Finite Group Databasebeta

← #29581307

Subgroups — #29581307

Order 15616.

648 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16840124C211yesmaximal, characteristic, complemented
2#21453864C211yesmaximal, characteristic
3#21446793C211yesmaximal, characteristic, complemented
4C2×#16839502C211yesmaximal, characteristic, complemented
5C2×#16839501C211yesmaximal, characteristic, complemented
6#21443701C211yesmaximal, characteristic, complemented
7C2×C61×#497C211yesmaximal, characteristic, complemented
8C2×#14519765C2211yescharacteristic
9#16840124—22no—
10#16840124—22no—
11C2×#14519700—21no—
12C2×#14519697—21no—
13C2×#14519688C2211yescharacteristic
14C2×#14519688C2211yescharacteristic
15#16839678—22no—
16#16839674—21no—
17#16839673—21no—
18#16839671—21no—
19#16839670—21no—
20#16839628C2211yescharacteristic
21#16839524—22no—
22#16839502—22no—
23#16839501—22no—
24C22×Q16×C61—21no—
25C22×SD16×C61—21no—
26C2×C4⋊Q8×C61C2211yescharacteristic
27C2×#124×C61C2211yescharacteristic
28C61×#497—22no—
29C61×#497—22no—
30C2×#97×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C22⋊D61—22no—
34C22×#92418—21no—
35C22×#92418—21no—
36#14519765—21no—
37#14519765—21no—
38#14519765—21no—
39#14519765—21no—
40C2×C4⋊Dic61D412yes—
41C2×C22⋊Dic61—22no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—22no—
45#14519700—24no—
46#14519697—24no—
47#14519688—21no—
48#14519688—21no—
49#14519688—21no—
50#14519688D412yescomplemented
51#14519684—22no—
52#14519684—21no—
53#14519684—21no—
54C22×Q8×C61—21no—
55C22×Q8×C61D411yescharacteristic
56C22×D4×C61D411yescharacteristic
57C2×Q16×C61—24no—
58C2×Q16×C61—44no—
59C2×SD16×C61—24no—
60C2×SD16×C61—44no—
61C22×C8×C61—21no—
62C4⋊Q8×C61—21no—
63C4⋊Q8×C61—21no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61D412yescomplemented
66#124×C61—21no—
67#124×C61—21no—
68#124×C61—21no—
69#124×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74C2×C42×C61C2311yescharacteristic
75#97×C61—21no—
76#97×C61D412yescomplemented
77C24×C61—21no—
78C23×D61—21no—
79C2×Q8×C61—21no—
80C2×Q8×C61—22no—
81C2×Q8×C61—21no—
82C2×Q8×C61—42no—
83C2×Q8×C61—42no—
84C2×Q8×C61—42no—
85C2×Q8×C61—41no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—42no—
89C22×C4×C61—21no—
90C22×C4×C61—41no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—82no—
95C2×C22⋊D61—82no—
96C2×C22⋊D61—82no—
97C22×Dic61—22no—
98C22×Dic61—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—42no—
102C2×Dic122—21no—
103C2×Dic122—21no—
104C2×Dic122—42no—
105Q16×C61—48no—
106SD16×C61—48no—
107C2×C8×C61—24no—
108C2×C8×C61—22no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C22⋊Dic61—82no—
122#11758198—41no—
123#11758198—41no—
124#11758198—41no—
125#11758198—41no—
126#11758198—82no—
127C4⋊Dic61—42no—
128C2×#92418—22no—
129C2×#92418—22no—
130C2×#92418—41no—
131C2×#92418—42no—
132C2×#92418—41no—
133C23×C61—41no—
134C23×C61—42no—
135C23×C61—42no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61#12011yescharacteristic
139C22×D61—41no—
140C22×D61—42no—
141C22×D61—42no—
142C22×D61—41no—
143C22×D61—41no—
144Q8×C61—42no—
145Q8×C61—42no—
146Q8×C61—42no—
147Q8×C61—82no—
148Q8×C61—81no—
149D4×C61—42no—
150D4×C61—81no—
151C2×C4×C61—22no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61—21no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—82no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12711yescharacteristic
165C22⋊D61—88no—
166C2×Dic61—41no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171D244—42no—
172D244—81no—
173Dic122—42no—
174Dic122—81no—
175C8×C61—24no—
176#92418—42no—
177#92418—42no—
178#10081—611nomaximal, Sylow
179C22×C61—82no—
180C22×C61—82no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61#39211yescharacteristic
185C22×C61#45312yes—
186C22×C61#45511yescharacteristic
187C22×C61#45512yes—
188C22×C61#45611yescharacteristic
189C2×D61—82no—
190C2×D61—82no—
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—81no—
201C4×C61—82no—
202Dic61—82no—
203Dic61—81no—
204C2×#497D6111yescharacteristic, complemented
205C2×#489—611no—
206#2819—611no—
207#1993—611no—
208C2×#329—611no—
209C2×#329—611no—
210#1295—611no—
211C2×C61—161no—
212C2×C61#165912yes—
213C2×C61#166112yes—
214C2×C61#200211yescharacteristic
215C2×C61#200411yescharacteristic
216C2×C61#200711yescharacteristic
217D61—161no—
218C22×Q16—21no—
