Finite Group Databasebeta

← #29575243

Subgroups — #29575243

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454768C211yesmaximal, characteristic, complemented
2#21446739C211yesmaximal, characteristic
3#21444318C211yesmaximal, characteristic, complemented
4#21444218C211yesmaximal, characteristic
5#21443659C211yesmaximal, characteristic, complemented
6#21443656C211yesmaximal, characteristic, complemented
7C61×#3167C211yesmaximal, characteristic
8#16839893—21no—
9#16839893—22no—
10Q8×#92418—21no—
11#16839815C2211yescharacteristic
12#16839789—21no—
13C2×C4⋊#92418C2211yescharacteristic
14C2×#14519704—21no—
15C2×#14519704—21no—
16#16839718—21no—
17C4×#11758194C2211yescharacteristic
18C4×C4⋊Dic61C2211yescharacteristic
19#16839499—22no—
20D4⋊#92418—21no—
21D4⋊#92418—21no—
22C61×#533C2211yescharacteristic
23C61×#532C2211yescharacteristic
24C61×#498C411yescharacteristic, complemented
25C61×#492C411yescharacteristic, complemented
26C61×#487C411yescharacteristic, complemented
27C61×#486C411yescharacteristic, complemented
28C61×#465—22no—
29C4×SD16×C61—22no—
30C2×C4×C8×C61C2211yescharacteristic
31C2×#11758194D411yescharacteristic
32C2×#11758194D411yescharacteristic
33C22×#92418—21no—
34C2×C4⋊Dic61—21no—
35C2×C4×Dic61—21no—
36C4⋊#92418—22no—
37C4⋊#92418—22no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418D411yescharacteristic
41C4⋊#92418D411yescharacteristic
42#14519704—42no—
43#14519704—42no—
44C22⋊#92418—21no—
45#14519688—22no—
46#14519688—21no—
47C4×#92418—22no—
48C4×#92418—21no—
49C4×#92418—21no—
50#14519684—21no—
51C2×#41×C61—21no—
52C2×SD16×C61—22no—
53C22×C8×C61—21no—
54C4⋊Q8×C61C2×C411yescharacteristic
55C4⋊Q8×C61C2×C411yescharacteristic
56C4⋊Q8×C61C2×C411yescharacteristic
57#127×C61C2×C411yescharacteristic
58#125×C61—21no—
59#124×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61D412yes—
62#121×C61—21no—
63#121×C61—21no—
64C4×Q8×C61—21no—
65C4×Q8×C61D412yes—
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68#117×C61—21no—
69C2×C42×C61C2311yescharacteristic
70#106×C61—22no—
71#103×C61—22no—
72#102×C61—22no—
73C4×C8×C61—21no—
74C4×C8×C61C2×C411yescharacteristic
75C4×C8×C61C2×C411yescharacteristic
76#41×C61—41no—
77#41×C61—41no—
78#41×C61—41no—
79#41×C61—41no—
80C2×Q8×C61—21no—
81C2×Q8×C61—21no—
82C2×Q8×C61C22⋊C412yes—
83C2×D4×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C22×Dic61—41no—
92SD16×C61—44no—
93C2×C8×C61—21no—
94C2×C8×C61—21no—
95C2×C8×C61—42no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61C22⋊C412yes—
104C22⋊C4×C61—22no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C42×C61—22no—
108C42×C61C22×C411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61#4111yescharacteristic
112C4⋊Dic61—22no—
113C4⋊Dic61—41no—
114C4×Dic61—41no—
115C4×Dic61—41no—
116#11758194—21no—
117#11758194—21no—
118#11758194—21no—
119#11758194SD1612yescomplemented
120C2×#92418—21no—
121C2×#92418—21no—
122C2×#92418—21no—
123C2×#92418—21no—
124C2×#92418—41no—
125C2×#92418—41no—
126C2×#92418—41no—
127C23×C61—41no—
128C23×C61#12111yescharacteristic
129Q8×C61—42no—
130Q8×C61—41no—
131Q8×C61—41no—
132Q8×C61—41no—
133Q8×C61—41no—
134D4×C61—42no—
135D4×C61—41no—
136D4×C61—41no—
137D4×C61—41no—
138D4×C61—41no—
139D4×C61—81no—
140D4×C61—81no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—81no—
148C2×C4×C61—81no—
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C4×D411yescharacteristic
151C2×C4×C61C4×D411yescharacteristic
152C2×C4×C61#12011yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12311yescharacteristic
155C2×C4×C61C2×SD1611yescharacteristic
156C2×C4×C61#13511yescharacteristic
157C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C61#13711yescharacteristic
159C2×Dic61—42no—
160C2×Dic61—81no—
161C2×Dic61—81no—
162C8×C61—22no—
163C8×C61—41no—
164#92418—22no—
165#92418—41no—
166#92418—41no—
167#92418—41no—
168#92418—81no—
169#9054—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61#38611yescharacteristic
176C22×C61#46011yescharacteristic
177C22×C61#46411yescharacteristic
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—42no—
182C4×C61—42no—
183C4×C61—81no—
184C4×C61C4×SD1611yescharacteristic
185C4×C61#44011yescharacteristic
186C4×C61#44311yescharacteristic
187C4×C61#44411yescharacteristic
188C4×C61#44811yescharacteristic
189C4×C61#44911yescharacteristic
190Dic61—82no—
191#3167D6111yescharacteristic
