Finite Group Databasebeta

← #29575851

Subgroups — #29575851

Order 15616.

486 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 95 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#21455919C211yesmaximal, characteristic
2#21454856C211yesmaximal, characteristic
3#21445647C211yesmaximal, characteristic
4#21444435C211yesmaximal, characteristic, complemented
5#21444317C211yesmaximal, characteristic, complemented
6#21443717C211yesmaximal, characteristic, complemented
7C61×#1285C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9C2×C4×D244C2211yescharacteristic
10#16840098—21no—
11#16840043C411yescharacteristic, complemented
12#16840043C411yescharacteristic, complemented
13#16839968C411yescharacteristic, complemented
14#16839968C411yescharacteristic, complemented
15#16839915—21no—
16#16839864—21no—
17#16839864—21no—
18#16839839—21no—
19#16839835—22no—
20#16839827—21no—
21#16839815C2211yescharacteristic
22C2×#14519697—21no—
23#16839724—21no—
24C22⋊C8×D61—21no—
25C2×#14519688C2211yescharacteristic
26#16839524—21no—
27#16839503—22no—
28#16839498—21no—
29#16839498—21no—
30C61×#432C2211yescharacteristic
31C61×#432C2211yescharacteristic
32C4×C4⋊C4×C61C2211yescharacteristic
33C22×D244D411yescharacteristic
34C22×C4×D61—21no—
35C2×#11758190—21no—
36C2×#11758190—21no—
37C2×#11758189—22no—
38C22×#92418—21no—
39C2×C8×D61—22no—
40#14519777C2×C411yescharacteristic
41#14519777C2×C411yescharacteristic
42#14519769—21no—
43#14519756—21no—
44#14519756—21no—
45C4×Dic122—21no—
46C4×Dic122—21no—
47C4×D244—21no—
48C4×D244—21no—
49C4×D244C2×C411yescharacteristic
50C4×D244C2×C411yescharacteristic
51C4×D244D411yescharacteristic
52C4×D244D411yescharacteristic
53#14519736—21no—
54C2×C4⋊Dic61D411yescharacteristic
55C2×#11758198—21no—
56#14519711—21no—
57#14519711—21no—
58C4⋊#92418—21no—
59#14519702—21no—
60#14519702—21no—
61#14519699—21no—
62#14519699—21no—
63#14519697—24no—
64C22⋊#92418D411yescharacteristic
65C22⋊#92418D411yescharacteristic
66#14519689—22no—
67#14519689—22no—
68#14519688—21no—
69#14519688—21no—
70C4×#92418—21no—
71C2×C4⋊C4×C61—21no—
72C2×C42×C61—21no—
73C2×C42×C61C2311yescharacteristic
74C4⋊C8×C61—21no—
75C4⋊C8×C61C2×C411yescharacteristic
76C4⋊C8×C61C2×C411yescharacteristic
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79#97×C61—21no—
80#97×C61D412yes—
81#95×C61—21no—
82C23×D61—21no—
83C22×C4×C61—41no—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C22×Dic61—21no—
88C2×D244—22no—
89C2×D244—21no—
90C2×D244—41no—
91C2×D244C22⋊C411yescharacteristic
92C2×D244C22⋊C411yescharacteristic
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—82no—
98C2×Dic122—21no—
99C2×Dic122—21no—
100C2×C8×C61—21no—
101C2×C8×C61—21no—
102C2×C8×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—41no—
105C42×C61—81no—
106C42×C61C22×C411yescharacteristic
107C42×C61C2×D411yescharacteristic
108C42×C61C2×D411yescharacteristic
109C42×C61#4111yescharacteristic
110C22⋊Dic61—41no—
111#11758198—21no—
112#11758198—21no—
113#11758198—21no—
114#11758198—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61C22⋊C411yescharacteristic
121C4⋊Dic61C22⋊C411yescharacteristic
122#11758196—41no—
123#11758196—41no—
124C4×Dic61—41no—
125C2×#92418—22no—
126C2×#92418—21no—
127C2×#92418—21no—
128C2×#92418—41no—
129C2×#92418—41no—
130#11758190—42no—
131#11758190—42no—
132#11758189—82no—
133C8×D61—82no—
134C23×C61#12111yescharacteristic
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—42no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—81no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—81no—
148C2×C4×C61—81no—
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C4×D411yescharacteristic
151C2×C4×C61C4×D411yescharacteristic
152C2×C4×C61#12011yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12311yescharacteristic
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—81no—
160D244—21no—
161D244—21no—
162D244—21no—
163D244—42no—
164D244#10412yes—
165C4×D61—42no—
166C4×D61—42no—
167C4×D61—81no—
168C4×D61—81no—
169Dic122—41no—
170Dic122—41no—
171Dic122—81no—
172C8×C61—42no—
173C8×C61—41no—
174C8×C61—41no—
175#92418—41no—
176#92418—41no—
177#92418—42no—
178#9165—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61#38611yescharacteristic
182C22×C61#47111yescharacteristic
183C22×C61#47311yescharacteristic
184C2×D61—22no—
185C2×D61—41no—
186C2×D61—41no—
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—41no—
190C2×D61—82no—
191C2×D61—82no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—82no—
198C4×C61C2×#10411yescharacteristic
199C4×C61#42111yescharacteristic
200Dic61—41no—
201Dic61—41no—
202Dic61—81no—
203Dic61—81no—
