Finite Group Databasebeta

← #29594969

Subgroups — #29594969

Order 15616.

768 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 147 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#410×D61C212yesmaximal
2#21454261C211yesmaximal, characteristic
3#21443940C211yesmaximal, characteristic, complemented
4#21443939C212yesmaximal, complemented
5C61×#1391C211yesmaximal, characteristic, complemented
6C2×#41×D61C2211yescharacteristic
7C22×#11758189C411yescharacteristic, complemented
8C2×M16×D61C411yescharacteristic, complemented
9#117×D61—22no—
10#16839736C2212yes—
11#16839735—22no—
12#16839734—22no—
13#16839734C414yescomplemented
14#99×D61—22no—
15#99×D61C414yescomplemented
16#16839730C411yescharacteristic, complemented
17#16839729C2211yescharacteristic
18#16839728C411yescharacteristic, complemented
19#16839727—22no—
20#16839724—22no—
21C61×#410C2212yes—
22C61×#408C2211yescharacteristic
23C2×#11758226Q811yescharacteristic
24C2×#11758222C2×C411yescharacteristic
25C2×#11758222D412yes—
26C2×#11758224C2×C412yes—
27C2×#11758224D411yescharacteristic
28#41×D61—42no—
29#41×D61—42no—
30C2×Q8×D61Q811yescharacteristic
31C2×D4×D61C2×C412yes—
32C2×D4×D61D411yescharacteristic
33C22×C4×D61C2×C411yescharacteristic
34C22×C4×D61D412yes—
35C2×#11758194C2×C411yescharacteristic, complemented
36C2×#11758189—21no—
37C2×#11758189—21no—
38C2×#11758189—21no—
39C2×#11758189—42no—
40M16×D61—42no—
41C22×#92418C2×C411yescharacteristic, complemented
42C2×C8×D61—21no—
43#14519736—22no—
44#14519735—42no—
45#14519734—42no—
46#14519733—24no—
47C4⋊C4×D61—42no—
48C22⋊C4×D61—24no—
49C42×D61—42no—
50#14519692—22no—
51#14519692C2×C414yescomplemented
52#14519691—24no—
53#14519691—24no—
54C22⋊#92418—22no—
55#14519689—22no—
56#14519689—22no—
57C2×#41×C61C2311yescharacteristic
58C2×M16×C61C2×C411yescharacteristic, complemented
59C22×C8×C61C2×C411yescharacteristic, complemented
60#117×C61—22no—
61#99×C61—22no—
62#99×C61C2×C414yescomplemented
63C22⋊C8×C61—22no—
64C23×D61C4211yescharacteristic
65C23×D61C22⋊C412yes—
66#41×C61—41no—
67#41×C61—41no—
68C2×Q8×C61C2×Q811yescharacteristic
69C2×D4×C61C22×C412yes—
70C2×D4×C61C2×D411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C2×D412yes—
73C2×C22⋊D61C4212yes—
74C2×C22⋊D61C22⋊C412yes—
75C2×C22⋊D61C22⋊C412yes—
76C22×Dic61C4211yescharacteristic
77C22×Dic61C22⋊C412yes—
78#11758226—42no—
79Q8×D61—42no—
80#11758224—42no—
81#11758224—42no—
82#11758224—42no—
83D4×D61—42no—
84D4×D61—42no—
85D4×D61—42no—
86#11758222—41no—
87#11758222—41no—
88#11758222—42no—
89#11758222—42no—
90C2×D244C4⋊C412yes—
91C2×D244C4⋊C411yescharacteristic
92C2×C4×D61—22no—
93C2×C4×D61—22no—
94C2×C4×D61—22no—
95C2×C4×D61—22no—
96C2×C4×D61—22no—
97C2×C4×D61—22no—
98C2×C4×D61—44no—
99C2×C4×D61C22⋊C411yescharacteristic
100C2×C4×D61C22⋊C412yes—
101C2×C4×D61C22⋊C411yescharacteristic
102C2×C4×D61C4⋊C411yescharacteristic
103C2×C4×D61C4⋊C411yescharacteristic
104C2×C4×D61C4⋊C412yes—
105C2×C4×D61C4⋊C412yes—
106C2×Dic122C4⋊C412yes—
107C2×Dic122C4⋊C411yescharacteristic
108M16×C61—41no—
109C2×C8×C61—21no—
110C2×C8×C61—21no—
111C2×C8×C61—41no—
112C4⋊C4×C61—42no—
113C22⋊C4×C61—24no—
114C42×C61—42no—
115C22⋊Dic61—24no—
116#11758198—44no—
117C4⋊Dic61—42no—
118#11758196—44no—
119C4×Dic61—44no—
120C4×Dic61—42no—
