Finite Group Databasebeta

← #29577050

Subgroups — #29577050

Order 15616.

576 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 103 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#21448976C211yesmaximal, characteristic, complemented
2#21446656C211yesmaximal, characteristic, complemented
3#21444348C211yesmaximal, characteristic, complemented
4#21444335C211yesmaximal, characteristic, complemented
5#21443726C211yesmaximal, characteristic, complemented
6#21443673C211yesmaximal, characteristic, complemented
7C61×#3192C211yesmaximal, characteristic
8C2×#14519692—21no—
9C2×#14519692—21no—
10#16839726C2211yescharacteristic
11#16839665—21no—
12#16839655—21no—
13#16839648—21no—
14#16839642—21no—
15#16839642—21no—
16#16839637—21no—
17#16839625C2211yescharacteristic
18#16839622C2211yescharacteristic
19C4×C22⋊Dic61C2211yescharacteristic
20#16839496—21no—
21#16839496—21no—
22#16839496—22no—
23C2×#121×C61C2211yescharacteristic
24C2×#121×C61C2211yescharacteristic
25C61×#481—21no—
26C61×#481C412yescomplemented
27C61×#469—22no—
28C61×#447—21no—
29C61×#447C412yescomplemented
30C61×#407C2211yescharacteristic
31C23×Dic61—21no—
32C2×#11758194—21no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61D411yescharacteristic
41C2×C22⋊Dic61D411yescharacteristic
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44#14519692—42no—
45#14519692—42no—
46C22⋊#92418—21no—
47C22⋊#92418—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684D411yescharacteristic
54#14519684D411yescharacteristic
55C22×D4×C61—21no—
56C22×D4×C61—21no—
57C22×D4×C61C2×C411yescharacteristic, complemented
58C22×D4×C61D411yescharacteristic
59C23×C4×C61C2311yescharacteristic
60C2×D8×C61—22no—
61C2×M16×C61—21no—
62#121×C61—21no—
63#121×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66#121×C61—21no—
67#121×C61—21no—
68#121×C61C2×C412yes—
69#121×C61D412yes—
70C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—21no—
74#106×C61—22no—
75#102×C61—22no—
76#102×C61—22no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61C2×C412yescomplemented
79C24×C61—21no—
80C24×C61—21no—
81C24×C61C2×D411yescharacteristic
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C2×D4×C61—41no—
90C2×D4×C61—41no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×D4×C61—81no—
94C2×D4×C61—81no—
95C2×D4×C61—81no—
96C2×D4×C61—81no—
97C2×D4×C61C22⋊C412yes—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—81no—
101C22×C4×C61C22×C411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—41no—
112C22×Dic61—81no—
113C22×Dic61—81no—
114C22×Dic61—81no—
115D8×C61—82no—
116M16×C61—81no—
117C2×C8×C61—22no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61C22⋊C412yes—
122C22⋊C4×C61—22no—
123C22⋊C4×C61—22no—
124C22⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊Dic61—22no—
127C22⋊Dic61—41no—
128C22⋊Dic61—41no—
129C22⋊Dic61—41no—
130C22⋊Dic61—81no—
131C22⋊Dic61—81no—
132C22⋊Dic61—81no—
133C22⋊Dic61—81no—
134C4⋊Dic61—81no—
135C4⋊Dic61—81no—
136C4×Dic61—81no—
137C4×Dic61—81no—
138#11758194—81no—
139C2×#92418—41no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—22no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—81no—
155C23×C61—81no—
156C23×C61C4×D411yescharacteristic
157C23×C61#12011yescharacteristic
158C23×C61#12111yescharacteristic
159D4×C61—42no—
160D4×C61—42no—
161D4×C61—81no—
162D4×C61—81no—
163D4×C61—82no—
164D4×C61—82no—
165D4×C61—81no—
166D4×C61—81no—
167D4×C61—81no—
168D4×C61—81no—
169C2×C4×C61—22no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61C2×C22⋊C411yescharacteristic
182C2×C4×C61C4×D411yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12311yescharacteristic
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—81no—
199C2×Dic61—81no—
200C2×Dic61—84no—
201C8×C61—42no—
202#92418—81no—
203#9343—611nomaximal, Sylow
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—42no—
215C22×C61—81no—
216C22×C61—81no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—161no—
225C22×C61—161no—
226C22×C61#38611yescharacteristic
227C22×C61#45311yescharacteristic
228C22×C61#45411yescharacteristic
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—42no—
234C4×C61—81no—
235Dic61—81no—
236Dic61—81no—
237Dic61—81no—
238Dic61—81no—
239Dic61—82no—
240#3192D6111yescharacteristic
241#2377—611no—
242#1964—611no—
243#1510—611no—
244#1507—611no—
245#1305—611no—
246#1284—611no—
247C2×C61—21no—
248C2×C61—21no—
249C2×C61—41no—
250C2×C61—41no—
251C2×C61—82no—
252C2×C61—161no—
253C2×C61#188311yescharacteristic
254C2×C61#190411yescharacteristic
