Finite Group Databasebeta

← #29590772

Subgroups — #29590772

Order 15616.

626 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 83 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#21449034C211yesmaximal, characteristic
2#21444160C211yesmaximal, characteristic, complemented
3#21443839C212yesmaximal, complemented
4#21443815C212yesmaximal, complemented
5C61×#1799C211yesmaximal, characteristic, complemented
6C22×#11758222C411yescharacteristic, complemented
7C2×C4×C22⋊D61—21no—
8#16839819—21no—
9#16839682C411yescharacteristic, complemented
10#16839679C411yescharacteristic, complemented
11#16839677—21no—
12#16839674C2211yescharacteristic
13#16839671C2211yescharacteristic
14#16839670—21no—
15C4×#11758198—21no—
16C4×C22⋊Dic61C411yescharacteristic, complemented
17#16839486—21no—
18#16839486C2212yes—
19C2×C4×D4×C61C2211yescharacteristic
20C2×#99×C61—21no—
21C61×#380—21no—
22C61×#342C2212yes—
23C2×#11758222—44no—
24C2×#11758222—41no—
25C2×#11758222—41no—
26C22×Dic122C2×C411yescharacteristic
27C22×D244C2×C411yescharacteristic
28C22×C22⋊D61—21no—
29C22×C22⋊D61—21no—
30C22×C4×D61—21no—
31C4×C22⋊D61—44no—
32C2×#11758196—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×C4×Dic61—21no—
39C4⋊#92418—41no—
40C22⋊#92418—22no—
41#14519688—41no—
42#14519684—21no—
43#14519684—21no—
44#14519684C2×C411yescharacteristic
45#14519684C2×C411yescharacteristic
46C22×D4×C61D411yescharacteristic
47C23×C4×C61C2×C411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C4×D4×C61—41no—
50C4×D4×C61—41no—
51C2×C4⋊C4×C61—41no—
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C22⋊C4×C61—22no—
54C2×C22⋊C4×C61D411yescharacteristic
55C2×C22⋊C4×C61C2311yescharacteristic
56C2×C42×C61D411yescharacteristic
57#99×C61—42no—
58#95×C61—41no—
59C24×C61C22⋊C411yescharacteristic
60C24×C61C2×D411yescharacteristic
61C23×D61—21no—
62C2×D4×C61—21no—
63C2×D4×C61—21no—
64C2×D4×C61—41no—
65C2×D4×C61—41no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C2×C22⋊D61—42no—
79C2×C22⋊D61—42no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88#11758222—44no—
89#11758222—44no—
90#11758222—44no—
91#11758222—44no—
92C2×D244—41no—
93C2×D244—42no—
94C2×C4×D61—24no—
95C2×C4×D61—42no—
96C2×C4×D61—41no—
97C2×C4×D61—41no—
98C2×Dic122—41no—
99C2×Dic122—42no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—82no—
102C22⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—42no—
105C22⋊C4×C61—42no—
106C42×C61—21no—
107C42×C61—21no—
108C22⋊Dic61—42no—
109C22⋊Dic61—42no—
110#11758198—42no—
111#11758198—42no—
112#11758198—42no—
113#11758196—42no—
114C4×Dic61—42no—
115C2×#92418—42no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61C2×C22⋊C411yescharacteristic
126C23×C61AGL(1,ℤ/8)11yescharacteristic
127C23×C61#13711yescharacteristic
128C22×D61—22no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135D4×C61—42no—
136D4×C61—42no—
137C2×C4×C61—22no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—81no—
151C2×C4×C61—82no—
152C2×C4×C61—82no—
153C2×C4×C61—82no—
154C2×C4×C61#9911yescharacteristic
155C2×C4×C61#9911yescharacteristic
156C22⋊D61—48no—
157C22⋊D61—44no—
158C22⋊D61—44no—
159C2×Dic61—22no—
160C2×Dic61—22no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168D244—42no—
169D244—42no—
170C4×D61—44no—
171C4×D61—44no—
172Dic122—42no—
173Dic122—42no—
174#92418—82no—
175#11820—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61C2×#9911yescharacteristic
191C22×C61#41711yescharacteristic
192C22×C61#42111yescharacteristic
193C2×D61—42no—
194C2×D61—42no—
195C2×D61—42no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C4×C61—21no—
201C4×C61—41no—
202C4×C61—41no—
203C4×C61—41no—
204C4×C61—41no—
205C4×C61—162no—
206C4×C61AΣL(1,𝔽16)12yes—
207Dic61—42no—
208Dic61—42no—
209Dic61—81no—
210#2388—611no—
211#1799D6111yescharacteristic, complemented
212#1466—611no—
213#1351—612no—
214#1343—612no—
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—41no—
218C2×C61—41no—
