Finite Group Databasebeta

← #29575845

Subgroups — #29575845

Order 15616.

458 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454908C211yesmaximal, characteristic, complemented
2#21445649C211yesmaximal, characteristic, complemented
3#21444430C211yesmaximal, characteristic
4#21444314C211yesmaximal, characteristic
5#21443717C211yesmaximal, characteristic, complemented
6#21443676C211yesmaximal, characteristic, complemented
7C61×#3112C211yesmaximal, characteristic
8#16839893—22no—
9#16839893—21no—
10Q8×#92418—21no—
11#16839819—21no—
12#16839819C2211yescharacteristic
13#16839815—21no—
14#16839815C2211yescharacteristic
15C2×#14519701—21no—
16C2×#14519688C2211yescharacteristic
17C4×C4⋊Dic61C2211yescharacteristic
18#16839524—21no—
19#16839502—22no—
20#16839496—21no—
21#16839496—21no—
22C61×#532C2211yescharacteristic
23C2×C4×Q8×C61C2211yescharacteristic
24C61×#480—21no—
25C61×#477C411yescharacteristic, complemented
26C61×#477C411yescharacteristic, complemented
27C61×#460C411yescharacteristic, complemented
28C61×#460C411yescharacteristic, complemented
29C61×#448—21no—
30C61×#440—21no—
31C61×#440—21no—
32C61×#432C2211yescharacteristic
33C22×#92418—21no—
34C2×C4⋊Dic61—21no—
35C2×C4×Dic61—21no—
36C4⋊#92418—22no—
37C4⋊#92418—22no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40#14519701—24no—
41C22⋊#92418—21no—
42C22⋊#92418D411yescharacteristic
43C22⋊#92418D411yescharacteristic
44#14519688—22no—
45#14519688—21no—
46#14519688—21no—
47#14519688D412yes—
48C4×#92418—22no—
49C4×#92418—21no—
50#14519684—21no—
51C2×#41×C61—21no—
52C22×Q8×C61D411yescharacteristic
53C2×SD16×C61—21no—
54C2×SD16×C61—21no—
55C4⋊Q8×C61C2×C411yescharacteristic
56#127×C61C2×C411yescharacteristic
57#124×C61—21no—
58#121×C61—21no—
59#121×C61—21no—
60C4×Q8×C61—21no—
61C4×Q8×C61—21no—
62C4×Q8×C61C2×C411yescharacteristic
63C4×Q8×C61C2×C411yescharacteristic
64C4×Q8×C61D411yescharacteristic
65C4×Q8×C61D411yescharacteristic
66C4×D4×C61—21no—
67C4×D4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C42×C61—21no—
71C2×C42×C61C2311yescharacteristic
72#107×C61—21no—
73#107×C61—21no—
74C4⋊C8×C61C2×C411yescharacteristic
75C4⋊C8×C61C2×C411yescharacteristic
76#103×C61—21no—
77#103×C61—21no—
78#102×C61—21no—
79#102×C61—21no—
80C22⋊C8×C61—21no—
81#97×C61—21no—
82#41×C61—81no—
83#41×C61—81no—
84C2×Q8×C61—21no—
85C2×Q8×C61—22no—
86C2×Q8×C61—21no—
87C2×Q8×C61—41no—
88C2×Q8×C61C22⋊C411yescharacteristic
89C2×Q8×C61C22⋊C411yescharacteristic
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—41no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C22×Dic61—41no—
101SD16×C61—42no—
102SD16×C61—42no—
103C2×C8×C61—21no—
104C2×C8×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61C22⋊C411yescharacteristic
113C4⋊C4×C61C22⋊C411yescharacteristic
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C42×C61—21no—
117C42×C61—21no—
118C42×C61—21no—
119C42×C61—21no—
120C42×C61C22×C411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124C4⋊Dic61—22no—
125C4⋊Dic61—41no—
126C4×Dic61—81no—
127C2×#92418—22no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—22no—
131C2×#92418—41no—
132C2×#92418—41no—
133C23×C61—41no—
134C23×C61#12111yescharacteristic
135Q8×C61—21no—
136Q8×C61—21no—
137Q8×C61—21no—
138Q8×C61—42no—
139Q8×C61—81no—
140Q8×C61#10412yes—
141D4×C61—41no—
142D4×C61—41no—
143D4×C61—81no—
144D4×C61—81no—
145D4×C61—81no—
146D4×C61—81no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—82no—
161C2×C4×C61C2×C22⋊C411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12311yescharacteristic
167C2×Dic61—42no—
168C2×Dic61—81no—
169C2×Dic61—81no—
170C8×C61—41no—
171C8×C61—41no—
172#92418—41no—
173#92418—41no—
174#92418—42no—
175#92418—42no—
176#9164—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61#38611yescharacteristic
183C22×C61#47111yescharacteristic
184C22×C61#47211yescharacteristic
185C4×C61—21no—
186C4×C61—22no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—41no—
192C4×C61—41no—
193C4×C61—41no—
194C4×C61—81no—
195C4×C61C2×#10411yescharacteristic
196C4×C61#42111yescharacteristic
197Dic61—82no—
198#3112D6111yescharacteristic
199#2984—611no—
200#1763—611no—
201#1530—611no—
202#1502—611no—
203#1302—611no—
204#1285—611no—
205C2×C61—21no—
206C2×C61—21no—
207C2×C61—161no—