121#11758194—41no—
122C2×#92418—21no—
123C2×#92418—21no—
124C2×#92418—41no—
125#11758189—42no—
126#11758189—42no—
127#11758189—42no—
128C8×D61—42no—
129C23×C61C2×C4211yescharacteristic
130C23×C61C2×C22⋊C412yes—
131C22×D61—22no—
132C22×D61—22no—
133C22×D61—22no—
134C22×D61—82no—
135C22×D61—81no—
136C22×D61#9512yes—
137C22×D61#9511yescharacteristic
138C22×D61#9511yescharacteristic
139Q8×C61—41no—
140D4×C61—42no—
141D4×C61—41no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—44no—
147C2×C4×C61C2×C22⋊C411yescharacteristic
148C2×C4×C61C2×C4⋊C411yescharacteristic
149C2×C4×C61C2×C4⋊C412yes—
150C22⋊D61—42no—
151C22⋊D61—42no—
152C22⋊D61—42no—
153C2×Dic61—22no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—44no—
157C2×Dic61#9511yescharacteristic
158C2×Dic61#9512yes—
159C2×Dic61#9511yescharacteristic
160D244—42no—
161D244—41no—
162C4×D61—22no—
163C4×D61—22no—
164C4×D61—22no—
165C4×D61—21no—
166C4×D61—21no—
167C4×D61—22no—
168C4×D61—22no—
169C4×D61—22no—
170C4×D61—22no—
171C4×D61—22no—
172C4×D61—88no—
173C4×D61#9912yes—
174C4×D61#9912yes—
175Dic122—42no—
176Dic122—41no—
177C8×C61—41no—
178C8×C61—41no—
179#92418—41no—
180#92418—41no—
181#12577—611nomaximal, Sylow
182C22×C61—22no—
183C22×C61—21no—
184C22×C61—41no—
185C22×C61—42no—
186C22×C61C2×#9511yescharacteristic
187C2×D61—22no—
188C2×D61—21no—
189C2×D61—41no—
190C2×D61—44no—
191C2×D61—42no—
192C2×D61—41no—
193C2×D61—82no—
194C2×D61—82no—
195C2×D61#34212yes—
196C4×C61—21no—
197C4×C61—22no—
198C4×C61—84no—
199C4×C61C2×#9911yescharacteristic
200C4×C61C2×#9911yescharacteristic
201Dic61—21no—
202Dic61—22no—
203Dic61—84no—
204Dic61#34212yes—
205C2×#410—612no—
206#2879—611no—
207#1391—612no—
208#1391—611no—
209#1391D6111yescharacteristic, complemented
210C2×C61—21no—
211C2×C61—42no—
212C2×C61—41no—
213C2×C61C2×#34211yescharacteristic
214D61—22no—
215D61—42no—
216D61—41no—
217C22×#41—611no—
218C22×M16—611no—
219C22×M16—611no—
220C2×#117—1222no—
221#410—612no—
222#410—614no—
223#410—1222no—
224#410—1222no—
225#410C2×D6112yes—
226C2×#99—614no—
227C2×#99—1222no—
228#408—611no—
229#408—611no—
230#408—611no—
231#408—1222no—
232#408C2×D6111yescharacteristic
233#407—1222no—
234C61#1257711yescharacteristic, complemented, Sylow
235C2×#41—611no—
236C2×#41—612no—
237C2×#41—611no—
238C2×#41—611no—
239C2×#41—612no—
240C2×#41—2442no—
241C2×#41—2442no—
242C2×#41C22×D6111yescharacteristic
243C22×Q8—611no—
244C22×D4—612no—
245C22×D4—611no—
246C23×C4—611no—
247C23×C4—612no—
248C2×M16—611no—
249C2×M16—1221no—
250C2×M16—1221no—
251C2×M16—1221no—
252C2×M16—2442no—
253C2×M16—2442no—
254C2×M16C4×D6111yescharacteristic, complemented
255C22×C8—611no—
256C22×C8—1221no—
257C22×C8C4×D6111yescharacteristic, complemented
258#117—22no—
259#117—1224no—
260#117—1222no—
261#117—2442no—
262#117—2442no—
263C2×C4⋊C4—2442no—
264C2×C22⋊C4—1224no—
265C2×C42—2442no—
266#99—22no—
267#99—614no—
268#99—1224no—
269#99—1224no—
270#99—1222no—
271#99C4×D6114yescomplemented
272C22⋊C8—22no—
273C22⋊C8—1222no—
274C22⋊C8—1222no—
275C22⋊C8—1222no—
276C24—611no—
277C24—612no—
278#41—41no—
279#41—41no—
280#41—2442no—
281#41—2441no—
282#41—2442no—
283#41—2441no—
284#41—2442no—
285#41—2442no—
286#41—2442no—
287#41—2442no—
288C2×Q8—611no—
289C2×Q8—612no—
