Finite Group Databasebeta

← #29575844

Subgroups — #29575844

Order 15616.

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

normal subgroups only

This group has 458 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#21455942C211yesmaximal, characteristic
2#21454909C211yesmaximal, characteristic, complemented
3#21445652C211yesmaximal, characteristic, complemented
4#21444431C211yesmaximal, characteristic
5#21444313C211yesmaximal, characteristic
6#21443717C211yesmaximal, characteristic, complemented
7C61×#1285C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C2×C4×Dic122C2211yescharacteristic
10#16840076—21no—
11#16840058C411yescharacteristic, complemented
12#16840058C411yescharacteristic, complemented
13#16839963C411yescharacteristic, complemented
14#16839963C411yescharacteristic, complemented
15#16839903—21no—
16#16839890—22no—
17#16839889—21no—
18#16839886—21no—
19#16839864—21no—
20#16839864—21no—
21#16839816—21no—
22#16839815C2211yescharacteristic
23#16839812—21no—
24C2×#14519700—21no—
25C2×#14519688C2211yescharacteristic
26#16839524—21no—
27#16839501—22no—
28#16839495—21no—
29#16839495—21no—
30C61×#432C2211yescharacteristic
31C61×#432C2211yescharacteristic
32C4×C4⋊C4×C61C2211yescharacteristic
33C2×#11758222—21no—
34C22×Dic122D411yescharacteristic
35C2×#11758190—21no—
36C2×#11758190—21no—
37C22×#92418—21no—
38#14519777C2×C411yescharacteristic
39#14519774C2×C411yescharacteristic
40#14519765—21no—
41#14519751—21no—
42#14519751—21no—
43C4×Dic122—21no—
44C4×Dic122—21no—
45C4×Dic122C2×C411yescharacteristic
46C4×Dic122C2×C411yescharacteristic
47C4×Dic122D411yescharacteristic
48C4×Dic122D411yescharacteristic
49C4×D244—21no—
50C4×D244—21no—
51C2×C4⋊Dic61D411yescharacteristic
52C2×#11758196—21no—
53C2×C4×Dic61—21no—
54#14519711—21no—
55#14519711—21no—
56C4⋊#92418—21no—
57#14519706—22no—
58#14519706—22no—
59#14519702—21no—
60#14519702—21no—
61#14519700—24no—
62#14519699—21no—
63#14519699—21no—
64C22⋊#92418D411yescharacteristic
65C22⋊#92418D411yescharacteristic
66#14519688—21no—
67#14519688—21no—
68#14519687—22no—
69C4×#92418—21no—
70C8×Dic61—22no—
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—
82C22×C4×C61—41no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61#4111yescharacteristic
86C2×C22⋊D61—41no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89#11758222—81no—
90#11758222—81no—
91C2×D244—21no—
92C2×D244—21no—
93C2×D244—21no—
94C2×C4×D61—41no—
95C2×Dic122—22no—
96C2×Dic122—21no—
97C2×Dic122—21no—
98C2×Dic122—41no—
99C2×Dic122C22⋊C411yescharacteristic
100C2×Dic122C22⋊C411yescharacteristic
101C2×C8×C61—21no—
102C2×C8×C61—21no—
103C2×C8×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—41no—
106C42×C61—81no—
107C42×C61C22×C411yescharacteristic
108C42×C61C2×D411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61#4111yescharacteristic
111#11758198—41no—
112#11758198—41no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61C22⋊C411yescharacteristic
117C4⋊Dic61C22⋊C411yescharacteristic
118#11758196—21no—
119#11758196—21no—
120#11758196—21no—
121#11758196—21no—
122C4×Dic61—21no—
123C4×Dic61—21no—
124C4×Dic61—21no—
125C4×Dic61—21no—
126C2×#92418—22no—
127C2×#92418—21no—
128C2×#92418—21no—
129C2×#92418—41no—
130C2×#92418—41no—
131#11758190—42no—
132#11758190—42no—
133C23×C61#12111yescharacteristic
134C22×D61—41no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—81no—
142C2×C4×C61—81no—
143C2×C4×C61C2×C22⋊C411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C2×C4×C61C4×D411yescharacteristic
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12311yescharacteristic
149C22⋊D61—81no—
150C22⋊D61—81no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—42no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159D244—41no—
160D244—41no—
161D244—81no—
162D244—81no—
163C4×D61—82no—
164Dic122—21no—
165Dic122—21no—
166Dic122—21no—
167Dic122—42no—
168Dic122—81no—
169Dic122#10412yes—
170C8×C61—42no—
171C8×C61—41no—
172C8×C61—41no—
173#92418—41no—
174#92418—41no—
175#92418—42no—
176#9164—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61#38611yescharacteristic
180C22×C61#47111yescharacteristic
181C22×C61#47211yescharacteristic
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—82no—
191C4×C61C2×#10411yescharacteristic
192C4×C61#42111yescharacteristic
193Dic61—22no—
194Dic61—41no—
195Dic61—41no—
196Dic61—41no—
197Dic61—81no—
198#3112—611no—
