Finite Group Databasebeta

← #29574724

Subgroups — #29574724

Order 15616.

514 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 99 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#21455880C211yesmaximal, characteristic
2#21454948C211yesmaximal, characteristic, complemented
3#21446647C211yesmaximal, characteristic, complemented
4C4×#14519705C211yesmaximal, characteristic, complemented
5#21444235C211yesmaximal, characteristic, complemented
6#21443710C211yesmaximal, characteristic, complemented
7C61×#1276C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C4×#11758222C2211yescharacteristic
10#16840087—21no—
11#16840058C411yescharacteristic, complemented
12#16840053C411yescharacteristic, complemented
13#16839963C411yescharacteristic, complemented
14#16839958C411yescharacteristic, complemented
15#16839909—21no—
16C8×Dic122—21no—
17C4×#11758190—21no—
18C4×D488—21no—
19#16839841—21no—
20#16839837—22no—
21#16839814—21no—
22#16839786C2211yescharacteristic
23#16839784—21no—
24C2×#14519705—22no—
25C2×#14519705—21no—
26C4×#11758194C2211yescharacteristic
27#16839522—21no—
28#16839489—21no—
29#16839489—21no—
30C61×#431C2211yescharacteristic
31C61×#431C2211yescharacteristic
32C43×C61C2211yescharacteristic
33C2×#11758222—21no—
34C2×#11758222D411yescharacteristic
35C2×#11758190—21no—
36C2×D488—21no—
37C2×#11758194—21no—
38C2×#11758194D412yes—
39C2×#11758189—22no—
40#14519777C2×C411yescharacteristic
41#14519774C2×C411yescharacteristic
42#14519765—21no—
43#14519751—21no—
44#14519751—21no—
45C4×Dic122C2×C411yescharacteristic
46C4×Dic122D411yescharacteristic
47C4×D244—21no—
48C4×D244—21no—
49C4×D244C2×C411yescharacteristic
50C4×D244D411yescharacteristic
51C4×C22⋊D61—21no—
52C4×C22⋊D61—21no—
53#14519736D411yescharacteristic
54#14519734—21no—
55C42×D61—21no—
56#14519711—21no—
57#14519709—21no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60#14519706—22no—
61#14519705—24no—
62#14519705—44no—
63#14519702—21no—
64#14519699—21no—
65#14519699—21no—
66#14519699—21no—
67#14519689—22no—
68#14519688—21no—
69C4×#92418—21no—
70C8×Dic61—22no—
71C2×C42×C61—21no—
72C2×C42×C61—21no—
73C2×C42×C61—21no—
74C2×C42×C61C2311yescharacteristic
75C4⋊C8×C61C2×C411yescharacteristic
76C4⋊C8×C61C2×C411yescharacteristic
77C4⋊C8×C61D411yescharacteristic
78C4⋊C8×C61D411yescharacteristic
79C22⋊C8×C61—21no—
80C22⋊C8×C61—21no—
81C4×C8×C61—21no—
82C4×C8×C61—21no—
83C22×C4×C61—41no—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C2×C22⋊D61—21no—
88C2×C22⋊D61—41no—
89#11758222—22no—
90#11758222—41no—
91#11758222—42no—
92#11758222—42no—
93C2×D244—21no—
94C2×D244—21no—
95C2×D244—21no—
96C2×D244C22⋊C411yescharacteristic
97C2×C4×D61—21no—
98C2×C4×D61—21no—
99C2×C4×D61—21no—
100C2×C4×D61—41no—
101C2×Dic122—21no—
102C2×Dic122C22⋊C411yescharacteristic
103C2×C8×C61—21no—
104C2×C8×C61—21no—
105C2×C8×C61—21no—
106C2×C8×C61—21no—
107C42×C61—22no—
108C42×C61—22no—
109C42×C61—22no—
110C42×C61—41no—
111C42×C61—41no—
112C42×C61—41no—
113C42×C61C22×C411yescharacteristic
114C42×C61C2×D411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61#4111yescharacteristic
117C22⋊Dic61—21no—
118#11758198—21no—
119#11758198—21no—
