Finite Group Databasebeta

← #29580315

Subgroups — #29580315

Order 15616.

610 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16840097C211yesmaximal, characteristic
2#21451635C211yesmaximal, characteristic
3C2×#16839700C211yesmaximal, characteristic
4C2×#16839500C211yesmaximal, characteristic
5C2×#16839487C211yesmaximal, characteristic
6#21443697C211yesmaximal, characteristic
7C61×#1824C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9C22×#11758196C2211yescharacteristic
10#16840097—22no—
11#16840097—22no—
12C2×#14519710—21no—
13C2×#14519700—21no—
14C2×C22⋊#92418C2211yescharacteristic
15#16839702—21no—
16#16839700—22no—
17#16839700—22no—
18#16839692—22no—
19#16839683—21no—
20#16839637—21no—
21#16839623—21no—
22C2×#14519684—21no—
23C2×#14519684—21no—
24C2×#14519684C2211yescharacteristic
25#16839500—22no—
26#16839487—22no—
27C22×C4⋊C4×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C61×#380C2211yescharacteristic
30C61×#340—22no—
31C22×Dic122D411yescharacteristic
32C23×Dic61—21no—
33C23×Dic61D411yescharacteristic
34C23×Dic61D411yescharacteristic
35C22×#92418—21no—
36#14519756—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50#14519710—24no—
51#14519700—24no—
52C22⋊#92418—21no—
53C22⋊#92418D412yes—
54#14519684—22no—
55#14519684—22no—
56#14519684—22no—
57#14519684—24no—
58#14519684—24no—
59#14519684—21no—
60#14519684D412yes—
61C23×C4×C61—21no—
62C23×C4×C61C2311yescharacteristic
63C22×C8×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D412yes—
70C2×C4⋊C4×C61D412yes—
71C2×C22⋊C4×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74#95×C61—21no—
75C24×C61C2×D411yescharacteristic
76C22×C4×C61—22no—
77C22×C4×C61—22no—
78C22×C4×C61—42no—
79C22×C4×C61—44no—
80C22×C4×C61—41no—
81C22×C4×C61—41no—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—22no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—42no—
101C22×Dic61—42no—
102C22×Dic61—42no—
103C22×Dic61—41no—
104C22×Dic61—41no—
105C22×Dic61Q1612yes—
106C2×Dic122—21no—
107C2×Dic122—21no—
108C2×Dic122—42no—
109C2×C8×C61—42no—
110C2×C8×C61—41no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—42no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊Dic61—41no—
119C22⋊Dic61—41no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—82no—
123#11758196—41no—
124#11758196—41no—
125#11758196—41no—
126#11758196—41no—
127#11758196—41no—
128#11758196—41no—
129C2×#92418—22no—
130C2×#92418—42no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61#12011yescharacteristic
134C23×C61#12011yescharacteristic
135C23×C61#12111yescharacteristic
136C23×C61C2×Q1611yescharacteristic
137C23×C61#13511yescharacteristic
138C23×C61#13711yescharacteristic
139C23×C61#13711yescharacteristic
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—44no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—82no—
152C2×C4×C61—82no—
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12711yescharacteristic
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—42no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—82no—
175C2×Dic61—82no—
176C2×Dic61—81no—
177Dic122—42no—
178Dic122—81no—
179C8×C61—81no—
180#92418—42no—
181#9929—611nomaximal, Sylow
182C22×C61—22no—
183C22×C61—22no—
184C22×C61—22no—
185C22×C61—22no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61#39211yescharacteristic
191C22×C61#45111yescharacteristic
192C22×C61#45111yescharacteristic
193C22×C61#45211yescharacteristic
194C22×C61#45211yescharacteristic
195C22×C61#45412yes—
196C22×C61#45812yes—
197C22×C61#46711yescharacteristic
198C22×C61#47011yescharacteristic
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—82no—
204C4×C61—82no—
205Dic61—42no—
206Dic61—82no—
207Dic61—81no—
208Dic61—81no—
209Dic61—81no—
210C2×#484—611no—
211#2627—611no—
212C2×#400—611no—
213#1824D6111yescharacteristic
214C2×#328—611no—
215C2×#324—611no—
216#1294—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—21no—