192#2962—611no—
193#1980—611no—
194#1503—611no—
195#1480—611no—
196#1279—611no—
197#1278—611no—
198C2×C61—21no—
199C2×C61—21no—
200C2×C61—161no—
201C2×C61#187611yescharacteristic
202C2×C61#188011yescharacteristic
203C2×C61#188411yescharacteristic
204#533C2×D6111yescharacteristic
205#532C2×D6111yescharacteristic
206#498Dic6111yescharacteristic, complemented
207#492Dic6111yescharacteristic, complemented
208#487Dic6111yescharacteristic, complemented
209#486Dic6111yescharacteristic, complemented
210#465—22no—
211#446—1221no—
212#446—1222no—
213C8×Q8—1221no—
214C4×SD16—22no—
215#431—611no—
216#424—1221no—
217C2×C4⋊C8—611no—
218C2×#104—1221no—
219C2×#104—1221no—
220#405—1221no—
221C4×M16—611no—
222C2×C4×C8C2×D6111yescharacteristic
223C4×C4⋊C4—611no—
224#328—1222no—
225D4⋊C8—1221no—
226D4⋊C8—1221no—
227C61#905411yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C2×SD16—22no—
230C2×M16—611no—
231C2×M16—611no—
232C22×C8—21no—
233C22×C8—1221no—
234C4⋊Q8C2×Dic6111yescharacteristic
235C4⋊Q8C2×Dic6111yescharacteristic
236C4⋊Q8C2×Dic6111yescharacteristic
237#127C2×Dic6111yescharacteristic
238#125—21no—
239#124—21no—
240C22⋊Q8—21no—
241C22⋊Q8C22⋊D6112yes—
242#121—21no—
243#121—21no—
244C4×Q8—21no—
245C4×Q8C22⋊D6112yes—
246C4×D4—21no—
247C4×D4—21no—
248#117—21no—
249C2×C4⋊C4—1221no—
250C2×C42—1221no—
251C2×C42C22×D6111yescharacteristic
252#106—22no—
253C4⋊C8—611no—
254C4⋊C8—611no—
255C4⋊C8—1222no—
256C4⋊C8—1222no—
257C4⋊C8—1221no—
258C4⋊C8—1221no—
259#104—2442no—
260#104—2442no—
261#103—22no—
262#102—22no—
263C22⋊C8—1221no—
264#97—1222no—
265#97—1221no—
266C4×C8—21no—
267C4×C8—1222no—
268C4×C8—1221no—
269C4×C8—1221no—
270C4×C8C2×Dic6111yescharacteristic
271C4×C8C2×Dic6111yescharacteristic
272#95—1221no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8C22⋊Dic6112yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—2441no—
286C22×C4D4×D6111yescharacteristic
287C22×C4#1175822411yescharacteristic
288C22×C4C2×C22⋊D6111yescharacteristic
289SD16—44no—
290M16—612no—
291M16—1221no—
292M16—1221no—
293M16—1221no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—42no—
297C2×C8—1221no—
298C2×C8—1221no—
299C2×C8—1221no—
300C2×C8—1221no—
301C2×C8—2441no—
302C2×C8—2441no—
303C2×C8—2441no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—1222no—
312C4⋊C4—2441no—
313C4⋊C4C22⋊Dic6112yes—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—22no—
318C42—2441no—
319C42—2441no—
320C42D4×D6111yescharacteristic
321C42#1175822411yescharacteristic
322C42C22×Dic6111yescharacteristic
323C42C2×C22⋊D6111yescharacteristic
324C23—41no—
325C23#1451974911yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—2442no—
349C2×C4C2×C22⋊Dic6111yescharacteristic
350C2×C4D4×Dic6111yescharacteristic
351C2×C4D4×Dic6111yescharacteristic
352C2×C4#1451974511yescharacteristic
353C2×C4#1451974911yescharacteristic
354C2×C4#1451975811yescharacteristic
355C2×C4SD16×D6111yescharacteristic
356C2×C4#1451979811yescharacteristic
357C2×C4#1451980411yescharacteristic
358C2×C4#1451980611yescharacteristic
359C8—22no—
360C8—41no—
361C8—1222no—
362C8—4881no—
363C8—2441no—
364C8—2441no—
365C8—2441no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1683964211yescharacteristic
372C22#1683996011yescharacteristic
373C22#1683998211yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—4882no—
381C4SD16×Dic6111yescharacteristic
382C4#1683986311yescharacteristic
383C4#1683987911yescharacteristic
384C4#1683988411yescharacteristic
385C4#1683989911yescharacteristic
386C4#1683990611yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2144614811yescharacteristic
391C2#2144617611yescharacteristic
392C2#2144620311yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3167C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#3167D61
hypocenterC61#9054
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21454768C2semidirect
3. #21444318C2semidirect
5. #21443659C2semidirect
6. #21443656C2semidirect
24. C61×#498C4semidirect
25. C61×#492C4semidirect
26. C61×#487C4semidirect
27. C61×#486C4semidirect
119. #11758194SD16semidirect
206. #498Dic61semidirect
207. #492Dic61semidirect
208. #487Dic61semidirect
209. #486Dic61semidirect
227. C61#9054semidirect

Maximal subgroups

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

subgroupindex
#214547682
#214467392
#214443182
#214442182
#214436592
#214436562
C61×#31672
#905461