199#2984—611no—
200#1763—611no—
201#1530—611no—
202#1502—611no—
203#1302—611no—
204#1285D6111yescharacteristic, complemented
205C2×C61—21no—
206C2×C61—21no—
207C2×C61#187111yescharacteristic
208C2×C61#190511yescharacteristic
209C2×C61#190711yescharacteristic
210D61—161no—
211#532—611no—
212C2×C4×Q8—611no—
213#480—1221no—
214#477—611no—
215#477—611no—
216#460—611no—
217#460—611no—
218#448—1221no—
219#446—1221no—
220#446—1222no—
221C8×Q8—1221no—
222#440—1221no—
223#440—1221no—
224#432—1221no—
225#432C2×D6111yescharacteristic
226#432C2×D6111yescharacteristic
227#431—611no—
228#431—1221no—
229C2×#103—1221no—
230C2×#97—611no—
231C4×C4⋊C4C2×D6111yescharacteristic
232#340—1221no—
233#329—1222no—
234#327—1221no—
235#327—1221no—
236C61#916411yescharacteristic, complemented, Sylow
237C2×#41—1221no—
238C22×Q8—611no—
239C2×SD16—1221no—
240C2×SD16—1221no—
241C22×C8—1221no—
242C4⋊Q8—611no—
243#127—611no—
244#124—1221no—
245#121—1221no—
246#121—1221no—
247C4×Q8—611no—
248C4×Q8—611no—
249C4×Q8—611no—
250C4×Q8—611no—
251C4×Q8—1221no—
252C4×Q8—1221no—
253C4×D4—1221no—
254C4×D4—1221no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1221no—
258C2×C42—21no—
259C2×C42—1221no—
260C2×C42C22×D6111yescharacteristic
261#107—1221no—
262#107—1221no—
263C4⋊C8—21no—
264C4⋊C8—1222no—
265C4⋊C8—1222no—
266C4⋊C8—1221no—
267C4⋊C8C4×D6111yescharacteristic
268C4⋊C8C4×D6111yescharacteristic
269#103—1224no—
270#103—1221no—
271#103—1221no—
272#102—1221no—
273#102—1221no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—611no—
277C22⋊C8—611no—
278#97—21no—
279#97—1222no—
280#97—1221no—
281#97—1221no—
282#97D24412yes—
283C4×C8—1222no—
284C4×C8—1221no—
285#95—21no—
286#41—4881no—
287#41—4881no—
288C2×Q8—611no—
289C2×Q8—611no—
290C2×Q8—1222no—
291C2×Q8—1221no—
292C2×Q8—1221no—
293C2×Q8—2441no—
294C2×D4—1221no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—2441no—
298C22×C4—41no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—2441no—
302C22×C4#1175822211yescharacteristic
303C22×C4D4×D6111yescharacteristic
304C22×C4D4×D6111yescharacteristic
305SD16—2442no—
306SD16—2442no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—1222no—
311C2×C8—1221no—
312C2×C8—1221no—
313C2×C8—2441no—
314C2×C8—2441no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—611no—
318C4⋊C4—611no—
319C4⋊C4—1221no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C22⋊C4—2441no—
327C22⋊C4—2441no—
328C42—81no—
329C42—1221no—
330C42—1221no—
331C42—1221no—
332C42—1221no—
333C42C2×C4×D6111yescharacteristic
334C42C2×D24411yescharacteristic
335C42D4×D6111yescharacteristic
336C42#1175822611yescharacteristic
337C23—2441no—
338C23#1451975011yescharacteristic
339Q8—612no—
340Q8—1221no—
341Q8—1221no—
342Q8—1221no—
343Q8—4881no—
344Q8—2442no—
345D4—4881no—
346D4—4881no—
347D4—4881no—
348D4—4881no—
349D4—2441no—
350D4—2441no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—1221no—
363C2×C4—4882no—
364C2×C4—2442no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4C22⋊C4×D6111yescharacteristic
369C2×C4#1451973911yescharacteristic
370C2×C4C4×D24411yescharacteristic
371C2×C4#1451974711yescharacteristic
372C2×C4#1451975211yescharacteristic
373C2×C4#1451976211yescharacteristic
374C8—42no—
375C8—41no—
376C8—41no—
377C8—2441no—
378C8—2441no—
379C8—2442no—
380C22—21no—
381C22—21no—
382C22—4881no—
383C22—4881no—
384C22—4881no—
385C22#1683964011yescharacteristic
386C22#1684002811yescharacteristic
387C22#1684003111yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—82no—
394C4—1222no—
395C4—4881no—
396C4—2441no—
397C4—2441no—
398C4—2441no—
399C4#104×D6111yescharacteristic
400C4#1683977711yescharacteristic
401C2—21no—
402C2—21no—
403C2—9761no—
404C2#2144612011yescharacteristic
405C2#2144631411yescharacteristic
406C2#2144632811yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1285C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1285D61
hypocenterC61#9164
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
2. #21454909C2semidirect
3. #21445652C2semidirect
6. #21443717C2semidirect
7. C61×#1285C2semidirect
11. #16840058C4semidirect
12. #16840058C4semidirect
13. #16839963C4semidirect
14. #16839963C4semidirect
204. #1285D61semidirect
236. C61#9164semidirect

Maximal subgroups

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

subgroupindex
#214559422
#214549092
#214456522
#214444312
#214443132
#214437172
C61×#12852
#916461