Finite Group Databasebeta

← #29583243

Subgroups — #29583243

Order 15616.

602 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 75 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#21451480C211yesmaximal, characteristic, complemented
2#21445763C211yesmaximal, characteristic
3#21445737C211yesmaximal, characteristic
4#21443728C211yesmaximal, characteristic, complemented
5#21443673C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#1789C211yesmaximal, characteristic
8C2×#14519704—21no—
9C2×#14519704—21no—
10C2×#14519692—21no—
11C2×#14519692—21no—
12#16839726C2211yescharacteristic
13#16839698—21no—
14#16839674—21no—
15#16839668—21no—
16#16839666—21no—
17#16839646—21no—
18#16839645—21no—
19#16839625C2211yescharacteristic
20#16839625C2211yescharacteristic
21C4×C22⋊Dic61C2211yescharacteristic
22C22×#41×C61C2211yescharacteristic
23C2×#104×C61—21no—
24C2×#104×C61—21no—
25C61×#407C2211yescharacteristic
26C4×C22⋊C4×C61C2211yescharacteristic
27C22×Dic122—21no—
28C2×#11758194—21no—
29C2×C4⋊Dic61—21no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61D411yescharacteristic
35C2×C22⋊Dic61D411yescharacteristic
36C2×C4×Dic61D411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38#14519704—42no—
39#14519704—42no—
40#14519692—42no—
41#14519692—42no—
42C22⋊#92418—21no—
43C22⋊#92418—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C2×#41×C61—21no—
54C2×#41×C61—21no—
55C2×#41×C61—21no—
56C2×#41×C61—21no—
57C2×#41×C61—42no—
58C2×#41×C61—42no—
59C22×Q8×C61Q811yescharacteristic, complemented
60C22×D4×C61—21no—
61C22×D4×C61Q811yescharacteristic, complemented
62C23×C4×C61—21no—
63C23×C4×C61C2311yescharacteristic
64C2×M16×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C42×C61D411yescharacteristic
67C2×C42×C61D411yescharacteristic
68#104×C61—42no—
69#104×C61—42no—
70C22⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72#95×C61—21no—
73C24×C61—21no—
74C24×C61C2×D411yescharacteristic
75#41×C61—22no—
76#41×C61—22no—
77#41×C61—22no—
78#41×C61—22no—
79#41×C61—41no—
80#41×C61—41no—
81#41×C61—41no—
82#41×C61—41no—
83#41×C61—42no—
84#41×C61—42no—
85#41×C61—42no—
86#41×C61—42no—
87C2×Q8×C61—21no—
88C2×Q8×C61—21no—
89C2×Q8×C61—42no—
90C2×D4×C61—22no—
91C2×D4×C61—22no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—42no—
95C2×D4×C61—41no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—41no—
103C22×C4×C61—41no—
104C22×C4×C61—42no—
105C22×C4×C61—42no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×Q811yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61—41no—
119C2×Dic122—42no—
120C2×Dic122—42no—
121C2×Dic122—81no—
122M16×C61—41no—
123M16×C61—41no—
124C2×C8×C61—41no—
125C22⋊C4×C61—42no—
126C42×C61—41no—
127C42×C61—41no—
128C22⋊Dic61—42no—
129C22⋊Dic61—41no—
130C22⋊Dic61—41no—
131C22⋊Dic61—41no—
132C22⋊Dic61—41no—
133C4⋊Dic61—42no—
134#11758196—42no—
135#11758196—81no—
136C4×Dic61—41no—
137C4×Dic61—41no—
138#11758194—41no—
139#11758194—41no—
140C2×#92418—41no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—22no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—81no—
149C23×C61—81no—
150C23×C61#12011yescharacteristic
151C23×C61C22⋊Q811yescharacteristic
152C23×C61#12411yescharacteristic
153Q8×C61—22no—
154Q8×C61—22no—
155Q8×C61—41no—
156Q8×C61—41no—
157D4×C61—22no—
158D4×C61—22no—
159D4×C61—41no—
160D4×C61—41no—
161D4×C61—42no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—42no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61C22⋊Q811yescharacteristic
187C2×C4×C61C22⋊Q811yescharacteristic
188C2×C4×C61#12411yescharacteristic
189C2×C4×C61#12411yescharacteristic
190C2×Dic61—22no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—81no—
203C2×Dic61—81no—
204C2×Dic61—81no—
205Dic122—84no—
206C8×C61—81no—
207#92418—81no—
208#10386—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—42no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—82no—
221C22×C61—82no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61#39311yescharacteristic
225C22×C61B(4,𝔽2)11yescharacteristic
226C22×C61B(4,𝔽2)11yescharacteristic
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—81no—
234Dic61—42no—
235Dic61—81no—
236Dic61—81no—
237Dic61—81no—
238Dic61—161no—
239#2607—611no—
240#1789—611no—
241#1789D6111yescharacteristic
242#1782—611no—
243#1306—611no—
244#1284—611no—
