Finite Group Databasebeta

← #29834298

Subgroups — #29834298

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840204C212yesmaximal, complemented
2#21456248C211yesmaximal, characteristic
3C4×Q16×D61C211yesmaximal, characteristic, complemented
4#21447884C212yesmaximal, complemented
5#21447883C212yesmaximal, complemented
6#21447866C211yesmaximal, characteristic, complemented
7#21447860C211yesmaximal, characteristic, complemented
8#21447859C211yesmaximal, characteristic, complemented
9#21447573C211yesmaximal, characteristic, complemented
10#21447530C212yesmaximal, complemented
11C61×#2209C211yesmaximal, characteristic, complemented
12C2×#14519800—21no—
13C2×#14519800C2212yes—
14C2×Q16×D61C2211yescharacteristic
15#16840313—21no—
16C4×Q8×D61—21no—
17#16840204—28no—
18C2×#14519723C2212yes—
19C2×#14519722C2212yes—
20C2×#14519720C2212yes—
21C2×#14519716C2212yes—
22#16840147—21no—
23#16840147C2212yes—
24#16840139C2211yescharacteristic
25#16840136—21no—
26C4×Dic244C2211yescharacteristic
27Q16×Dic61C2211yescharacteristic
28#16839856C2211yescharacteristic
29C4×#11758202—21no—
30#16839854—21no—
31#107×D61C2211yescharacteristic
32#103×D61—21no—
33C4×C8×D61C2211yescharacteristic
34C4⋊#11758185C2211yescharacteristic
35#16839580C2212yes—
36#16839566C2212yes—
37#16839564C2212yes—
38#16839562C2212yes—
39C2×Q32×C61C2212yes—
40C61×#494C2211yescharacteristic
41C4×Q16×C61C2211yescharacteristic
42C61×C4⋊C16C2211yescharacteristic
43C61×#358C2212yes—
44C2×#11758226—22no—
45C2×Q8×D61—21no—
46#14519800—24no—
47#14519800—24no—
48#14519800—44no—
49C2×Dic244C2311yescharacteristic
50C2×#11758202—21no—
51Q16×D61—22no—
52Q16×D61D414yescomplemented
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×D488—21no—
56C2×D488C2312yes—
57C2×C8×D61D411yescharacteristic
58C2×C8×D61C2311yescharacteristic
59#14519774—21no—
60#14519766—22no—
61#14519752—41no—
62#14519752—41no—
63Q8×Dic61—21no—
64C4×Dic122—21no—
65C4×Dic122—41no—
66#14519742—21no—
67#14519742—41no—
68#14519739—41no—
69#14519739—41no—
70C4⋊C4×D61—21no—
71C4⋊C4×D61—41no—
72C42×D61—41no—
73C42×D61D411yescharacteristic
74#14519723—24no—
75#14519722—24no—
76#14519720—24no—
77#14519716—24no—
78C2×#11758185C2312yes—
79#14519711C2311yescharacteristic
80#14519710—21no—
81#14519702—21no—
82#14519701—21no—
83#14519700—21no—
84C4×#92418D411yescharacteristic
85C8×Dic61D411yescharacteristic
86C8×Dic61C2311yescharacteristic
87C2×Q16×C61—21no—
88C2×Q16×C61C2312yes—
89C2×Q16×C61C2311yescharacteristic
90C4⋊Q8×C61—21no—
91C4×Q8×C61—21no—
92Q32×C61—24no—
93C2×C16×C61C2312yes—
94#107×C61C2311yescharacteristic
95#103×C61—21no—
96C4×C8×C61C2311yescharacteristic
97C2×Q8×C61—21no—
98C2×Q8×C61—22no—
99#11758226—44no—
100#11758226—44no—
101Q8×D61—24no—
102Q8×D61—42no—
103C2×D244—22no—
104C2×D244—42no—
105C2×D244—41no—
106C2×D244—41no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—42no—
110C2×C4×D61D811yescharacteristic
111C2×C4×D61D811yescharacteristic
112C2×C4×D61C2×D411yescharacteristic
113C2×Dic122—21no—
114C2×Dic122—41no—
115Q16×C61—22no—
116Q16×C61—21no—
117Q16×C61—22no—
118Q16×C61—42no—
119Q16×C61C2×D412yes—
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2411yescharacteristic
122C4⋊C4×C61—21no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61—41no—
125C4⋊C4×C61—41no—
126C42×C61—41no—
127C42×C61C2×D411yescharacteristic
128#11758202—24no—
129#11758201—44no—
