Finite Group Databasebeta

← #29589189

Subgroups — #29589189

Order 15616.

562 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 93 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#21448430C211yesmaximal, characteristic
2#323×D61C211yesmaximal, characteristic
3#323×D61C211yesmaximal, characteristic
4#21443768C211yesmaximal, characteristic, complemented
5#21443767C211yesmaximal, characteristic, complemented
6#21443766C211yesmaximal, characteristic, complemented
7C61×#1326C211yesmaximal, characteristic, complemented
8C22×#11758198C411yescharacteristic
9C2×C22⋊C4×D61C411yescharacteristic
10C2×C22⋊C4×D61C2211yescharacteristic
11C22⋊C8×D61—21no—
12C22⋊C8×D61—21no—
13#16839637C411yescharacteristic
14#16839636—21no—
15#16839635C411yescharacteristic
16#16839634C2211yescharacteristic
17#16839633—21no—
18#16839621—21no—
19#16839619—21no—
20#16839486C2211yescharacteristic
21#16839486C2211yescharacteristic
22#16839485—21no—
23#16839485—21no—
24C61×#385C2211yescharacteristic
25C61×#323C2211yescharacteristic
26C61×#323C2211yescharacteristic
27C24×D61C2×C411yescharacteristic
28C23×Dic61C2×C411yescharacteristic
29C22×C4×D61—21no—
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C2×C8×D61—41no—
33C2×C8×D61—41no—
34C2×C4⋊Dic61—21no—
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×C22⋊Dic61C2×C411yescharacteristic
38C2×C22⋊Dic61C2×C411yescharacteristic
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—41no—
43C2×#11758198—41no—
44C2×#11758198D411yescharacteristic
45C2×#11758198D411yescharacteristic
46C22⋊C4×D61—22no—
47C22⋊C4×D61—41no—
48C22⋊C4×D61—41no—
49C22⋊C4×D61C814yescomplemented
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52#14519689—41no—
53#14519689—41no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C23×C4×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61—21no—
59C2×C22⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61C2311yescharacteristic
61C22⋊C8×C61—21no—
62C22⋊C8×C61—21no—
63#95×C61—21no—
64C24×C61C22×C411yescharacteristic
65C23×D61—21no—
66C23×D61—21no—
67C23×D61—22no—
68C23×D61—81no—
69C23×D61—81no—
70C23×D61C22⋊C411yescharacteristic
71C23×D61C22⋊C411yescharacteristic
72C23×D61C2×C812yes—
73C23×D61M1612yes—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—41no—
77C22×C4×C61—81no—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—41no—
83C22×Dic61—81no—
84C22×Dic61C22⋊C411yescharacteristic
85C22×Dic61C22⋊C411yescharacteristic
86C2×C4×D61—21no—
87C2×C4×D61—21no—
88C2×C4×D61—24no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—81no—
92C2×C4×D61—81no—
93C2×C4×D61—81no—
94C2×C4×D61—81no—
95C2×C8×C61—41no—
96C2×C8×C61—41no—
97C4⋊C4×C61—81no—
98C22⋊C4×C61—21no—
99C22⋊C4×C61—41no—
100C22⋊C4×C61C2×C812yescomplemented
101C22⋊Dic61—21no—
102C22⋊Dic61—41no—
103C22⋊Dic61C2×C812yescomplemented
104#11758198—24no—
105#11758198—41no—
106#11758198—41no—
107#11758198—42no—
108#11758198—41no—
109#11758198—41no—
110C4⋊Dic61—81no—
111#11758196—81no—
112#11758196—81no—
113C2×#92418—41no—
114C2×#92418—41no—
115C8×D61—82no—
116C8×D61—82no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61C2×C22⋊C411yescharacteristic
122C23×C61C22×C811yescharacteristic
123C23×C61C2×M1611yescharacteristic
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—22no—
127C22×D61—22no—
128C22×D61—22no—
129C22×D61—22no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—81no—
135C22×D61—81no—
136C22×D61—81no—
137C22×D61—81no—
138C22×D61—84no—
139C22×D61—82no—
140C22×D61—82no—
141C22×D61C22⋊C812yes—
142C22×D61C22⋊C812yes—
143C22×D61#9911yescharacteristic
144C22×D61#9911yescharacteristic
145C22×D61#10112yes—
146C2×C4×C61—22no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—82no—
157C2×Dic61—22no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—82no—
168C4×D61—42no—
169C4×D61—42no—
170C4×D61—81no—
171C4×D61—81no—
172C4×D61—84no—
173C8×C61—81no—
174C8×C61—81no—
175#92418—81no—
176#92418—81no—
177#11453—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—42no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61C2×C22⋊C811yescharacteristic
188C22×C61C2×#9911yescharacteristic
189C22×C61C2×#10111yescharacteristic
190C2×D61—21no—
191C2×D61—21no—
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—22no—
195C2×D61—42no—
196C2×D61—42no—
197C2×D61—42no—
198C2×D61—41no—
199C2×D61—41no—
200C2×D61—42no—
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—82no—
205C2×D61#32412yes—
206C2×D61#32412yes—
207C4×C61—42no—
208C4×C61—41no—
209C4×C61—41no—
210C4×C61—81no—
211Dic61—42no—
212Dic61—41no—
213Dic61—41no—
214Dic61—81no—
215#2315—611no—
216C2×#323—611no—