120#11758198—41no—
121#11758198—41no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61C22⋊C411yescharacteristic
125C4⋊Dic61C22⋊C411yescharacteristic
126#11758196—21no—
127#11758196—21no—
128C4×Dic61—21no—
129C4×Dic61—21no—
130C4×Dic61—21no—
131#11758194—22no—
132#11758194—41no—
133#11758194—42no—
134C2×#92418—22no—
135C2×#92418—41no—
136D488—42no—
137#11758190—42no—
138#11758189—82no—
139C23×C61#12111yescharacteristic
140C22×D61—21no—
141C22×D61—41no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61C2×C22⋊C411yescharacteristic
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12311yescharacteristic
157C22⋊D61—42no—
158C22⋊D61—82no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—41no—
163D244—21no—
164D244—21no—
165D244—42no—
166D244—41no—
167D244—41no—
168D244—41no—
169D244—41no—
170C4×D61—42no—
171C4×D61—41no—
172C4×D61—41no—
173C4×D61—41no—
174C4×D61—41no—
175C4×D61—81no—
176C4×D61—81no—
177Dic122—21no—
178Dic122—42no—
179Dic122#10412yescomplemented
180C8×C61—41no—
181C8×C61—41no—
182C8×C61—41no—
183C8×C61—41no—
184#92418—42no—
185#92418—42no—
186#8970—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61#38611yescharacteristic
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—41no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—44no—
201C4×C61C2×#10411yescharacteristic
202C4×C61#42111yescharacteristic
203C4×C61#44311yescharacteristic
204C4×C61#44411yescharacteristic
205C4×C61#45311yescharacteristic
206C4×C61#45411yescharacteristic
207Dic61—22no—
208Dic61—41no—
209Dic61—41no—
210Dic61—81no—
211#3106—611no—
212#2991—611no—
213#1962—611no—
214C4×#104—611no—
215#1484—611no—
216#1299—611no—
217#1276D6111yescharacteristic, complemented
218C2×C61—41no—
219C2×C61#187111yescharacteristic
220C2×C61#187611yescharacteristic
221C2×C61#188311yescharacteristic
222D61—81no—
223D61—161no—
224#532—611no—
225C4×#41—611no—
226#482—1221no—
227#477—611no—
228#476—611no—
229#460—611no—
230#459—611no—
231#449—1221no—
232C8×Q8—1221no—
233C4×SD16—1221no—
234C4×D8—1221no—
235#436—1221no—
236#436—1222no—
237#431—1221no—
238#431C2×D6111yescharacteristic
239#431C2×D6111yescharacteristic
240#423—611no—
241#423—1221no—
242C2×#104—1221no—
243C2×#104—1222no—
244C4×M16—611no—
245C43C2×D6111yescharacteristic
246#339—1221no—
247D4⋊C8—1221no—
248D4⋊C8—1221no—
249C61#897011yescharacteristic, complemented, Sylow
250C2×#41—611no—
251C2×#41—1221no—
252C2×SD16—1221no—
253C2×D8—1221no—
254C2×M16—612no—
255C2×M16—1222no—
256C2×M16—1221no—
257C4⋊Q8—611no—
258#127—611no—
259#124—1221no—
260#121—1221no—
261#121—1221no—
262C4×Q8—611no—
263C4×Q8—611no—
264C4×D4—611no—
265C4×D4—611no—
266C4×D4—1221no—
267C4×D4—1221no—
268C4×D4—1221no—
269C4×D4—1221no—
270#117—611no—
271#117—1221no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—1221no—
276C2×C42C22×D6111yescharacteristic
277#107—1221no—
278#106—1221no—
279C4⋊C8—1222no—
280C4⋊C8—1221no—
281C4⋊C8—1221no—
282C4⋊C8C4×D6111yescharacteristic
283C4⋊C8C4×D6111yescharacteristic
284C4⋊C8D24411yescharacteristic
285C4⋊C8D24411yescharacteristic
286#104—1224no—
287#104—2444no—
288#103—1221no—
289#102—1221no—
290#102—1221no—
