Finite Group Databasebeta

← #29576970

Subgroups — #29576970

Order 15616.

464 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 85 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#21456635C211yesmaximal, characteristic
2#21454879C211yesmaximal, characteristic, complemented
3#21444373C211yesmaximal, characteristic, complemented
4#21444347C211yesmaximal, characteristic, complemented
5#21444180C211yesmaximal, characteristic, complemented
6#21443695C211yesmaximal, characteristic, complemented
7C61×#1306C211yesmaximal, characteristic, complemented
8#16840273C2211yescharacteristic
9C2×#14519751C2211yescharacteristic
10#16840093C411yescharacteristic, complemented
11#16840087—21no—
12#16840076C411yescharacteristic, complemented
13#16840015—21no—
14#16840008—21no—
15#16839915C411yescharacteristic, complemented
16#16839909—21no—
17#16839898C411yescharacteristic, complemented
18#16839838—21no—
19#16839837—21no—
20#16839836—21no—
21#16839831—21no—
22#16839820—21no—
23#16839727—21no—
24#16839726C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26#16839498—21no—
27#16839495—21no—
28#16839495—21no—
29#16839495—21no—
30#16839485—21no—
31#16839485—21no—
32C61×#407C2211yescharacteristic
33C61×#407C2211yescharacteristic
34C61×#381C2211yescharacteristic
35C2×#11758222D411yescharacteristic
36C22×D244C2×C411yescharacteristic
37C22×C22⋊D61—21no—
38C2×#11758190—21no—
39C2×D488—21no—
40C2×#11758194—21no—
41C2×#11758189—21no—
42C22×#92418—21no—
43C2×C8×D61—21no—
44#14519760—21no—
45#14519756C2×C411yescharacteristic
46#14519756D411yescharacteristic
47#14519751—21no—
48#14519751—21no—
49#14519751—21no—
50#14519751—21no—
51#14519751C2×C411yescharacteristic
52#14519751D411yescharacteristic
53#14519746—21no—
54C4×C22⋊D61—21no—
55C4×C22⋊D61—21no—
56#14519736D411yescharacteristic
57C2×C4⋊Dic61C2×C411yescharacteristic
58C2×#11758198—21no—
59#14519711—21no—
60#14519709—21no—
61#14519706—21no—
62#14519706—21no—
63#14519702—21no—
64#14519699—21no—
65#14519699—21no—
66#14519699—21no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C22⋊#92418—21no—
70C22⋊#92418D411yescharacteristic
71C22⋊#92418D411yescharacteristic
72#14519689—21no—
73#14519689—21no—
74#14519687—21no—
75C8×Dic61—21no—
76C23×C4×C61C2311yescharacteristic
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×C22⋊C4×C61—21no—
80C22⋊C8×C61—21no—
81C22⋊C8×C61—21no—
82C22⋊C8×C61C2×C411yescharacteristic
83C22⋊C8×C61C2×C411yescharacteristic
84C22⋊C8×C61D411yescharacteristic
85C22⋊C8×C61D411yescharacteristic
86#95×C61—21no—
87#95×C61—21no—
88C24×C61#4111yescharacteristic
89C23×D61—21no—
90C22×C4×C61—41no—
91C22×C4×C61—81no—
92C22×C4×C61C22×C411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C2×C22⋊D61—21no—
99C2×C22⋊D61—21no—
100C2×C22⋊D61—21no—
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—82no—
104C22×Dic61—21no—
105#11758222—81no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—81no—
109C2×D244C22⋊C411yescharacteristic
110C2×C4×D61—21no—
111C2×Dic122C22⋊C411yescharacteristic
112M16×C61—81no—
113M16×C61—81no—
114C2×C8×C61—21no—
115C2×C8×C61—21no—
116C2×C8×C61—21no—
117C2×C8×C61—21no—
118C22⋊C4×C61—81no—
119C22⋊Dic61—21no—
