Finite Group Databasebeta

← #30606513

Subgroups — #30606513

Order 15616.

636 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#16840663C212yesmaximal
2#21461569C211yesmaximal, characteristic, complemented
3#21461566C211yesmaximal, characteristic, complemented
4#21460903C212yesmaximal, complemented
5#21448476C211yesmaximal, characteristic
6C2×#14519837C2212yes—
7C2×#14519835C2212yes—
8C4×#11758217—21no—
9C4×C22⋊#6923—21no—
10#16840663C418yescomplemented
11#16840642—21no—
12#16840640—21no—
13C2×#14519747C2211yescharacteristic
14C2×#14519737C2211yescharacteristic
15C22×#11758198C411yescharacteristic, complemented
16#16839663—21no—
17#16839652—21no—
18#16839651—21no—
19#16839641C2211yescharacteristic
20#16839640—21no—
21#16839617C411yescharacteristic, complemented
22C22⋊C4×Dic61C411yescharacteristic, complemented
23C61×#385C411yescharacteristic, complemented
24C2×#11758217—22no—
25C2×C22⋊#6923—22no—
26C2×C4×#6923—41no—
27#14519837—22no—
28#14519837C2×C414yes—
29#14519835—22no—
30#14519835C2×C414yes—
31#14519832—41no—
32C4×#92420—41no—
33(C2×#6923)⋊C4—41no—
34C24×D61C2×C411yescharacteristic
35C22×D244—21no—
36C22×C22⋊D61C2311yescharacteristic
37C23×Dic61C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39#14519747—22no—
40#14519747C2×C414yes—
41#14519737—41no—
42#14519737—41no—
43C2×#11758196D411yescharacteristic
44C2×C22⋊Dic61C2×C411yescharacteristic
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51C2×#11758198—41no—
52C2×#11758198—41no—
53C2×#11758198D411yescharacteristic
54C2×C4×Dic61—21no—
55C2×C4×Dic61D411yescharacteristic
56#14519684—21no—
57#14519684—21no—
58C23×C4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61—21no—
60C2×C22⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61C2×C411yescharacteristic
62#95×C61—21no—
63C24×C61C22×C411yescharacteristic
64C23×D61—21no—
65C23×D61—21no—
66C23×D61—41no—
67C23×D61—41no—
68C23×D61—81no—
69C23×D61—81no—
70C23×D61C4212yescomplemented
71C23×D61C2×D411yescharacteristic
72C22×#6923—42no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—41no—
76C22×C4×C61—81no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C2×C22⋊D61—41no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61C22×C412yes—
82C2×C22⋊D61#4112yes—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—81no—
87C22×Dic61C22⋊C411yescharacteristic
88C22×Dic61C22⋊C411yescharacteristic
89C22×Dic61C2×D411yescharacteristic
90C2×D244—24no—
91C2×D244—81no—
92C2×D244—81no—
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—81no—
96C22⋊#6923—24no—
97C22⋊#6923—42no—
98#11758217—24no—
99#11758217—42no—
100C2×#92420—42no—
101C4×#6923—82no—
102C4⋊C4×C61—81no—
103C22⋊C4×C61—21no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61C4212yescomplemented
106C22⋊Dic61—41no—
107#11758198—24no—
108#11758198—41no—
109#11758198—42no—
110#11758198—41no—
111#11758198—41no—
112#11758198—81no—
113#11758198—81no—
114#11758198—81no—
115#11758196—41no—
116C4×Dic61—21no—
117C4×Dic61—21no—
118C4×Dic61—81no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61C2×C4211yescharacteristic
124C23×C61C2×C22⋊C411yescharacteristic
125C23×C61#11711yescharacteristic
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—22no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—42no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61—42no—
139C22×D61—82no—
140C22×D61—82no—
141C22×D61—84no—
142C22×D61—82no—
143C22×D61—82no—
144C22×D61C4×D412yes—
145C2×#6923—44no—
146C2×#6923—82no—
147C2×#6923—82no—
148C2×C4×C61—22no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—82no—
159C22⋊D61—42no—
160C22⋊D61—42no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171C2×Dic61#9911yescharacteristic
172C2×Dic61#9911yescharacteristic
173C2×Dic61C4×D412yes—
174D244—84no—
175C4×D61—42no—
176C4×D61—81no—
177#92420—44no—
178#12622—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—42no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61C4×C22⋊C411yescharacteristic
189C22×C61C2×#9911yescharacteristic
190C22×C61#41011yescharacteristic
191C2×D61—22no—
192C2×D61—41no—
193C2×D61—41no—
194C2×D61—42no—
195C2×D61—42no—
196C2×D61—42no—
197C2×D61—42no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—84no—
202C2×D61—81no—
203C2×D61—81no—
204C2×D61—84no—
205#6923—84no—
206C4×C61—42no—
207C4×C61—41no—
208C4×C61—41no—
209C4×C61—81no—
210Dic61—21no—
211Dic61—21no—
212Dic61—22no—
213Dic61—41no—
214Dic61—81no—
215#2321—611no—
216C2×AΣL(1,𝔽16)—612no—
217C4×#100—611no—
218C4×#99—611no—
219#1390—612no—