208C2×C61#187111yescharacteristic
209C2×C61#190511yescharacteristic
210C2×C61#190711yescharacteristic
211#532C2×D6111yescharacteristic
212C2×C4×Q8C2×D6111yescharacteristic
213#480—21no—
214#477Dic6111yescharacteristic, complemented
215#477Dic6111yescharacteristic, complemented
216#460Dic6111yescharacteristic, complemented
217#460Dic6111yescharacteristic, complemented
218#448—21no—
219#446—1222no—
220#446—1221no—
221C8×Q8—1221no—
222#440—21no—
223#440—21no—
224#432—611no—
225#432—1221no—
226#432C2×D6111yescharacteristic
227#431—611no—
228#431—1221no—
229C2×#103—1221no—
230C2×#97—611no—
231C4×C4⋊C4—611no—
232#340—1221no—
233#329—1222no—
234#327—1221no—
235#327—1221no—
236C61#916411yescharacteristic, complemented, Sylow
237C2×#41—21no—
238C22×Q8C22⋊D6111yescharacteristic
239C2×SD16—21no—
240C2×SD16—21no—
241C22×C8—1221no—
242C4⋊Q8C2×Dic6111yescharacteristic
243#127C2×Dic6111yescharacteristic
244#124—21no—
245#121—21no—
246#121—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C2×Dic6111yescharacteristic
250C4×Q8C2×Dic6111yescharacteristic
251C4×Q8C22⋊D6111yescharacteristic
252C4×Q8C22⋊D6111yescharacteristic
253C4×D4—21no—
254C4×D4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4C22⋊D6111yescharacteristic
258C2×C42—21no—
259C2×C42—1221no—
260C2×C42C22×D6111yescharacteristic
261#107—21no—
262#107—21no—
263C4⋊C8—1222no—
264C4⋊C8—1222no—
265C4⋊C8—1221no—
266C4⋊C8—1221no—
267C4⋊C8C2×Dic6111yescharacteristic
268C4⋊C8C2×Dic6111yescharacteristic
269#103—21no—
270#103—21no—
271#103—1224no—
272#102—21no—
273#102—21no—
274C22⋊C8—21no—
275C22⋊C8—611no—
276C22⋊C8—611no—
277C22⋊C8—1221no—
278#97—21no—
279#97—612no—
280#97—1222no—
281#97—1221no—
282#97—1221no—
283C4×C8—1222no—
284C4×C8—1221no—
285#95—1221no—
286#41—81no—
287#41—81no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—21no—
291C2×Q8—41no—
292C2×Q8C22⋊Dic6111yescharacteristic
293C2×Q8C22⋊Dic6111yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—2441no—
302C22×C4D4×D6111yescharacteristic
303C22×C4#1175822411yescharacteristic
304C22×C4C2×C22⋊D6111yescharacteristic
305SD16—42no—
306SD16—42no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—1222no—
310C2×C8—1221no—
311C2×C8—1222no—
312C2×C8—1221no—
313C2×C8—2441no—
314C2×C8—2441no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—1222no—
323C4⋊C4—2441no—
324C4⋊C4C22⋊Dic6111yescharacteristic
325C4⋊C4C22⋊Dic6111yescharacteristic
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—4881no—
333C42D4×D6111yescharacteristic
334C42#1175822411yescharacteristic
335C42C22×Dic6111yescharacteristic
336C42C2×C22⋊D6111yescharacteristic
337C23—41no—
338C23#1451974911yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—42no—
343Q8—81no—
344Q8#1451970412yes—
345D4—41no—
346D4—41no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—4881no—
366C2×C4—4881no—
367C2×C4—2442no—
368C2×C4C2×C22⋊Dic6111yescharacteristic
369C2×C4D4×Dic6111yescharacteristic
370C2×C4D4×Dic6111yescharacteristic
371C2×C4#1451974511yescharacteristic
372C2×C4#1451974911yescharacteristic
373C2×C4#1451975811yescharacteristic
374C8—41no—
375C8—41no—
376C8—2441no—
377C8—2442no—
378C8—2441no—
379C8—2442no—
380C22—21no—
381C22—21no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#1683964211yescharacteristic
386C22#1684002311yescharacteristic
387C22#1684003011yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—4882no—
399C4C2×#1451970411yescharacteristic
400C4#1683977911yescharacteristic
401C2—21no—
402C2—21no—
403C2—161no—
404C2#2144612211yescharacteristic
405C2#2144631511yescharacteristic
406C2#2144632911yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3112C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#3112D61
hypocenterC61#9164
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
1. #21454908C2semidirect
2. #21445649C2semidirect
5. #21443717C2semidirect
6. #21443676C2semidirect
25. C61×#477C4semidirect
26. C61×#477C4semidirect
27. C61×#460C4semidirect
28. C61×#460C4semidirect
214. #477Dic61semidirect
215. #477Dic61semidirect
216. #460Dic61semidirect
217. #460Dic61semidirect
236. C61#9164semidirect

Maximal subgroups

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

subgroupindex
#214549082
#214456492
#214444302
#214443142
#214437172
#214436762
C61×#31122
#916461