120C22⋊Dic61—21no—
121C22⋊Dic61—21no—
122#11758198—21no—
123#11758198—21no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61C22⋊C411yescharacteristic
129C4⋊Dic61C22⋊C411yescharacteristic
130#11758196—21no—
131#11758196—21no—
132C4×Dic61—21no—
133#11758194—81no—
134C2×#92418—21no—
135C2×#92418—21no—
136C2×#92418—41no—
137C2×#92418—81no—
138D488—81no—
139#11758190—81no—
140#11758189—81no—
141C8×D61—81no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61C4×D411yescharacteristic
146C23×C61#12111yescharacteristic
147C23×C61#12311yescharacteristic
148C22×D61—21no—
149C22×D61—41no—
150C22×D61—42no—
151C22×D61—81no—
152C22×D61—81no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C2×C4×C61—81no—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C22⋊D61—81no—
167C22⋊D61—81no—
168C22⋊D61—81no—
169C22⋊D61—82no—
170C22⋊D61—82no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—41no—
175C2×Dic61—81no—
176D244—41no—
177D244—41no—
178D244—41no—
179D244—81no—
180C4×D61—41no—
181C4×D61—41no—
182Dic122—41no—
183C8×C61—41no—
184C8×C61—41no—
185C8×C61—41no—
186C8×C61—41no—
187#92418—41no—
188#92418—41no—
189#92418—81no—
190#9332—611nomaximal, Sylow
191C22×C61—21no—
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—82no—
201C22×C61#38611yescharacteristic
202C2×D61—41no—
203C2×D61—41no—
204C2×D61—41no—
205C2×D61—81no—
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—162no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—161no—
214Dic61—41no—
215Dic61—41no—
216Dic61—41no—
217Dic61—81no—
218#3194—611no—
219#2978—611no—
220#1516—611no—
221#1510—611no—
222#1471—611no—
223#1306D6111yescharacteristic, complemented
224#1293—611no—
225C2×C61—41no—
226C2×C61—41no—
227C2×C61—41no—
228C2×C61#188111yescharacteristic
229C2×C61#190311yescharacteristic
230C2×C61#191311yescharacteristic
231D61—81no—
232D61—161no—
233#525—611no—
234C2×#121—611no—
235#483—611no—
236#482—1221no—
237#480—611no—
238#469—1221no—
239#468—1221no—
240#450—611no—
241#449—1221no—
242#447—611no—
243#436—1221no—
244#436—1221no—
245#435—1221no—
246#435—1221no—
247#433—1221no—
248#408—1221no—
249#407—611no—
250#407C2×D6111yescharacteristic
251#407C2×D6111yescharacteristic
252C2×C22⋊C8—611no—
253#381C2×D6111yescharacteristic
254#328—1221no—
255#327—1221no—
256#327—1221no—
257#327—1221no—
258#323—1221no—
259#323—1221no—
260C61#933211yescharacteristic, complemented, Sylow
261C2×#41—611no—
262C22×D4—611no—
263C22×D4—1221no—
264C23×C4C22×D6111yescharacteristic
265C2×SD16—1221no—
266C2×D8—1221no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16—1221no—
270C2×M16—1221no—
271C22×C8—1221no—
272C22×C8—1221no—
273#123—1221no—
274C22⋊Q8—611no—
275C22⋊Q8—611no—
276#121—611no—
277#121—611no—
278#121—1221no—
279#121—1221no—
280#121—1221no—
281#121—1221no—
282#120—1221no—
283C4×D4—1221no—
284C4×D4—1221no—
285#117—611no—
286C2×C4⋊C4—611no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—1221no—
289#107—1221no—
290#106—1221no—
291C4⋊C8—1221no—
292C4⋊C8—1221no—
293#103—1221no—
294#102—1221no—
295#102—1221no—
296#102—1221no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—611no—
300C22⋊C8—611no—
301C22⋊C8—1221no—