219C22×SD16—21no—
220C2×C4⋊Q8C2×D6111yescharacteristic
221C2×#124—611no—
222C2×#124C2×D6111yescharacteristic
223#497—22no—
224#497—22no—
225#489—1222no—
226#489—1222no—
227C2×#103—1221no—
228C2×#102—1221no—
229C2×#97—611no—
230C2×#97—611no—
231C2×#97C2×D6111yescharacteristic
232#394—1222no—
233#393—1221no—
234#393—1221no—
235#392—1221no—
236#392—1221no—
237#382—611no—
238#340—1222no—
239#329—1222no—
240#329—1222no—
241C61#1008111yescharacteristic, complemented, Sylow
242C22×Q8—21no—
243C22×Q8—611no—
244C22×Q8C22⋊D6111yescharacteristic
245C22×D4—611no—
246C22×D4—1222no—
247C22×D4C22⋊D6111yescharacteristic
248C2×Q16—24no—
249C2×Q16—44no—
250C2×SD16—24no—
251C2×SD16—44no—
252C22×C8—21no—
253C22×C8—1221no—
254C22×C8—1221no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C22⋊D6112yescomplemented
259#124—21no—
260#124—21no—
261#124—21no—
262#124—21no—
263#124—1221no—
264#124—1221no—
265#124—1221no—
266#124—1221no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—612no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—1222no—
273C2×C22⋊C4—1222no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C42C22×D6111yescharacteristic
277#103—1224no—
278#102—1224no—
279#97—21no—
280#97—612no—
281#97—1221no—
282#97—1221no—
283#97—1221no—
284#97C22⋊D6112yescomplemented
285#95—1222no—
286#95—1221no—
287#95—1221no—
288C24—21no—
289C24—1221no—
290C2×Q8—21no—
291C2×Q8—22no—
292C2×Q8—21no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—41no—
297C2×Q8—1221no—
298C2×Q8—1221no—
299C2×Q8—2442no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—1221no—
304C2×D4—1221no—
305C2×D4—4882no—
306C2×D4—4882no—
307C2×D4—4882no—
308C2×D4—2442no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—1222no—
312C22×C4—1221no—
313C22×C4D4×D6111yescharacteristic
314C22×C4D4×D6111yescharacteristic
315C22×C4C2×C22⋊D6111yescharacteristic
316Q16—48no—
317SD16—48no—
318C2×C8—24no—
319C2×C8—22no—
320C2×C8—1222no—
321C2×C8—1222no—
322C2×C8—2441no—
323C2×C8—2442no—
324C2×C8—2441no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—2442no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—4882no—
335C22⋊C4—4882no—
336C22⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C22⋊C4—2441no—
340C42D4×D6111yescharacteristic
341C42#1175822411yescharacteristic
342C42C2×C22⋊D6111yescharacteristic
343C42C2×C22⋊D6111yescharacteristic
344C23—42no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—42no—
349C23—2441no—
350C23—2442no—
351C23—2441no—
352C23—2441no—
353C23—2442no—
354C23#1451974511yescharacteristic
355Q8—42no—
356Q8—42no—
357Q8—42no—
358Q8—81no—
359Q8—82no—
360Q8—4881no—
361Q8—2442no—
362D4—42no—
363D4—81no—
364D4—4881no—
365D4—4888no—
366D4—2442no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—2442no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2441no—
379C2×C4—2442no—
380C2×C4#1451974511yescharacteristic
381C2×C4#1451974611yescharacteristic
382C2×C4#1451974811yescharacteristic
383C2×C4#1451974911yescharacteristic
384C2×C4#1451974911yescharacteristic
385C2×C4#1451977311yescharacteristic
386C8—24no—
387C8—2442no—
388C8—2442no—
389C22—82no—
390C22—82no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—4882no—
395C22—4882no—
396C22—4881no—
397C22—4881no—
398C22—4881no—
399C22#1683967011yescharacteristic
400C22#1683992912yes—
401C22#1683993811yescharacteristic
402C22#1683993812yes—
403C22#1683994311yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—81no—
411C4—82no—
412C4—4882no—
413C4—4881no—
414C2—161no—
415C2—9761no—
416C2#2144512312yes—
417C2#2144513312yes—
418C2#2144683411yescharacteristic
419C2#2144684511yescharacteristic
420C2#2144686411yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC2×C61×#497C2
socleC23×C61#120
prime radical#21453864C2
hypercenterC2×#497D61
hypocenterC61#10081
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
1. C2×#16840124C2semidirect
3. #21446793C2semidirect
4. C2×#16839502C2semidirect
5. C2×#16839501C2semidirect
6. #21443701C2semidirect
7. C2×C61×#497C2semidirect
50. #14519688D4semidirect
65. C4⋊Q8×C61D4semidirect
76. #97×C61D4semidirect
204. C2×#497D61semidirect
241. C61#10081semidirect
258. C4⋊Q8C22⋊D61semidirect
284. #97C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401242
#214538642
#214467932
C2×#168395022
C2×#168395012
#214437012
C2×C61×#4972
#1008161