255C2×C61#191311yescharacteristic
256C2×#121C2×D6111yescharacteristic
257C2×#121C2×D6111yescharacteristic
258#481—21no—
259#481Dic6112yescomplemented
260#469—22no—
261#447—21no—
262#447Dic6112yescomplemented
263C2×#99—1221no—
264C2×#99—1221no—
265#407—611no—
266#407C2×D6111yescharacteristic
267#390—1221no—
268#388—1221no—
269#387—1221no—
270#386—1221no—
271#386—1221no—
272#385—1221no—
273#381—611no—
274#380—611no—
275C4×C22⋊C4—611no—
276#327—1221no—
277#327—1221no—
278#327—1222no—
279C61#934311yescharacteristic, complemented, Sylow
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2×Dic6111yescharacteristic, complemented
283C22×D4C22⋊D6111yescharacteristic
284C23×C4—1221no—
285C23×C4C22×D6111yescharacteristic
286C2×D8—22no—
287C2×M16—21no—
288C2×M16—1221no—
289#121—21no—
290#121—21no—
291#121—21no—
292#121—21no—
293#121—21no—
294#121—21no—
295#121C2×Dic6112yes—
296#121C22⋊D6112yes—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4—1221no—
299C2×C4⋊C4C2×Dic6111yescharacteristic, complemented
300C2×C4⋊C4C22⋊D6111yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—611no—
304C2×C22⋊C4—611no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1221no—
307C2×C22⋊C4—1221no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C42—1221no—
311C2×C42—1221no—
312#106—22no—
313#102—22no—
314#102—22no—
315#99—2442no—
316#99—2442no—
317C22⋊C8—21no—
318C22⋊C8—1221no—
319C22⋊C8—1221no—
320C22⋊C8C2×Dic6112yescomplemented
321#95—611no—
322#95—611no—
323#95—1221no—
324#95—1221no—
325#95—1221no—
326#95—1221no—
327#95—1221no—
328C24—21no—
329C24—21no—
330C24D4×D6111yescharacteristic
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—81no—
343C2×D4—81no—
344C2×D4—81no—
345C2×D4—81no—
346C2×D4C22⋊Dic6112yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—81no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—4881no—
355C22×C4—4881no—
356C22×C4—4881no—
357C22×C4—2441no—
358C22×C4D4×D6111yescharacteristic
359C22×C4#1175822411yescharacteristic
360C22×C4#1175822411yescharacteristic
361C22×C4C22×Dic6111yescharacteristic
362C22×C4C2×C22⋊D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364D8—82no—
365M16—81no—
366M16—4881no—
367C2×C8—22no—
368C2×C8—2441no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—4881no—
373C4⋊C4—4881no—
374C4⋊C4C22⋊Dic6112yes—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—1222no—
380C22⋊C4—4881no—
381C22⋊C4—4881no—
382C22⋊C4—4881no—
383C22⋊C4—4881no—
384C22⋊C4—2441no—
385C22⋊C4—2441no—
386C22⋊C4—2441no—
387C42—4881no—
388C42—4881no—
389C23—21no—
390C23—21no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—81no—
404C23—81no—
405C23D4×Dic6111yescharacteristic
406C23#1451974511yescharacteristic
407C23#1451974911yescharacteristic
408D4—42no—
409D4—42no—
410D4—81no—
411D4—81no—
412D4—82no—
413D4—82no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—4881no—
431C2×C4—4881no—
432C2×C4—4884no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2442no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4C2×C22⋊Dic6111yescharacteristic
447C2×C4D4×Dic6111yescharacteristic
448C2×C4#1451974911yescharacteristic
449C2×C4#1451975811yescharacteristic
450C8—42no—
451C8—4881no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—161no—
473C22—161no—
474C22#1683964211yescharacteristic
475C22#1683992811yescharacteristic
476C22#1683993311yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—81no—
483C4—4881no—
484C4—4881no—
485C4—4881no—
486C4—4881no—
487C4—4882no—
488C2—21no—
489C2—21no—
490C2—41no—
491C2—41no—
492C2—82no—
493C2—161no—
494C2#2144619711yescharacteristic
495C2#2144630811yescharacteristic
496C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#3192C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenter#3192D61
hypocenterC61#9343
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3192

Complements

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

NH ≅ G/Ntype
1. #21448976C2semidirect
2. #21446656C2semidirect
3. #21444348C2semidirect
4. #21444335C2semidirect
5. #21443726C2semidirect
6. #21443673C2semidirect
26. C61×#481C4semidirect
29. C61×#447C4semidirect
57. C22×D4×C61C2×C4semidirect
70. C2×C4⋊C4×C61C2×C4semidirect
78. C22⋊C8×C61C2×C4semidirect
259. #481Dic61semidirect
262. #447Dic61semidirect
279. C61#9343semidirect
282. C22×D4C2×Dic61semidirect
299. C2×C4⋊C4C2×Dic61semidirect
320. C22⋊C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214489762
#214466562
#214443482
#214443352
#214437262
#214436732
C61×#31922
#934361