220C2×C61—21no—
221C2×C61#161512yes—
222C2×C61#161712yes—
223C2×C61#200111yescharacteristic
224C2×C61#200311yescharacteristic
225C2×C61#202211yescharacteristic
226C2×C22⋊Q8—611no—
227C22×C4⋊C4—611no—
228C22×C4⋊C4C2×D6111yescharacteristic
229#484—1222no—
230#484—1222no—
231C2×#107—1221no—
232C2×#103—1221no—
233C2×C22⋊C8—611no—
234C2×C22⋊C8C2×D6111yescharacteristic
235#400—1221no—
236#400—1222no—
237#400—1222no—
238#398—1222no—
239#395—1221no—
240#385—1221no—
241#381—1221no—
242#380C2×D6111yescharacteristic
243C2×#95—611no—
244C2×#95—1221no—
245C2×#95—1221no—
246#340—22no—
247#328—1222no—
248#324—1222no—
249C61#992911yescharacteristic, complemented, Sylow
250C22×Q8—611no—
251C23×C4—21no—
252C23×C4—611no—
253C23×C4—611no—
254C23×C4—1221no—
255C23×C4C22×D6111yescharacteristic
256C22×C8—21no—
257C22×C8—1221no—
258C22⋊Q8—1221no—
259C22⋊Q8—1221no—
260C22⋊Q8—1221no—
261C22⋊Q8—1221no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—611no—
268C2×C4⋊C4—1222no—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4D24412yes—
277C2×C4⋊C4C22⋊D6112yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—1221no—
280#107—1224no—
281#103—1224no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—612no—
285C22⋊C8—1221no—
286#95—21no—
287#95—612no—
288#95—1224no—
289#95—1224no—
290#95—1222no—
291#95—1222no—
292#95—1222no—
293#95—1221no—
294C24D4×D6111yescharacteristic
295C2×Q8—1221no—
296C2×Q8—1221no—
297C2×Q8—2442no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—612no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1222no—
316C22×C4—1221no—
317C22×C4—2442no—
318C22×C4—2442no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4—2442no—
322C22×C4C2×D24411yescharacteristic
323C22×C4#1175822211yescharacteristic
324C22×C4D4×D6111yescharacteristic
325C22×C4D4×D6111yescharacteristic
326C22×C4D4×D6111yescharacteristic
327C22×C4C2×C22⋊D6111yescharacteristic
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—1222no—
331C2×C8—2442no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—4882no—
340C4⋊C4—2441no—
341C4⋊C4—2441no—
342C4⋊C4—2441no—
343C4⋊C4—2441no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—2441no—
349C22⋊C4—2441no—
350C23—22no—
351C23—22no—
352C23#1451974511yescharacteristic
353C23#1451974711yescharacteristic
354C23#1451975011yescharacteristic
355C23Q16×D6111yescharacteristic
356C23#1451979811yescharacteristic
357C23#1451980611yescharacteristic
358C23#1451980911yescharacteristic
359Q8—4881no—
360Q8—2442no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—1222no—
375C2×C4—1222no—
376C2×C4—1222no—
377C2×C4—4882no—
378C2×C4—4882no—
379C2×C4—4881no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4#1451974711yescharacteristic
395C2×C4#1451975011yescharacteristic
396C2×C4#1451975111yescharacteristic
397C2×C4#1451977311yescharacteristic
398C8—81no—
399C8—2442no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22#1683967111yescharacteristic
409C22#1683991611yescharacteristic
410C22#1683991911yescharacteristic
411C22#1683992111yescharacteristic
412C22#1683992711yescharacteristic
413C22#1683993312yes—
414C22#1683995112yes—
415C22#1684000211yescharacteristic
416C22#1684001911yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—82no—
422C4—82no—
423C4—4882no—
424C4—4881no—
425C4—4881no—
426C4—4881no—
427C4—2442no—
428C2—21no—
429C2—21no—
430C2—21no—
431C2—21no—
432C2#2144484712yes—
433C2#2144485912yes—
434C2#2144682811yescharacteristic
435C2#2144684211yescharacteristic
436C2#2144695011yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1824C2
socleC23×C61#120
prime radicalC24×C61C2×D4
hypercenter#1824D61
hypocenterC61#9929
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
249. C61#9929semidirect

Maximal subgroups

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

subgroupindex
C2×#168400972
#214516352
C2×#168397002
C2×#168395002
C2×#168394872
#214436972
C61×#18242
#992961