217C2×#323—611no—
218#1326—611no—
219#1326—611no—
220#1326—611no—
221#1326D6111yescharacteristic, complemented
222C2×C61—21no—
223C2×C61—21no—
224C2×C61—41no—
225C2×C61—41no—
226C2×C61C2×#32411yescharacteristic
227C2×C61#212111yescharacteristic
228C2×C61#212111yescharacteristic
229D61—22no—
230D61—22no—
231D61—42no—
232C22×C22⋊C4—611no—
233C22×C22⋊C4—611no—
234C22×C22⋊C4—611no—
235C2×C22⋊C8—1221no—
236C2×C22⋊C8—1221no—
237#385—611no—
238#385—611no—
239#385—611no—
240#385—1221no—
241#385—1221no—
242#385C2×D6111yescharacteristic
243#380—1221no—
244#380—1221no—
245#323—611no—
246#323—611no—
247#323—1221no—
248#323—1221no—
249#323C2×D6111yescharacteristic
250#323C2×D6111yescharacteristic
251C61#1145311yescharacteristic, complemented, Sylow
252C25—611no—
253C23×C4—611no—
254C23×C4—611no—
255C23×C4—611no—
256C23×C4—1221no—
257C23×C4C4×D6111yescharacteristic
258C22×C8—2441no—
259C22×C8—2441no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1221no—
264C2×C22⋊C4—611no—
265C2×C22⋊C4—614no—
266C2×C22⋊C4—611no—
267C2×C22⋊C4—611no—
268C2×C22⋊C4—611no—
269C2×C22⋊C4—1222no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—2441no—
274C2×C22⋊C4—2441no—
275C2×C22⋊C4—2441no—
276C2×C22⋊C4—2441no—
277C2×C22⋊C4C4×D6111yescharacteristic
278C2×C22⋊C4C22×D6111yescharacteristic
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—1221no—
282C22⋊C8—1221no—
283C22⋊C8—2441no—
284C22⋊C8—2441no—
285#95—21no—
286#95—1221no—
287#95—1221no—
288#95—1221no—
289C24—611no—
290C24—612no—
291C24—612no—
292C24—611no—
293C24—1221no—
294C24—1221no—
295C24—1222no—
296C24—4881no—
297C24—4881no—
298C24C2×C4×D6111yescharacteristic
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—81no—
303C22×C4—611no—
304C22×C4—611no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1224no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—4881no—
313C22×C4—4881no—
314C22×C4—4881no—
315C22×C4—4881no—
316C22×C4—4881no—
317C22×C4—2441no—
318C22×C4D4×D6111yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—4882no—
323C2×C8—4882no—
324C2×C8—2441no—
325C2×C8—2441no—
326C4⋊C4—81no—
327C4⋊C4—4881no—
328C4⋊C4—4881no—
329C4⋊C4—4881no—
330C22⋊C4—21no—
331C22⋊C4—41no—
332C22⋊C4—612no—
333C22⋊C4—1224no—
334C22⋊C4—1221no—
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C22⋊C4—2442no—
340C22⋊C4—2441no—
341C22⋊C4C8×D6112yescomplemented
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23—611no—
347C23—611no—
348C23—612no—
349C23—612no—
350C23—612no—
351C23—1222no—
352C23—1222no—
353C23—1222no—
354C23—1222no—
355C23—1221no—
356C23—1221no—
357C23—4881no—
358C23—4881no—
359C23—4881no—
360C23—4881no—
361C23—4884no—
362C23—4882no—
363C23—4882no—
364C23—2441no—
365C23—2441no—
366C23—2441no—
367C23—2441no—
368C23C22⋊C4×D6111yescharacteristic
369C23C2×C8×D6111yescharacteristic
370C23M16×D6111yescharacteristic
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—82no—
382C2×C4—1222no—
383C2×C4—1221no—
384C2×C4—1221no—
385C2×C4—4881no—
386C2×C4—4881no—
387C2×C4—4884no—
388C2×C4—4882no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C8—81no—
399C8—81no—
400C8—4881no—
401C8—4881no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—612no—
412C22—612no—
413C22—1221no—
414C22—1221no—
415C22—1221no—
416C22—1221no—
417C22—1222no—
418C22—4882no—
419C22—4882no—
420C22—4882no—
421C22—4882no—
422C22—2442no—
423C22—2442no—
424C22—2442no—
425C22—2442no—
426C22—2441no—
427C22—2441no—
428C22C22⋊C8×D6111yescharacteristic
429C22#99×D6111yescharacteristic
430C22#101×D6111yescharacteristic
431C4—42no—
432C4—41no—
433C4—41no—
434C4—81no—
435C4—4881no—
436C4—2442no—
437C4—2441no—
438C4—2441no—
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—1222no—
444C2—1222no—
445C2—2442no—
446C2#324×D6111yescharacteristic
447C2#2144746211yescharacteristic
448C2#2144746411yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC23×C61C22×C8
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1326C2
socleC22×C61C2×#99
prime radicalC24×D61C2×C4
cosocleC2×C22⋊C4×C61C23
hypercenter#1326D61
hypocenterC61#11453
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1326

Complements

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

NH ≅ G/Ntype
4. #21443768C2semidirect
5. #21443767C2semidirect
6. #21443766C2semidirect
7. C61×#1326C2semidirect
49. C22⋊C4×D61C8semidirect
100. C22⋊C4×C61C2×C8semidirect
103. C22⋊Dic61C2×C8semidirect
221. #1326D61semidirect
251. C61#11453semidirect
341. C22⋊C4C8×D61semidirect

Maximal subgroups

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

subgroupindex
#214484302
#323×D612
#214437682
#214437672
#214437662
C61×#13262
#1145361