220C2×C61—21no—
221C2×C61—21no—
222C2×C61—41no—
223C2×C61—41no—
224C2×C61C4×#9911yescharacteristic
225C2×C61#212111yescharacteristic
226C2×C61#212211yescharacteristic
227D61—42no—
228D61—82no—
229C2×#120—611no—
230C2×C4×D4—611no—
231C22×C22⋊C4—611no—
232C2×#100—612no—
233C2×#99—612no—
234C4×M16—1221no—
235#390—1221no—
236#388—1221no—
237#388—1221no—
238#386—611no—
239#386—1221no—
240#385#692311yescharacteristic, complemented
241#379—611no—
242C4×C22⋊C4—611no—
243C4×C22⋊C4—1221no—
244AΣL(1,𝔽16)—618no—
245#343—1221no—
246#342—1221no—
247C61#1262211yescharacteristic, complemented, Sylow
248C25—611no—
249C22×D4—611no—
250C22×D4—1221no—
251C23×C4—611no—
252C23×C4—611no—
253C23×C4C2×#692311yescharacteristic
254C2×M16—1222no—
255#120—614no—
256#120—1222no—
257C4×D4—2441no—
258C4×D4—2441no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—611no—
261C2×C22⋊C4—611no—
262C2×C22⋊C4—611no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1222no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—2441no—
271C2×C22⋊C4—2441no—
272C2×C22⋊C4C2×#692311yescharacteristic
273C2×C22⋊C4C2×#692311yescharacteristic
274C2×C42—611no—
275C2×C42—1221no—
276C2×C42—2441no—
277#100—614no—
278#100—1222no—
279#99—614no—
280#99—1222no—
281#97—2441no—
282C4×C8—2441no—
283#95—21no—
284#95—1221no—
285#95—1221no—
286#95—2441no—
287C24—611no—
288C24—612no—
289C24—1221no—
290C24—1221no—
291C24—4881no—
292C24—4881no—
293C24—2441no—
294C24—2441no—
295C24C22×#692311yescharacteristic
296C2×D4—612no—
297C2×D4—612no—
298C2×D4—1224no—
299C2×D4—4881no—
300C2×D4—4881no—
301C2×D4—2441no—
302C2×D4—2441no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—81no—
307C22×C4—611no—
308C22×C4—611no—
309C22×C4—611no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—4881no—
316C22×C4—4881no—
317C22×C4—2442no—
318C22×C4C22⋊#692311yescharacteristic
319C22×C4C22⋊#692311yescharacteristic
320M16—1224no—
321M16—2442no—
322C2×C8—2442no—
323C4⋊C4—81no—
324C4⋊C4—2441no—
325C22⋊C4—21no—
326C22⋊C4—41no—
327C22⋊C4—1224no—
328C22⋊C4—1224no—
329C22⋊C4—4881no—
330C22⋊C4—4881no—
331C22⋊C4—4881no—
332C22⋊C4—2441no—
333C22⋊C4—2441no—
334C22⋊C4—2441no—
335C22⋊C4—2442no—
336C22⋊C4—2442no—
337C22⋊C4—2441no—
338C22⋊C4C4×#692312yescomplemented
339C42—1221no—
340C42—1221no—
341C42—4881no—
342C42—4882no—
343C23—21no—
344C23—21no—
345C23—41no—
346C23—41no—
347C23—612no—
348C23—1221no—
349C23—1221no—
350C23—1222no—
351C23—4882no—
352C23—4882no—
353C23—4884no—
354C23—4882no—
355C23—4882no—
356C23—2442no—
357C23—2441no—
358C23—2441no—
359C23—2441no—
360C23—2441no—
361C23—2442no—
362C23—2441no—
363C23—2441no—
364C23—2441no—
365C23—2441no—
366C23C2×C4×#692311yescharacteristic
367C23C2×C22⋊#692311yescharacteristic
368C23#1451985611yescharacteristic
369D4—4884no—
370D4—2442no—
371D4—2442no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—82no—
383C2×C4—611no—
384C2×C4—611no—
385C2×C4—612no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—4881no—
390C2×C4—4881no—
391C2×C4—4881no—
392C2×C4—4882no—
393C2×C4—4882no—
394C2×C4—2442no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2444no—
399C2×C4—2441no—
400C2×C4—2441no—
401C8—2444no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—42no—
409C22—41no—
410C22—41no—
411C22—1222no—
412C22—4881no—
413C22—4881no—
414C22—4881no—
415C22—4884no—
416C22—4884no—
417C22—4881no—
418C22—4881no—
419C22—2442no—
420C22—2442no—
421C22—2442no—
422C22—2442no—
423C22—2441no—
424C22—2441no—
425C22C4×C22⋊#692311yescharacteristic
426C22D61⋊#9911yescharacteristic
427C22#1684074311yescharacteristic
428C4—42no—
429C4—41no—
430C4—41no—
431C4—81no—
432C4—1222no—
433C4—1221no—
434C4—1221no—
435C4—4881no—
436C4—4884no—
437C4—2441no—
438C2—21no—
439C2—21no—
440C2—41no—
441C2—41no—
442C2—4882no—
443C2—2442no—
444C2#2146156411yescharacteristic
445C2#2146207711yescharacteristic
446C2#2146207911yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC61×#385C4
socleC22×C61C2×#99
prime radicalC2×#14519747C22
cosocleC22×C22⋊D61C23
hypercenter#385#6923
hypocenterC61#12622
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #21461569C2semidirect
3. #21461566C2semidirect
4. #21460903C2semidirect
10. #16840663C4semidirect
15. C22×#11758198C4semidirect
21. #16839617C4semidirect
22. C22⋊C4×Dic61C4semidirect
23. C61×#385C4semidirect
70. C23×D61C42semidirect
105. C22⋊C4×C61C42semidirect
240. #385#6923semidirect
247. C61#12622semidirect
338. C22⋊C4C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406632
#214615692
#214615662
#214609032
#214484762
#1262261