130#11758201—44no—
131#11758198—42no—
132#11758198—42no—
133C4⋊Dic61—21no—
134C4⋊Dic61—41no—
135#11758196—41no—
136#11758196—41no—
137#11758196—41no—
138C4×Dic61—41no—
139C4×Dic61—41no—
140C4×Dic61—41no—
141C4×Dic61D811yescharacteristic
142C4×Dic61D811yescharacteristic
143C4×Dic61C2×D411yescharacteristic
144C2×#92418C2×D411yescharacteristic
145C2×#92418C2×D411yescharacteristic
146Dic244—21no—
147Dic244C2×D412yes—
148D488—22no—
149D488—22no—
150D488—42no—
151C8×D61—22no—
152C8×D61C2×D412yes—
153C8×D61#4112yes—
154C16×C61—22no—
155#11758185—22no—
156C22×D61—42no—
157C22×D61C2×D811yescharacteristic
158Q8×C61—22no—
159Q8×C61—41no—
160Q8×C61—42no—
161Q8×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61C2×D811yescharacteristic
166C2×C4×C61C2×D811yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61C2×D811yescharacteristic
171C2×Dic61C2×D811yescharacteristic
172C2×Dic61C2×D811yescharacteristic
173D244—42no—
174D244—42no—
175D244—82no—
176D244—84no—
177D244—82no—
178C4×D61—22no—
179C4×D61—41no—
180C4×D61—44no—
181C4×D61—82no—
182C4×D61—84no—
183C4×D61#12112yes—
184C4×D61AGL(1,ℤ/8)12yes—
185Dic122—22no—
186Dic122—41no—
187Dic122—44no—
188C8×C61—21no—
189C8×C61C22×D411yescharacteristic
190C8×C61C2×#4111yescharacteristic
191#92418—21no—
192#92418#12112yes—
193#33581—611nomaximal, Sylow
194C22×C61C22×D811yescharacteristic
195C2×D61—21no—
196C2×D61—21no—
197C2×D61—82no—
198C2×D61—82no—
199C2×D61—82no—
200C2×D61#45912yes—
201C4×C61—21no—
202C4×C61—42no—
203C4×C61—81no—
204C4×C61—82no—
205C4×C61#51012yescomplemented
206C4×C61C2×#12111yescharacteristic
207C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
208Dic61—21no—
209Dic61—21no—
210Dic61—42no—
211Dic61—81no—
212Dic61#45912yes—
213C2×#510—612no—
214#3147—611no—
215C2×C4×Q16—611no—
216#2211—612no—
217#2211—612no—
218#2209—611no—
219#2209D6111yescharacteristic, complemented
220#2207—611no—
221#2207—611no—
222#2150—611no—
223#2139—612no—
224C2×C61C2×#45911yescharacteristic
225C2×C61C2×#51011yescharacteristic
226C2×C61#341411yescharacteristic
227D61—22no—
228D61—162no—
229C2×#135—612no—
230C2×#135—1221no—
231C22×Q16—611no—
232#532—1221no—
233C2×C4×Q8—1221no—
234#510—1228no—
235C2×Q32—612no—
236C2×Q32C2×D6112yes—
237C2×SD32—612no—
238C2×SD32—612no—
239C2×M32—612no—
240#495—612no—
241#495—1221no—
242#494C2×D6111yescharacteristic
243#493—611no—
244#492—1221no—
245C4×Q16—611no—
246C4×Q16—611no—
247C4×Q16—611no—
248C4×Q16—1221no—
249C4×Q16—1221no—
250C4×Q16C2×D6111yescharacteristic
251C2×#107—611no—
252C2×#103—1221no—
253C2×C4×C8—611no—
254C4⋊C16—612no—
255C4⋊C16—611no—
256C4⋊C16C2×D6111yescharacteristic
257#358—612no—
258#358C2×D6112yes—
259#357—612no—
260#357—612no—
261C61#3358111yescharacteristic, complemented, Sylow
262C2×#41—1222no—
263C22×Q8—1221no—
264#135—1224no—
265#135—1224no—
266#135—2444no—
267C2×Q16—21no—
268C2×Q16—614no—
269C2×Q16—611no—
270C2×Q16—1222no—
271C2×Q16—1221no—
272C2×Q16C22×D6112yes—
273C2×Q16C22×D6111yescharacteristic
274C2×SD16—1222no—
275C2×SD16—1222no—
276C2×D8—612no—
277C2×D8—1221no—
278C22×C8—611no—
279C22×C8—611no—
280C4⋊Q8—21no—
281#127—1221no—
282#124—1222no—
283#121—2441no—
284#121—2441no—
285C4×Q8—21no—
286C4×Q8—1221no—
287C4×Q8—1221no—
288C4×Q8—1221no—
289C4×Q8—2441no—
290C4×Q8—2441no—
291C4×D4—2441no—
292C4×D4—2441no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—2441no—
295C2×C42—611no—