219C2×C61—81no—
220C2×C61#151311yescharacteristic
221C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
222C2×C61#212611yescharacteristic
223D61—82no—
224C22×#41—611no—
225C2×C4×D4—1221no—
226C2×C4×D4C2×D6111yescharacteristic
227#432—1221no—
228C2×#99—21no—
229#395—611no—
230#394—611no—
231#394—1221no—
232#393—611no—
233#392—611no—
234#392—1221no—
235#380—21no—
236C4×C22⋊C4—611no—
237C4×C22⋊C4—1221no—
238#342C2×D6112yes—
239#323—612no—
240#323—1221no—
241C61#1182011yescharacteristic, complemented, Sylow
242C2×#41—2441no—
243C2×#41—2441no—
244C2×#41—2444no—
245C22×Q8—611no—
246C22×D4—611no—
247C22×D4—1221no—
248C22×D4—1221no—
249C22×D4C22⋊D6111yescharacteristic
250C23×C4—1221no—
251C23×C4C4×D6111yescharacteristic
252C23×C4C22⋊D6111yescharacteristic
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—2444no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4C4×D6111yescharacteristic
259C2×C22⋊C4—22no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4D24411yescharacteristic
266C2×C22⋊C4C22×D6111yescharacteristic
267C2×C42—1221no—
268C2×C42D24411yescharacteristic
269C4⋊C8—2441no—
270#99—42no—
271C22⋊C8—1222no—
272#97—2441no—
273#95—41no—
274#95—611no—
275#95—611no—
276#95—1221no—
277#95—1221no—
278C24—1221no—
279C24#1175819811yescharacteristic
280C24C2×C22⋊D6111yescharacteristic
281#41—2444no—
282#41—2444no—
283#41—2444no—
284#41—2444no—
285C2×Q8—2441no—
286C2×Q8—2442no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—2441no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4—2442no—
295C2×D4—2442no—
296C2×D4—2442no—
297C2×D4—2442no—
298C2×D4—2441no—
299C2×D4—2441no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—1224no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—2442no—
312C22×C4—2441no—
313C22×C4—2441no—
314C22×C4#1175819811yescharacteristic
315C22×C4#1175819811yescharacteristic
316C22×C4#1175819811yescharacteristic
317C22×C4C2×C4×D6111yescharacteristic
318C22×C4C2×D24411yescharacteristic
319C2×C8—2442no—
320C4⋊C4—41no—
321C4⋊C4—82no—
322C4⋊C4—2442no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—2442no—
328C22⋊C4—2442no—
329C22⋊C4—2442no—
330C22⋊C4—2442no—
331C22⋊C4—2442no—
332C42—21no—
333C42—21no—
334C42—2442no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—1222no—
345C23—2441no—
346C23—2441no—
347C23—2441no—
348C23—2441no—
349C23—2441no—
350C23—2441no—
351C23C2×#1175819811yescharacteristic
352C23#1451980311yescharacteristic
353C23#1451980811yescharacteristic
354Q8—2442no—
355Q8—2442no—
356D4—42no—
357D4—42no—
358D4—2442no—
359D4—2442no—
360D4—2444no—
361D4—2444no—
362D4—2448no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—1222no—
381C2×C4—1222no—
382C2×C4—2444no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2444no—
391C2×C4#1451969111yescharacteristic
392C2×C4#1451969111yescharacteristic
393C8—4882no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22—2442no—
413C22—2442no—
414C22—2442no—
415C22C2×#1451969111yescharacteristic
416C22#1683976111yescharacteristic
417C22#1683978011yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—162no—
424C4—4881no—
425C4—2442no—
426C4—2442no—
427C4#1683954812yes—
428C2—21no—
429C2—21no—
430C2—41no—
431C2—41no—
432C2—81no—
433C2—4882no—
434C2#2144436011yescharacteristic
435C2C2×#1683954811yescharacteristic
436C2#2144748211yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1799C2
socleC22×C61C2×#99
prime radical#21449034C2
cosocleC2×C22⋊C4×C61C23
hypercenter#1799D61
hypocenterC61#11820
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #21444160C2semidirect
3. #21443839C2semidirect
4. #21443815C2semidirect
5. C61×#1799C2semidirect
6. C22×#11758222C4semidirect
9. #16839682C4semidirect
10. #16839679C4semidirect
16. C4×C22⋊Dic61C4semidirect
211. #1799D61semidirect
241. C61#11820semidirect

Maximal subgroups

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

subgroupindex
#214490342
#214441602
#214438392
#214438152
C61×#17992
#1182061