8—1224no—
383C22—84no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—84no—
388C22—4884no—
389C22—4884no—
390C22—2442no—
391C22—2441no—
392C22—2444no—
393C22—2442no—
394C22—2444no—
395C22—2442no—
396C22—2444no—
397C22C2×#1451974511yescharacteristic
398C22#1684031111yescharacteristic
399C22#1684033011yescharacteristic
400C22#1684056614yes—
401C4—24no—
402C4—44no—
403C4—4882no—
404C4—2441no—
405C4—2442no—
406C4#1684040512yes—
407C4#1684043112yes—
408C2—162no—
409C2—4884no—
410C2—4881no—
411C2#2144944111yescharacteristic
412C2#2145508612yes—
413C2#2145646812yes—
414C2#2145687012yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#490C2
socleC23×C61C22×D4
hypercenterC2×#490D61
hypocenterC61#25608
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
1. C2×#16840368C2semidirect
2. C2×#16840138C2semidirect
3. #21451570C2semidirect
4. #21446816C2semidirect
5. #21446785C2semidirect
6. #21446643C2semidirect
7. #21446164C2semidirect
8. #21445588C2semidirect
9. C2×C61×#490C2semidirect
46. #14519774D4semidirect
74. #125×C61D4semidirect
203. C2×#490D61semidirect
242. C61#25608semidirect
261. #125C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403682
C2×#168401382
#214515702
#214468162
#214467852
#214466432
#214461642
#214455882
C2×C61×#4902
#2560861