302C22⋊C8—1221no—
303C22⋊C8—1221no—
304C22⋊C8—1221no—
305C22⋊C8—1221no—
306C22⋊C8C4×D6111yescharacteristic
307C22⋊C8C4×D6111yescharacteristic
308C22⋊C8D24411yescharacteristic
309C22⋊C8D24411yescharacteristic
310#97—1221no—
311C4×C8—1221no—
312#95—21no—
313#95—21no—
314C24—1221no—
315C24#1175822211yescharacteristic
316#41—4881no—
317C2×Q8—611no—
318C2×D4—611no—
319C2×D4—1221no—
320C2×D4—1221no—
321C2×D4—1221no—
322C2×D4—1221no—
323C2×D4—1221no—
324C2×D4—4881no—
325C2×D4—4882no—
326C2×D4—2441no—
327C2×D4—2441no—
328C22×C4—41no—
329C22×C4—81no—
330C22×C4—1221no—
331C22×C4—1221no—
332C22×C4C2×C4×D6111yescharacteristic
333C22×C4C2×D24411yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4D4×D6111yescharacteristic
336C22×C4D4×D6111yescharacteristic
337C22×C4#1175822611yescharacteristic
338SD16—4881no—
339D8—4881no—
340M16—81no—
341M16—81no—
342M16—4881no—
343M16—4881no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—1221no—
349C2×C8—1221no—
350C2×C8—4881no—
351C2×C8—4881no—
352C2×C8—2441no—
353C4⋊C4—611no—
354C4⋊C4—611no—
355C4⋊C4—1221no—
356C4⋊C4—1221no—
357C4⋊C4—1221no—
358C4⋊C4—1221no—
359C22⋊C4—81no—
360C22⋊C4—1221no—
361C22⋊C4—1221no—
362C22⋊C4—1221no—
363C22⋊C4—1221no—
364C22⋊C4—1221no—
365C22⋊C4—2441no—
366C22⋊C4—2441no—
367C42—1221no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—1221no—
372C23—4881no—
373C23—4881no—
374C23—2442no—
375C23—2441no—
376C23C4×D24411yescharacteristic
377C23#1451975011yescharacteristic
378C23#1451976211yescharacteristic
379Q8—2441no—
380D4—4881no—
381D4—4882no—
382D4—4882no—
383D4—4881no—
384D4—4881no—
385D4—4881no—
386D4—2441no—
387D4—2441no—
388D4—2441no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—1221no—
401C2×C4—4881no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4C22⋊C4×D6111yescharacteristic
406C2×C4#1451973911yescharacteristic
407C2×C4#1451974711yescharacteristic
408C2×C4#1451975211yescharacteristic
409C8—41no—
410C8—41no—
411C8—41no—
412C8—41no—
413C8—4881no—
414C8—2441no—
415C8—2441no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—82no—
426C22—9762no—
427C22—4881no—
428C22—4881no—
429C22—4881no—
430C22—2441no—
431C22—2441no—
432C22—2441no—
433C22#1683964011yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—161no—
439C4—4881no—
440C4—2441no—
441C4—2441no—
442C4—2441no—
443C2—41no—
444C2—41no—
445C2—41no—
446C2—9761no—
447C2—4881no—
448C2#2144618011yescharacteristic
449C2#2144629811yescharacteristic
450C2#2144636011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1306C2
socleC22×C61#386
prime radical#21456635C2
cosocleC23×C4×C61C23
hypercenter#1306D61
hypocenterC61#9332
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
2. #21454879C2semidirect
3. #21444373C2semidirect
4. #21444347C2semidirect
5. #21444180C2semidirect
6. #21443695C2semidirect
7. C61×#1306C2semidirect
10. #16840093C4semidirect
12. #16840076C4semidirect
15. #16839915C4semidirect
17. #16839898C4semidirect
223. #1306D61semidirect
260. C61#9332semidirect

Maximal subgroups

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

subgroupindex
#214566352
#214548792
#214443732
#214443472
#214441802
#214436952
C61×#13062
#933261