245#1284—611no—
246C2×C61—21no—
247C2×C61—21no—
248C2×C61—41no—
249C2×C61—41no—
250C2×C61—82no—
251C2×C61#202911yescharacteristic
252C2×C61#203111yescharacteristic
253C2×C61#203111yescharacteristic
254C22×#41C2×D6111yescharacteristic
255C2×#104—21no—
256C2×#104—21no—
257C2×#104—1221no—
258C2×#104—1221no—
259C2×#99—1221no—
260C2×#99—1221no—
261#407—611no—
262#407C2×D6111yescharacteristic
263#399—1221no—
264#393—1221no—
265#391—1221no—
266#391—1221no—
267#387—1221no—
268#387—1221no—
269#381—611no—
270#381—611no—
271C4×C22⋊C4—611no—
272C4×C22⋊C4C2×D6111yescharacteristic
273C61#1038611yescharacteristic, complemented, Sylow
274C2×#41—21no—
275C2×#41—21no—
276C2×#41—21no—
277C2×#41—21no—
278C2×#41—42no—
279C2×#41—42no—
280C22×Q8—1221no—
281C22×Q8Dic12211yescharacteristic, complemented
282C22×D4—21no—
283C22×D4Dic12211yescharacteristic, complemented
284C23×C4—21no—
285C23×C4C22×D6111yescharacteristic
286C2×M16—21no—
287C2×M16—1221no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4—1221no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—611no—
293C2×C22⋊C4—611no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4—1221no—
296C2×C42—611no—
297C2×C42—611no—
298C2×C42C22⋊D6111yescharacteristic
299C2×C42C22⋊D6111yescharacteristic
300#104—42no—
301#104—42no—
302#104—2442no—
303#104—2442no—
304#99—2442no—
305#99—2442no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—1221no—
309C22⋊C8—1221no—
310#95—21no—
311#95—1221no—
312#95—1221no—
313#95—1221no—
314#95—1221no—
315#95—1221no—
316#95—1221no—
317#95—1221no—
318#95—1221no—
319#95—1221no—
320C24—21no—
321C24D4×D6111yescharacteristic
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—42no—
331#41—42no—
332#41—42no—
333#41—42no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—42no—
337C2×Q8—4881no—
338C2×Q8—2442no—
339C2×Q8—2442no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—1221no—
362C22×C4—2441no—
363C22×C4C2×Dic12211yescharacteristic
364C22×C4#1175822211yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4#1175822411yescharacteristic
367C22×C4#1175822411yescharacteristic
368C22×C4C2×C22⋊D6111yescharacteristic
369M16—41no—
370M16—41no—
371M16—2441no—
372M16—2441no—
373C2×C8—41no—
374C2×C8—2441no—
375C4⋊C4—4881no—
376C4⋊C4—2442no—
377C4⋊C4—2442no—
378C22⋊C4—42no—
379C22⋊C4—2442no—
380C22⋊C4—2441no—
381C22⋊C4—2441no—
382C22⋊C4—2441no—
383C22⋊C4—2441no—
384C42—41no—
385C42—41no—
386C42—2441no—
387C42—2441no—
388C23—21no—
389C23—21no—
390C23—22no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—81no—
396C23—81no—
397C23#1451974511yescharacteristic
398C23#1451975411yescharacteristic
399C23#1451976311yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—4884no—
405D4—22no—
406D4—22no—
407D4—41no—
408D4—41no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—42no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—1222no—
435C2×C4—4881no—
436C2×C4—4881no—
437C2×C4—4881no—
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×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4#1451975411yescharacteristic
450C2×C4#1451975611yescharacteristic
451C2×C4#1451976311yescharacteristic
452C2×C4#1451976411yescharacteristic
453C8—81no—
454C8—4881no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—42no—
464C22—81no—
465C22—81no—
466C22—82no—
467C22—82no—
468C22—81no—
469C22—81no—
470C22#1683967411yescharacteristic
471C22#1683994811yescharacteristic
472C22#1683994811yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—9761no—
481C4—4881no—
482C4—4881no—
483C4—4881no—
484C4—2442no—
485C2—21no—
486C2—21no—
487C2—41no—
488C2—41no—
489C2—82no—
490C2#2144698311yescharacteristic
491C2#2144698911yescharacteristic
492C2#2144699111yescharacteristic
subgroupG/H
centerC22#16839674
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1789C2
socleC22×C61#393
prime radicalC22×#41×C61C22
hypercenter#1789D61
hypocenterC61#10386
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
1. #21451480C2semidirect
4. #21443728C2semidirect
5. #21443673C2semidirect
6. #21443672C2semidirect
59. C22×Q8×C61Q8semidirect
61. C22×D4×C61Q8semidirect
273. C61#10386semidirect
281. C22×Q8Dic122semidirect
283. C22×D4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214514802
#214457632
#214457372
#214437282
#214436732
#214436722
C61×#17892
#1038661