204#3110—611no—
205#2974—611no—
206#1762—611no—
207#1531—611no—
208#1503—611no—
209#1302—611no—
210#1285D6111yescharacteristic, complemented
211C2×C61—21no—
212C2×C61—21no—
213C2×C61#187111yescharacteristic
214C2×C61#190411yescharacteristic
215C2×C61#190811yescharacteristic
216D61—42no—
217D61—81no—
218#533—611no—
219C2×C4×D4—611no—
220#484—1221no—
221#474—611no—
222#474—611no—
223#461—611no—
224#461—611no—
225#450—1221no—
226#440—1221no—
227#440—1221no—
228#436—1221no—
229#435—1222no—
230C8×D4—1221no—
231#432C2×D6111yescharacteristic
232#432C2×D6111yescharacteristic
233#431—611no—
234C2×#102—1221no—
235#407—1221no—
236C2×C22⋊C8—1221no—
237C2×#97—611no—
238C4×C4⋊C4C2×D6111yescharacteristic
239#340—1221no—
240#330—1222no—
241#328—1221no—
242#328—1221no—
243C61#916511yescharacteristic, complemented, Sylow
244C22×D4—611no—
245C23×C4—1221no—
246C2×SD16—1221no—
247C2×SD16—1221no—
248C2×M16—1222no—
249C22×C8—1222no—
250C22×C8—1221no—
251C4⋊Q8—611no—
252C4⋊Q8—611no—
253#125—1221no—
254C22⋊Q8—1221no—
255C22⋊Q8—1221no—
256C4×Q8—1221no—
257C4×Q8—1221no—
258C4×D4—611no—
259C4×D4—611no—
260C4×D4—611no—
261C4×D4—611no—
262C4×D4—1221no—
263C4×D4—1221no—
264#117—1221no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—611no—
267C2×C22⋊C4—1221no—
268C2×C42—21no—
269C2×C42C22×D6111yescharacteristic
270#107—1221no—
271#107—1221no—
272C4⋊C8—21no—
273C4⋊C8—1221no—
274C4⋊C8C4×D6111yescharacteristic
275C4⋊C8C4×D6111yescharacteristic
276#103—1221no—
277#103—1221no—
278#102—1224no—
279#102—1221no—
280#102—1221no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—611no—
284C22⋊C8—611no—
285C22⋊C8—1222no—
286C22⋊C8—1222no—
287#97—21no—
288#97—1221no—
289#97—1221no—
290#97D24412yes—
291C4×C8—1221no—
292#95—21no—
293C24—1221no—
294C2×Q8—1221no—
295C2×Q8—1221no—
296C2×D4—611no—
297C2×D4—611no—
298C2×D4—1222no—
299C2×D4—1221no—
300C2×D4—2441no—
301C22×C4—41no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—4882no—
308C22×C4#1175822211yescharacteristic
309C22×C4D4×D6111yescharacteristic
310C22×C4D4×D6111yescharacteristic
311SD16—2442no—
312SD16—2442no—
313M16—4882no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—1222no—
318C2×C8—1221no—
319C2×C8—1221no—
320C2×C8—4882no—
321C2×C8—2441no—
322C2×C8—2441no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—611no—
326C4⋊C4—611no—
327C4⋊C4—1221no—
328C4⋊C4—1221no—
329C4⋊C4—1221no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C4⋊C4—2441no—
333C4⋊C4—2441no—
334C22⋊C4—1221no—
335C22⋊C4—1221no—
336C22⋊C4—1221no—
337C22⋊C4—1221no—
338C22⋊C4—2441no—
339C42—81no—
340C42—2441no—
341C42C2×C4×D6111yescharacteristic
342C42C2×D24411yescharacteristic
343C42D4×D6111yescharacteristic
344C42#1175822611yescharacteristic
345C23—1221no—
346C23—1221no—
347C23—4881no—
348C23—2442no—
349C23—2441no—
350C23—2441no—
351C23#1451975011yescharacteristic
352Q8—4881no—
353Q8—2441no—
354Q8—2441no—
355D4—612no—
356D4—1221no—
357D4—1221no—
358D4—1221no—
359D4—2442no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—1221no—
369C2×C4—1221no—
370C2×C4—4881no—
371C2×C4—4881no—
372C2×C4—4881no—
373C2×C4—2442no—
374C2×C4—2442no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4C22⋊C4×D6111yescharacteristic
378C2×C4#1451973911yescharacteristic
379C2×C4C4×D24411yescharacteristic
380C2×C4#1451974711yescharacteristic
381C2×C4#1451975211yescharacteristic
382C2×C4#1451976211yescharacteristic
383C8—42no—
384C8—41no—
385C8—41no—
386C8—2441no—
387C8—2442no—
388C8—2441no—
389C22—21no—
390C22—21no—
391C22—1222no—
392C22—4882no—
393C22—4882no—
394C22—2441no—
395C22—2441no—
396C22—2441no—
397C22—2441no—
398C22—2441no—
399C22#1683964011yescharacteristic
400C22#1684002411yescharacteristic
401C22#1684003611yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—82no—
408C4—4881no—
409C4—4881no—
410C4—2441no—
411C4—2441no—
412C4#104×D6111yescharacteristic
413C4#1683977711yescharacteristic
414C2—21no—
415C2—21no—
416C2—4881no—
417C2—2442no—
418C2#2144612011yescharacteristic
419C2#2144630711yescharacteristic
420C2#2144633511yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1285C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1285D61
hypocenterC61#9165
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
4. #21444435C2semidirect
5. #21444317C2semidirect
6. #21443717C2semidirect
7. C61×#1285C2semidirect
11. #16840043C4semidirect
12. #16840043C4semidirect
13. #16839968C4semidirect
14. #16839968C4semidirect
210. #1285D61semidirect
243. C61#9165semidirect

Maximal subgroups

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

subgroupindex
#214559192
#214548562
#214456472
#214444352
#214443172
#214437172
C61×#12852
#916561