Finite Group Databasebeta

← #29578010

Subgroups — #29578010

Order 15616.

628 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#21454848C211yesmaximal, characteristic, complemented
2#21450705C211yesmaximal, characteristic
3#21444145C212yesmaximal, complemented
4#21443724C212yesmaximal, complemented
5C61×#1465C211yesmaximal, characteristic, complemented
6C2×#14519758C411yescharacteristic
7C2×#14519745C411yescharacteristic
8C22×#11758198C2211yescharacteristic
9#16839834—24no—
10#16839819—21no—
11#16839726C2212yes—
12C2×C22⋊#92418—21no—
13#16839684—21no—
14#16839678—21no—
15#16839653—21no—
16#16839641—21no—
17#16839623C411yescharacteristic
18#16839622C2211yescharacteristic
19#16839617C411yescharacteristic
20#16839485—24no—
21C2×C4×D4×C61C2211yescharacteristic
22C61×#407C2212yes—
23C61×#324—21no—
24C61×#323—21no—
25C24×D61C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C23×Dic61C2×C411yescharacteristic
28C2×#11758194—22no—
29C22×#92418—22no—
30#14519758—42no—
31#14519745—42no—
32C2×C4⋊Dic61—21no—
33C2×#11758196—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61C2×C411yescharacteristic
36C2×C22⋊Dic61C2×C411yescharacteristic
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—22no—
41C2×#11758198—21no—
42C2×#11758198—22no—
43C2×#11758198—21no—
44C4⋊#92418—22no—
45C22⋊#92418—22no—
46C22⋊#92418—22no—
47C22⋊#92418D414yescomplemented
48#14519688—22no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52C22×D4×C61C2×C411yescharacteristic
53C23×C4×C61D411yescharacteristic
54C23×C4×C61C2311yescharacteristic
55C2×M16×C61—22no—
56C4×D4×C61—22no—
57C4×D4×C61—22no—
58C2×C4⋊C4×C61D411yescharacteristic
59C2×C22⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61D411yescharacteristic
61C2×C42×C61D411yescharacteristic
62C22⋊C8×C61—22no—
63C22⋊C8×C61—22no—
64C24×C61C22⋊C411yescharacteristic
65C24×C61C22×C411yescharacteristic
66C23×D61—21no—
67C23×D61—22no—
68C23×D61—42no—
69C23×D61—41no—
70C23×D61—41no—
71C23×D61—81no—
72C2×D4×C61—22no—
73C2×D4×C61—22no—
74C2×D4×C61—41no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—81no—
78C22×C4×C61—81no—
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61#4112yes—
86C2×C22⋊D61—42no—
87C2×C22⋊D61—42no—
88C2×C22⋊D61—81no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—22no—
93C22×Dic61—41no—
94C22×Dic61—41no—
95M16×C61—82no—
96C2×C8×C61—42no—
97C4⋊C4×C61—22no—
98C22⋊C4×C61—22no—
99C22⋊C4×C61—22no—
100C42×C61—21no—
101C42×C61—21no—
102C22⋊Dic61—42no—
103C22⋊Dic61—41no—
104C22⋊Dic61—41no—
105#11758198—42no—
106#11758198—42no—
107#11758198—41no—
108#11758198—42no—
109#11758198—41no—
110#11758198—81no—
111C4⋊Dic61—81no—
112#11758196—42no—
113#11758194—82no—
114C2×#92418—24no—
115C2×#92418—82no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61#10012yes—
125C23×C61C2×C22⋊C411yescharacteristic
126C23×C61C4×D412yes—
127C22×D61—22no—
128C22×D61—24no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—42no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—84no—
137C22×D61—84no—
138C22×D61—82no—
139C22×D61—82no—
140D4×C61—44no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—81no—
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12112yes—
156C2×C4×C61#12311yescharacteristic
157C22⋊D61—84no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—82no—
168C8×C61—82no—
169#92418—44no—
170#9515—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—22no—
174C22×C61—22no—
175C22×C61—22no—
176C22×C61—22no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61—81no—
183C22×C61#38611yescharacteristic
184C22×C61C2×#10011yescharacteristic
185C22×C61#41311yescharacteristic
186C2×D61—42no—
187C2×D61—42no—
188C2×D61—42no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—84no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—84no—
195C4×C61—21no—
196C4×C61—22no—
197C4×C61—21no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201Dic61—81no—
202Dic61—81no—
203Dic61—81no—
204#2973—611no—
205#2534—611no—
206#1465D6111yescharacteristic, complemented
207#1462—612no—
208#1305—612no—
