Finite Group Databasebeta

← #29568992

Subgroups — #29568992

Order 15616.

478 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 81 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#21452778C211yesmaximal, characteristic
2#21447149C211yesmaximal, characteristic, complemented
3#21447144C211yesmaximal, characteristic, complemented
4#21443643C211yesmaximal, characteristic, complemented
5#21443641C211yesmaximal, characteristic, complemented
6#21443640C211yesmaximal, characteristic
7C61×#2013C211yesmaximal, characteristic
8C2×#14519770—21no—
9C2×#14519710—21no—
10C2×#14519708—21no—
11#16839698—21no—
12#16839685—21no—
13#16839676—21no—
14#16839660C2211yescharacteristic
15#16839660C2211yescharacteristic
16#16839657—21no—
17#16839655—21no—
18#16839524—21no—
19#16839524—21no—
20#16839517C2211yescharacteristic
21#16839517C2211yescharacteristic
22C22×SD16×C61—21no—
23C2×C4⋊Q8×C61C2211yescharacteristic
24C2×#103×C61—21no—
25C61×#390C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×#92418—21no—
28C22×#92418—21no—
29#14519770—42no—
30#14519770—42no—
31C2×C4⋊Dic61D411yescharacteristic
32C2×C4⋊Dic61D411yescharacteristic
33C2×C4⋊Dic61D411yescharacteristic
34C2×C4⋊Dic61D411yescharacteristic
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×#11758196—21no—
38C2×#11758196—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C4×Dic61—21no—
44#14519710—42no—
45#14519708—42no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49C22×Q8×C61—21no—
50C22×D4×C61D411yescharacteristic
51C2×SD16×C61—42no—
52C2×SD16×C61—42no—
53C2×SD16×C61—42no—
54C22×C8×C61—21no—
55C4⋊Q8×C61—42no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61D411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C42×C61C2311yescharacteristic
62#103×C61—42no—
63C24×C61—21no—
64C2×Q8×C61—22no—
65C2×Q8×C61—41no—
66C2×Q8×C61—41no—
67C2×Q8×C61—41no—
68C2×Q8×C61—41no—
69C2×Q8×C61—41no—
70C2×D4×C61—21no—
71C2×D4×C61—21no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C22×C4×C61—21no—
75C22×C4×C61—41no—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—21no—
83SD16×C61—48no—
84C2×C8×C61—22no—
85C2×C8×C61—41no—
86C2×C8×C61—41no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—41no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C42×C61—41no—
97C22⋊Dic61—42no—
98C22⋊Dic61—42no—
99C22⋊Dic61—81no—
100C22⋊Dic61—81no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—41no—
105C4⋊Dic61—41no—
106C4⋊Dic61SD1612yescomplemented
107#11758196—42no—
108#11758196—41no—
109#11758196—41no—
110#11758196—41no—
111#11758196—41no—
112#11758196—81no—
113C4×Dic61—42no—
114C2×#92418—22no—
115C2×#92418—22no—
116C2×#92418—41no—
117C2×#92418—41no—
118C2×#92418—41no—
119C2×#92418—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61#12011yescharacteristic
128Q8×C61—42no—
129Q8×C61—42no—
130Q8×C61—81no—
131Q8×C61—81no—
132D4×C61—42no—
133D4×C61—81no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—42no—
144C2×C4×C61—81no—
145C2×C4×C61—81no—
146C2×C4×C61C2×SD1611yescharacteristic
147C2×C4×C61#13511yescharacteristic
148C2×C4×C61#13511yescharacteristic
149C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C61#13711yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C8×C61—42no—
165#92418—42no—
166#92418—42no—
167#7780—611nomaximal, Sylow
168C22×C61—81no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61#44911yescharacteristic
176C22×C61#45011yescharacteristic
177C22×C61#45211yescharacteristic
178C22×C61#45311yescharacteristic
179C22×C61#45611yescharacteristic
180C22×C61#45611yescharacteristic
181C22×C61B(4,𝔽2)11yescharacteristic
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—82no—
189C4×C61—81no—
190Dic61—42no—
191Dic61—81no—
192Dic61—81no—
193Dic61—81no—
194#2727—611no—
195#2058—611no—
