Finite Group Databasebeta

← #29578077

Subgroups — #29578077

Order 15616.

538 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 89 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#21455759C211yesmaximal, characteristic
2#21448581C211yesmaximal, characteristic
3#21445918C211yesmaximal, characteristic, complemented
4#21444347C211yesmaximal, characteristic
5#21443727C211yesmaximal, characteristic, complemented
6#21443671C211yesmaximal, characteristic, complemented
7C61×#1820C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840086—22no—
10#16840080C412yescomplemented
11#16840075C412yescomplemented
12#16839805—21no—
13#16839766—21no—
14#16839726C2211yescharacteristic
15#16839726C2211yescharacteristic
16#16839662—21no—
17#16839651—21no—
18#16839647—21no—
19#16839640—21no—
20#16839638—21no—
21C4×C22⋊Dic61C2211yescharacteristic
22#16839609—21no—
23#16839521—21no—
24#16839521—21no—
25#16839495—22no—
26C2×#117×C61C2211yescharacteristic
27C61×#381C2211yescharacteristic
28C61×#381C2211yescharacteristic
29C61×#342—21no—
30C61×#342—21no—
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C22×C4×D61—21no—
34C2×#11758194—21no—
35C2×#11758194—21no—
36#14519751—21no—
37#14519751—21no—
38#14519751C2×C412yes—
39#14519751D412yes—
40C2×C4⋊Dic61D411yescharacteristic
41C2×#11758196—21no—
42C2×C22⋊Dic61—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50#14519710—22no—
51#14519708—22no—
52#14519697—22no—
53#14519697—22no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57C22⋊#92418C2×C412yescomplemented
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61C23×C4×C61C2311yescharacteristic
62#117×C61—21no—
63#117×C61—21no—
64#117×C61—21no—
65#117×C61—21no—
66C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
67C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
68C2×C22⋊C4×C61—21no—
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61D411yescharacteristic
71C2×C22⋊C4×C61D411yescharacteristic
72C2×C42×C61—21no—
73#95×C61—21no—
74#95×C61—21no—
75#95×C61—21no—
76#95×C61—21no—
77C24×C61C2×D411yescharacteristic
78C23×D61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61C22×C411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C22×C4×C61#4111yescharacteristic
92C2×C22⋊D61—22no—
93C2×C22⋊D61—22no—
94C2×C22⋊D61—81no—
95C2×C22⋊D61—81no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C2×D244—21no—
100C2×D244—41no—
101C2×D244—41no—
102C2×D244C22⋊C412yes—
103C2×C4×D61—42no—
104C2×C4×D61—81no—
105C2×C4×D61—81no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C42×C61—41no—
117C42×C61—41no—
118C22⋊Dic61—42no—
119#11758198—22no—
120#11758198—22no—
121#11758198—42no—
122#11758198—81no—
123#11758198—81no—
124C4⋊Dic61—21no—
125C4⋊Dic61C22⋊C412yes—
126#11758196—81no—
127C4×Dic61—41no—
128C4×Dic61—41no—
129C4×Dic61—81no—
130#11758194—41no—
131#11758194—41no—
132#11758194—41no—
133#11758194—41no—
134C2×#92418—22no—
135C2×#92418—41no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61C4×D411yescharacteristic
141C23×C61#12011yescharacteristic
142C23×C61#12111yescharacteristic
143C22×D61—22no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—81no—
167C2×C4×C61—81no—
168C2×C4×C61—81no—
169C2×C4×C61—81no—
170C2×C4×C61—81no—
171C2×C4×C61C2×C22⋊C411yescharacteristic
172C2×C4×C61C4×D411yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12311yescharacteristic
175C22⋊D61—82no—
176C22⋊D61—82no—
177C2×Dic61—22no—
178C2×Dic61—22no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186D244—42no—
187D244—81no—
188C4×D61—84no—
189#92418—42no—
190#92418—81no—
191#9525—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—81no—
201C22×C61—81no—
202C22×C61#38611yescharacteristic
203C22×C61#45811yescharacteristic
204C22×C61#45811yescharacteristic
205C2×D61—42no—
206C2×D61—42no—
207C2×D61—42no—
208C2×D61—81no—
209C2×D61—81no—
210C2×D61—81no—
211C2×D61—81no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—81no—
217C4×C61—81no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61—161no—
221Dic61—42no—
222Dic61—42no—
223Dic61—81no—
224#3094—611no—
225#2333—611no—
226#1826—611no—
227#1820D6111yescharacteristic, complemented
228#1510—611no—
229#1306—611no—
230#1284—611no—
231C2×C61—21no—
232C2×C61—21no—