291#102—1221no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—1222no—
295#97—1221no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—1222no—
299C4×C8—1221no—
300#41—1222no—
301#41—2442no—
302#41—2442no—
303#41—2441no—
304C2×Q8—611no—
305C2×Q8—1221no—
306C2×D4—611no—
307C2×D4—1221no—
308C2×D4—1221no—
309C2×D4—1221no—
310C2×D4—1221no—
311C2×D4—2441no—
312C22×C4—41no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—2441no—
317C22×C4#1175822211yescharacteristic
318C22×C4D4×D6111yescharacteristic
319C22×C4D4×D6111yescharacteristic
320SD16—2442no—
321D8—2442no—
322M16—1222no—
323M16—4882no—
324M16—2441no—
325M16—2442no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—1222no—
331C2×C8—2441no—
332C4⋊C4—611no—
333C4⋊C4—611no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C22⋊C4—1221no—
339C22⋊C4—1221no—
340C22⋊C4—1221no—
341C22⋊C4—2441no—
342C22⋊C4—2441no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—1221no—
350C42—1221no—
351C42—1221no—
352C42C2×C4×D6111yescharacteristic
353C42C2×D24411yescharacteristic
354C42D4×D6111yescharacteristic
355C42#1175822611yescharacteristic
356C23—1221no—
357C23—2441no—
358C23#1451975011yescharacteristic
359Q8—612no—
360Q8—1221no—
361Q8—2442no—
362D4—1221no—
363D4—1221no—
364D4—4882no—
365D4—2442no—
366D4—2442no—
367D4—2441no—
368D4—2441no—
369D4—2441no—
370D4—2441no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—4881no—
384C2×C4—4881no—
385C2×C4—2442no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4C22⋊C4×D6111yescharacteristic
392C2×C4#1451973911yescharacteristic
393C2×C4C4×D24411yescharacteristic
394C2×C4#1451974711yescharacteristic
395C2×C4#1451975211yescharacteristic
396C2×C4#1451976211yescharacteristic
397C8—41no—
398C8—41no—
399C8—41no—
400C8—41no—
401C8—2442no—
402C8—2442no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—4881no—
407C22—4881no—
408C22—4881no—
409C22—2441no—
410C22—2441no—
411C22—2441no—
412C22#1683964011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—44no—
417C4—1222no—
418C4—4881no—
419C4—2441no—
420C4—2441no—
421C4#104×D6111yescharacteristic
422C4#1683977711yescharacteristic
423C4#1683988011yescharacteristic
424C4#1683988511yescharacteristic
425C4#1683993111yescharacteristic
426C4#1683993611yescharacteristic
427C2—41no—
428C2—9761no—
429C2—4881no—
430C2#2144612011yescharacteristic
431C2#2144614511yescharacteristic
432C2#2144619611yescharacteristic
subgroupG/H
centerC2×C4#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1276C2
socleC22×C61#386
prime radical#21455880C2
cosocleC2×C42×C61C23
hypercenter#1276D61
hypocenterC61#8970
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #21454948C2semidirect
3. #21446647C2semidirect
4. C4×#14519705C2semidirect
5. #21444235C2semidirect
6. #21443710C2semidirect
7. C61×#1276C2semidirect
11. #16840058C4semidirect
12. #16840053C4semidirect
13. #16839963C4semidirect
14. #16839958C4semidirect
179. Dic122#104semidirect
217. #1276D61semidirect
249. C61#8970semidirect

Maximal subgroups

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

subgroupindex
#214558802
#214549482
#214466472
C4×#145197052
#214442352
#214437102
C61×#12762
#897061