209C2×C61—21no—
210C2×C61—21no—
211C2×C61—42no—
212C2×C61—42no—
213C2×C61#189111yescharacteristic
214C2×C61#190011yescharacteristic
215C2×C61#191311yescharacteristic
216D61—82no—
217C2×#123—611no—
218C2×#120—611no—
219C2×C4×D4C2×D6111yescharacteristic
220C22×C22⋊C4—611no—
221#435—1224no—
222#432—1221no—
223#407—612no—
224#407C2×D6112yes—
225C2×C22⋊C8—1221no—
226#396—1221no—
227#394—1221no—
228#388—1221no—
229#386—1221no—
230#381—611no—
231#380—611no—
232#379—611no—
233#324—21no—
234#323—21no—
235#323—1224no—
236C61#951511yescharacteristic, complemented, Sylow
237C25—611no—
238C22×D4—1221no—
239C22×D4C4×D6111yescharacteristic
240C23×C4—611no—
241C23×C4D24411yescharacteristic
242C23×C4C22×D6111yescharacteristic
243C2×M16—22no—
244C2×M16—1222no—
245C22×C8—1222no—
246#123—2442no—
247#120—2442no—
248C4×D4—22no—
249C4×D4—22no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4C22⋊D6111yescharacteristic
253C2×C22⋊C4—611no—
254C2×C22⋊C4—611no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1222no—
261C2×C22⋊C4—1222no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4C4×D6111yescharacteristic
264C2×C22⋊C4C22⋊D6111yescharacteristic
265C2×C42D24411yescharacteristic
266C4⋊C8—1222no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—614no—
270C22⋊C8—1222no—
271C22⋊C8—1222no—
272#97—1222no—
273#95—1221no—
274#95—1221no—
275#95—1221no—
276C24—1222no—
277C24—1221no—
278C24—4881no—
279C24—2441no—
280C24—2441no—
281C24—2442no—
282C24#1175819811yescharacteristic
283C24C2×C4×D6111yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—4881no—
288C2×D4—2442no—
289C2×D4—2442no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—1222no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—2441no—
299C22×C4—2441no—
300C22×C4#1175819811yescharacteristic
301C22×C4#1175819811yescharacteristic
302C22×C4#1175819811yescharacteristic
303C22×C4C2×D24411yescharacteristic
304C22×C4D4×D6112yes—
305C22×C4#1175822412yes—
306C22×C4C2×C22⋊D6111yescharacteristic
307M16—82no—
308M16—4882no—
309C2×C8—42no—
310C2×C8—1224no—
311C2×C8—4882no—
312C4⋊C4—22no—
313C4⋊C4—4881no—
314C4⋊C4—2442no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—4881no—
318C22⋊C4—2442no—
319C22⋊C4—2442no—
320C22⋊C4—2442no—
321C22⋊C4—2441no—
322C22⋊C4—2442no—
323C22⋊C4—2441no—
324C22⋊C4—2441no—
325C22⋊C4—2441no—
326C42—21no—
327C42—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—1224no—
337C23—1222no—
338C23—4882no—
339C23—4882no—
340C23—4884no—
341C23—4884no—
342C23—2441no—
343C23—2441no—
344C23—2441no—
345C23—2442no—
346C23—2441no—
347C23—2441no—
348C23—2441no—
349C23#1451969312yes—
350C23C2×#1175819811yescharacteristic
351C23#1451973712yes—
352D4—44no—
353D4—4884no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—4882no—
368C2×C4—2441no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4#1451974711yescharacteristic
378C2×C4#1451974912yes—
379C2×C4#1451976111yescharacteristic
380C8—82no—
381C8—2444no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—4881no—
395C22—4881no—
396C22—4884no—
397C22—4884no—
398C22—4881no—
399C22—4881no—
400C22—2442no—
401C22—2442no—
402C22—2442no—
403C22#1683963911yescharacteristic
404C22C2×#1451969311yescharacteristic
405C22#1683974411yescharacteristic
406C4—22no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—4881no—
413C4—4881no—
414C4—4881no—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—42no—
419C2—4882no—
420C2#2144623511yescharacteristic
421C2#2144628011yescharacteristic
422C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1465C2
socleC22×C61#386
prime radicalC2×#14519745C4
cosocleC23×C4×C61C23
hypercenter#1465D61
hypocenterC61#9515
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
1. #21454848C2semidirect
3. #21444145C2semidirect
4. #21443724C2semidirect
5. C61×#1465C2semidirect
47. C22⋊#92418D4semidirect
206. #1465D61semidirect
236. C61#9515semidirect

Maximal subgroups

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

subgroupindex
#214548482
#214507052
#214441452
#214437242
C61×#14652
#951561