233C2×C61—41no—
234C2×C61—41no—
235C2×C61#189011yescharacteristic
236C2×C61#190411yescharacteristic
237C2×C61#190611yescharacteristic
238D61—82no—
239C2×#121—611no—
240C2×#117C2×D6111yescharacteristic
241#482—1222no—
242#481—612no—
243#480—612no—
244#428—1221no—
245#418—1221no—
246#407—611no—
247#407—611no—
248#390—1221no—
249#388—1221no—
250#387—1221no—
251#386—1221no—
252#386—1221no—
253#381C2×D6111yescharacteristic
254#381C2×D6111yescharacteristic
255C4×C22⋊C4—611no—
256C4×C22⋊C4—1221no—
257#342—21no—
258#342—21no—
259#339—1221no—
260#339—1221no—
261#327—1222no—
262C61#952511yescharacteristic, complemented, Sylow
263C22×D4—611no—
264C22×D4—1221no—
265C23×C4—1221no—
266C23×C4C22×D6111yescharacteristic
267C2×M16—1221no—
268C2×M16—1221no—
269#121—612no—
270#121—612no—
271#121—1221no—
272#121—1221no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277C2×C4⋊C4—611no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4C4×D6111yescharacteristic, complemented
280C2×C4⋊C4C4×D6111yescharacteristic, complemented
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4C22⋊D6111yescharacteristic
289C2×C22⋊C4C22⋊D6111yescharacteristic
290C2×C42—21no—
291C2×C42—611no—
292C2×C42—611no—
293C2×C42—1221no—
294#107—1222no—
295#106—1222no—
296#102—1222no—
297#102—1222no—
298C22⋊C8—612no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301C22⋊C8—1221no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—1221no—
307#95—1221no—
308#95—1221no—
309C24—1221no—
310C24C2×C22⋊D6111yescharacteristic
311C2×D4—612no—
312C2×D4—1222no—
313C2×D4—1222no—
314C2×D4—1221no—
315C2×D4—4881no—
316C2×D4—4881no—
317C2×D4—2441no—
318C2×D4—2441no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—4881no—
330C22×C4—4881no—
331C22×C4—2442no—
332C22×C4C2×C4×D6111yescharacteristic
333C22×C4#1175822211yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4D4×D6111yescharacteristic
336C22×C4D4×D6111yescharacteristic
337C22×C4#1175822411yescharacteristic
338M16—2441no—
339M16—2441no—
340M16—2441no—
341M16—2441no—
342C2×C8—1222no—
343C2×C8—2441no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—612no—
349C4⋊C4—1221no—
350C4⋊C4—4881no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—1222no—
358C22⋊C4—1222no—
359C22⋊C4—4881no—
360C22⋊C4—4881no—
361C22⋊C4—2442no—
362C22⋊C4—2442no—
363C42—41no—
364C42—41no—
365C42—4881no—
366C42—2441no—
367C42—2441no—
368C23—21no—
369C23—21no—
370C23—41no—
371C23—41no—
372C23—1222no—
373C23—2441no—
374C23—2441no—
375C23—2441no—
376C23—2441no—
377C23—2441no—
378C23—2441no—
379C23C4×C22⋊D6111yescharacteristic
380C23#1451974511yescharacteristic
381C23#1451974811yescharacteristic
382D4—4881no—
383D4—4882no—
384D4—4882no—
385D4—2442no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—1222no—
408C2×C4—1222no—
409C2×C4—4884no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4C22⋊C4×D6111yescharacteristic
418C2×C4#1451973711yescharacteristic
419C2×C4#1451975011yescharacteristic
420C2×C4#1451975911yescharacteristic
421C8—4881no—
422C8—2442no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—81no—
432C22—81no—
433C22—4881no—
434C22—4881no—
435C22—4881no—
436C22—4881no—
437C22—2442no—
438C22—2442no—
439C22—2442no—
440C22#1683963811yescharacteristic
441C22#1683995111yescharacteristic
442C22#1683995111yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—161no—
452C4—4881no—
453C4—2442no—
454C4—2442no—
455C2—21no—
456C2—21no—
457C2—41no—
458C2—41no—
459C2—4882no—
460C2#2144622911yescharacteristic
461C2#2144630911yescharacteristic
462C2#2144631711yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1820C2
socleC22×C61#386
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1820D61
hypocenterC61#9525
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
3. #21445918C2semidirect
5. #21443727C2semidirect
6. #21443671C2semidirect
7. C61×#1820C2semidirect
10. #16840080C4semidirect
11. #16840075C4semidirect
57. C22⋊#92418C2×C4semidirect
66. C2×C4⋊C4×C61C2×C4semidirect
67. C2×C4⋊C4×C61C2×C4semidirect
227. #1820D61semidirect
262. C61#9525semidirect
279. C2×C4⋊C4C4×D61semidirect
280. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214557592
#214485812
#214459182
#214443472
#214437272
#214436712
C61×#18202
#952561