290C2×Q8—2442no—
291C2×Q8Q8×D6111yescharacteristic
292C2×D4—612no—
293C2×D4—612no—
294C2×D4—612no—
295C2×D4—611no—
296C2×D4—612no—
297C2×D4—2442no—
298C2×D4—2442no—
299C2×D4—2442no—
300C2×D4C2×C4×D6112yes—
301C2×D4D4×D6111yescharacteristic
302C22×C4—611no—
303C22×C4—611no—
304C22×C4—612no—
305C22×C4—612no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—612no—
309C22×C4—611no—
310C22×C4—612no—
311C22×C4—1222no—
312C22×C4—1222no—
313C22×C4—1222no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4—1222no—
317C22×C4—2444no—
318C22×C4C2×C4×D6111yescharacteristic
319C22×C4D4×D6112yes—
320M16—41no—
321M16—2442no—
322M16—2442no—
323M16—2442no—
324M16—2441no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—41no—
328C2×C8—1221no—
329C2×C8—1221no—
330C2×C8—2442no—
331C2×C8—2441no—
332C4⋊C4—42no—
333C4⋊C4—2442no—
334C4⋊C4—2444no—
335C22⋊C4—24no—
336C22⋊C4—1224no—
337C22⋊C4—2444no—
338C42—42no—
339C42—2442no—
340C42—2444no—
341C23—611no—
342C23—612no—
343C23—611no—
344C23—1222no—
345C23—1222no—
346C23—1222no—
347C23—4881no—
348C23—4882no—
349C23C42×D6111yescharacteristic
350C23C22⋊C4×D6112yes—
351Q8—41no—
352Q8—2442no—
353Q8—2441no—
354D4—42no—
355D4—41no—
356D4—2442no—
357D4—2441no—
358D4—2442no—
359D4—2442no—
360D4—2442no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—44no—
366C2×C4—611no—
367C2×C4—612no—
368C2×C4—612no—
369C2×C4—611no—
370C2×C4—612no—
371C2×C4—1222no—
372C2×C4—1222no—
373C2×C4—1222no—
374C2×C4—1222no—
375C2×C4—1222no—
376C2×C4—1222no—
377C2×C4—1222no—
378C2×C4—1222no—
379C2×C4—1222no—
380C2×C4—1221no—
381C2×C4—1222no—
382C2×C4—1221no—
383C2×C4—1222no—
384C2×C4—4888no—
385C2×C4—2444no—
386C2×C4C22⋊C4×D6111yescharacteristic
387C2×C4C4⋊C4×D6111yescharacteristic
388C2×C4C4⋊C4×D6112yes—
389C8—41no—
390C8—41no—
391C8—2441no—
392C8—2441no—
393C22—22no—
394C22—21no—
395C22—41no—
396C22—42no—
397C22—612no—
398C22—1221no—
399C22—1222no—
400C22—4882no—
401C22—4882no—
402C22—2441no—
403C22—2441no—
404C22—2442no—
405C22—2444no—
406C22#95×D6111yescharacteristic
407C4—22no—
408C4—21no—
409C4—84no—
410C4—612no—
411C4—1222no—
412C4—1221no—
413C4—4884no—
414C4#99×D6111yescharacteristic
415C4#99×D6111yescharacteristic
416C2—21no—
417C2—42no—
418C2—41no—
419C2—1222no—
420C2—2441no—
421C2—2442no—
422C2#342×D6111yescharacteristic
subgroupG/H
centerC4#99×D61
commutatorC23×C61C2×C42
FrattiniC2×#41C22×D61
FittingC61×#1391C2
socleC2×C61C2×#342
prime radicalC2×#41×D61C22
cosocleC2×#41×C61C23
hypercenter#1391D61
hypocenterC61#12577
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
3. #21443940C2semidirect
4. #21443939C2semidirect
5. C61×#1391C2semidirect
7. C22×#11758189C4semidirect
8. C2×M16×D61C4semidirect
13. #16839734C4semidirect
15. #99×D61C4semidirect
16. #16839730C4semidirect
18. #16839728C4semidirect
35. C2×#11758194C2×C4semidirect
41. C22×#92418C2×C4semidirect
51. #14519692C2×C4semidirect
58. C2×M16×C61C2×C4semidirect
59. C22×C8×C61C2×C4semidirect
62. #99×C61C2×C4semidirect
209. #1391D61semidirect
234. C61#12577semidirect
254. C2×M16C4×D61semidirect
257. C22×C8C4×D61semidirect
271. #99C4×D61semidirect

Maximal subgroups

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

subgroupindex
#410×D612
#214542612
#214439402
#214439392
C61×#13912
#1257761