296C2×C42—2441no—
297Q32—24no—
298Q32—1224no—
299SD32—1224no—
300SD32—1224no—
301M32—1224no—
302C2×C16—612no—
303C2×C16C22×D6112yes—
304#107—611no—
305#107—1221no—
306#107C22×D6111yescharacteristic
307#103—21no—
308#103—1221no—
309#103—1221no—
310#103—1221no—
311C4×C8—611no—
312C4×C8—611no—
313C4×C8—611no—
314C4×C8C22×D6111yescharacteristic
315#41—2444no—
316#41—2444no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—1224no—
320C2×Q8—1221no—
321C2×Q8—2441no—
322C2×Q8—2442no—
323C2×D4—1222no—
324C2×D4—2442no—
325C2×D4—2441no—
326C2×D4—2441no—
327C22×C4—611no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—2442no—
331C22×C4—2441no—
332C22×C4—2441no—
333Q16—22no—
334Q16—21no—
335Q16—22no—
336Q16—42no—
337Q16—612no—
338Q16—1224no—
339Q16—1221no—
340Q16D4×D6112yes—
341SD16—2444no—
342SD16—2444no—
343D8—1222no—
344D8—1222no—
345D8—2442no—
346C2×C8—611no—
347C2×C8—612no—
348C2×C8—612no—
349C2×C8—611no—
350C2×C8—1222no—
351C2×C8D4×D6111yescharacteristic
352C2×C8C23×D6111yescharacteristic
353C4⋊C4—21no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—1221no—
358C4⋊C4—2441no—
359C4⋊C4—2441no—
360C4⋊C4—2441no—
361C4⋊C4—2441no—
362C22⋊C4—2442no—
363C22⋊C4—2442no—
364C42—41no—
365C42—611no—
366C42—611no—
367C42—611no—
368C42—2441no—
369C42—2441no—
370C42—2441no—
371C42D4×D6111yescharacteristic
372C16—22no—
373C16—1222no—
374C23—611no—
375C23—2442no—
376Q8—22no—
377Q8—41no—
378Q8—42no—
379Q8—42no—
380Q8—1222no—
381Q8—2444no—
382Q8—2441no—
383D4—4882no—
384D4—4884no—
385D4—4882no—
386D4—2442no—
387D4—2442no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—611no—
392C2×C4—612no—
393C2×C4—612no—
394C2×C4—611no—
395C2×C4—611no—
396C2×C4—1222no—
397C2×C4—4884no—
398C2×C4—4882no—
399C2×C4—2441no—
400C2×C4—2444no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4D8×D6111yescharacteristic
404C2×C4D8×D6111yescharacteristic
405C2×C4C2×D4×D6111yescharacteristic
406C8—21no—
407C8—612no—
408C8—1221no—
409C8C2×D4×D6111yescharacteristic
410C8#41×D6111yescharacteristic
411C22—612no—
412C22—1221no—
413C22—1221no—
414C22—4882no—
415C22—4882no—
416C22—4882no—
417C22C2×D8×D6111yescharacteristic
418C4—21no—
419C4—42no—
420C4—81no—
421C4—82no—
422C4—612no—
423C4—1221no—
424C4—1221no—
425C4—4881no—
426C4—2442no—
427C4#1684020412yescomplemented
428C4#121×D6111yescharacteristic
429C4AGL(1,ℤ/8)×D6111yescharacteristic
430C2—1222no—
431C2—9762no—
432C2#459×D6111yescharacteristic
433C2C2×#1684020411yescharacteristic
434C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2209C2
socleC22×C61C22×D8
prime radical#21456248C2
hypercenter#2209D61
hypocenterC61#33581
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2209

Complements

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

NH ≅ G/Ntype
1. C2×#16840204C2semidirect
3. C4×Q16×D61C2semidirect
4. #21447884C2semidirect
5. #21447883C2semidirect
6. #21447866C2semidirect
7. #21447860C2semidirect
8. #21447859C2semidirect
9. #21447573C2semidirect
10. #21447530C2semidirect
11. C61×#2209C2semidirect
52. Q16×D61D4semidirect
205. C4×C61#510semidirect
219. #2209D61semidirect
261. C61#33581semidirect
427. C4#16840204semidirect

Maximal subgroups

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

subgroupindex
C2×#168402042
#214562482
C4×Q16×D612
#214478842
#214478832
#214478662
#214478602
#214478592
#214475732
#214475302
C61×#22092
#3358161