196#2057—611no—
197#2013D6111yescharacteristic
198#1271—611no—
199#1271—611no—
200#1270—611no—
201C2×C61—161no—
202C2×C61#160011yescharacteristic
203C2×C61#160311yescharacteristic
204C2×C61#160411yescharacteristic
205C2×C61#163111yescharacteristic
206C2×C61#165011yescharacteristic
207C2×C61#165411yescharacteristic
208C2×C61#166011yescharacteristic
209C22×SD16—21no—
210C2×C4⋊Q8C2×D6111yescharacteristic
211C2×#126—1221no—
212C2×#107—1221no—
213C2×#106—1221no—
214C2×#103—21no—
215#399—1221no—
216#396—1221no—
217#394—1221no—
218#390C2×D6111yescharacteristic
219#389—611no—
220#389—611no—
221#389—1221no—
222#388—1221no—
223#340—1221no—
224#340—1221no—
225#336—611no—
226#336—611no—
227#336C2×D6111yescharacteristic
228C61#778011yescharacteristic, complemented, Sylow
229C22×Q8—21no—
230C22×D4C22⋊D6111yescharacteristic
231C2×SD16—42no—
232C2×SD16—42no—
233C2×SD16—42no—
234C22×C8—21no—
235C22×C8—1221no—
236C22×C8—1221no—
237C4⋊Q8—42no—
238#126—2442no—
239#126—2442no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—611no—
243C2×C4⋊C4—611no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4C22⋊D6111yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C42—1221no—
258C2×C42C22×D6111yescharacteristic
259#107—2442no—
260#106—2442no—
261#103—42no—
262#95—1221no—
263#95—1221no—
264#95—1221no—
265C24—21no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4D4×D6111yescharacteristic
283C22×C4D4×D6111yescharacteristic
284C22×C4C2×C22⋊D6111yescharacteristic
285SD16—48no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—1222no—
290C2×C8—1222no—
291C2×C8—2441no—
292C2×C8—2441no—
293C2×C8—2441no—
294C2×C8—2441no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—612no—
301C4⋊C4—1221no—
302C4⋊C4—1221no—
303C4⋊C4—1221no—
304C4⋊C4—4881no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C4⋊C4—2442no—
308C4⋊C4—2441no—
309C4⋊C4—2441no—
310C4⋊C4—2441no—
311C4⋊C4—2441no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—4881no—
317C22⋊C4—4881no—
318C22⋊C4—2442no—
319C22⋊C4—2442no—
320C42—41no—
321C42—2442no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#1451974511yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—81no—
333Q8—81no—
334D4—42no—
335D4—81no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—1222no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4SD16×D6111yescharacteristic
361C2×C4#1451979711yescharacteristic
362C2×C4#1451979811yescharacteristic
363C2×C4#1451980311yescharacteristic
364C2×C4#1451980411yescharacteristic
365C2×C4#1451980611yescharacteristic
366C8—42no—
367C8—2442no—
368C8—2442no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#1683990511yescharacteristic
377C22#1683991111yescharacteristic
378C22#1683992111yescharacteristic
379C22#1683992911yescharacteristic
380C22#1683994311yescharacteristic
381C22#1683994311yescharacteristic
382C22#1683994811yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—81no—
390C4—82no—
391C4—4881no—
392C4—4881no—
393C4—4881no—
394C4—2442no—
395C2—161no—
396C2#2144476311yescharacteristic
397C2#2144477911yescharacteristic
398C2#2144478611yescharacteristic
399C2#2144494311yescharacteristic
400C2#2144507311yescharacteristic
401C2#2144510111yescharacteristic
402C2#2144512811yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2013C2
socleC23×C61#120
prime radicalC22×D4×C61D4
hypercenter#2013D61
hypocenterC61#7780
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. #21447149C2semidirect
3. #21447144C2semidirect
4. #21443643C2semidirect
5. #21443641C2semidirect
106. C4⋊Dic61SD16semidirect
228. C61#7780semidirect

Maximal subgroups

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

subgroupindex
#214527782
#214471492
#214471442
#214436432
#214436412
#214436402
C61×#20132
#778061