Finite Group Databasebeta

← #29663129

Subgroups — #29663129

Order 15616.

650 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#21456949C211yesmaximal, characteristic
2#21456947C211yesmaximal, characteristic
3C2×#16840165C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic
5#21445341C214yesmaximal, complemented
6#21445217C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444501C211yesmaximal, characteristic, complemented
9C61×#1548C211yesmaximal, characteristic, complemented
10C2×#14519785—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13#16840165—21no—
14#16840165—21no—
15#16840165C2212yes—
16#16840165C2212yes—
17#16840160—21no—
18#16840158—21no—
19#16840110—21no—
20#16840109—21no—
21#16840081—22no—
22#16840080—22no—
23#16840053C2214yes—
24#16840049C2214yes—
25#16839839—21no—
26#16839839C2212yes—
27#16839827—21no—
28#16839827C2212yes—
29#16839815C2211yescharacteristic
30C2×#14519710—21no—
31C2×#14519710—21no—
32#16839730—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yes—
35#16839512C2212yes—
36#16839510C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#333C2214yes—
41C2×#11758222—21no—
42C22×Dic122—21no—
43#14519785—44no—
44C2×#11758189—22no—
45C22×#92418D411yescharacteristic
46C22×#92418D411yescharacteristic
47C2×C8×D61—22no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777D411yescharacteristic
52#14519777C2312yes—
53#14519774D411yescharacteristic
54#14519765—21no—
55#14519751—21no—
56#14519751D412yes—
57C4×D244—21no—
58C4×D244C2312yes—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61#14519710—22no—
62#14519710—22no—
63#14519710—22no—
64#14519710—22no—
65#14519709—24no—
66#14519708—24no—
67C4⋊#92418C2312yes—
68#14519699—24no—
69#14519697—24no—
70C22⋊#92418—21no—
71#14519689—22no—
72#14519689—22no—
73C4×#92418—21no—
74C4×#92418D411yescharacteristic
75C4×#92418D411yescharacteristic
76C4×#92418C2311yescharacteristic
77C4×#92418C2311yescharacteristic
78C22×Q8×C61—21no—
79C2×M16×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C42×C61C2311yescharacteristic
87C4⋊C8×C61—21no—
88C4⋊C8×C61C2312yes—
89C4⋊C8×C61C2312yes—
90C22⋊C8×C61—21no—
91C4×C8×C61—21no—
92C2×Q8×C61—22no—
93C2×Q8×C61—41no—
94C2×Q8×C61—82no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C2×C22⋊D61—22no—
100C22×Dic61—41no—
101#11758222—42no—
102#11758222—42no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244C2×D412yes—
106C2×C4×D61—22no—
107C2×Dic122—21no—
108C2×Dic122—22no—
109C2×Dic122—41no—
110C2×Dic122—82no—
111M16×C61—42no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C42×C61C2411yescharacteristic
122#11758198—22no—
123#11758198—22no—
124C4⋊Dic61—21no—
125C4⋊Dic61—22no—
126C4⋊Dic61—22no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129C4⋊Dic61C2×D412yes—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418SD1614yes—
133C2×#92418C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136C2×#92418C2×D411yescharacteristic
137#11758189—44no—
138C8×D61—44no—
139C23×C61#12111yescharacteristic
140C22×D61—22no—
141Q8×C61—82no—
142Q8×C61—82no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—42no—
146C2×C4×C61—82no—
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61C2×SD1612yes—
151C2×C4×C61#13712yes—
152C2×C4×C61C22×D411yescharacteristic
153C2×C4×C61C22×D411yescharacteristic
154C2×C4×C61C2×#4111yescharacteristic
155C22⋊D61—44no—
156C2×Dic61—22no—
157C2×Dic61—42no—
158C2×Dic61—82no—
159D244—24no—
160D244—42no—
161D244—42no—
162C4×D61—42no—
163C4×D61—42no—
164Dic122—42no—
165Dic122—82no—
166Dic122—82no—
167C8×C61—42no—
168C8×C61—42no—
169#92418—21no—
170#92418—21no—
171#92418—42no—
172#92418C2×SD1612yes—
173#92418C2×SD1612yes—
174#19193—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61#46112yes—
178C22×C61C2×#12111yescharacteristic
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—42no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—82no—
185C4×C61#46112yes—
186C4×C61#47212yes—
187C4×C61C22×SD1611yescharacteristic
188C4×C61C2×#13711yescharacteristic
189C4×C61#57812yes—
190Dic61—42no—
191Dic61—82no—
192#3227—611no—
193#3227—611no—
194C2×#500—611no—
195#2931—611no—
196#1695—614no—
197#1675—614no—
198#1564—611no—
199#1548—611no—
200#1548D6111yescharacteristic, complemented
201C2×C61—22no—
202C2×C61C2×#46111yescharacteristic
203C2×C61#305911yescharacteristic
204C2×C61C2×#47211yescharacteristic
205D61—82no—
206C2×#131—1221no—
207#532—611no—
208C2×C4⋊Q8—611no—
209C2×C4⋊Q8C2×D6111yescharacteristic
210#500—612no—
211#500—612no—
212#500—1221no—
213#500—1221no—
214#498—1221no—
215#498—1221no—
216#486—1221no—
217#486—1221no—
218#481—1222no—
219#481—1222no—
220#476—614no—
221#476—614no—
222#436—612no—
223#436—1221no—
224C8×D4—612no—
225C8×D4—1221no—
226#431—611no—
227#431C2×D6111yescharacteristic
228C2×#107—1221no—
229C2×#107—1221no—
230#423C2×D6111yescharacteristic
231#408—1221no—
232C2×C4×C8—611no—
233#335—612no—
234#334—612no—
235#333—614no—
236#333C2×D6114yes—
237C61#1919311yescharacteristic, complemented, Sylow
238C2×#41—1221no—
239C22×Q8—21no—
240C22×Q8—1221no—
241#131—2444no—
242C2×M16—21no—
243C2×M16—1222no—
244C22×C8—611no—
245C22×C8—611no—
246C22×C8—1222no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—611no—
251C4⋊Q8—612no—
252C4⋊Q8—1221no—
253C4⋊Q8—1221no—
254C4⋊Q8—1221no—
255C4⋊Q8C22×D6112yes—
256#127—611no—
257#124—1221no—
258#121—612no—
259#121—1221no—
260C4×D4—612no—
261C4×D4—1221no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C42C22×D6111yescharacteristic
267#107—1222no—
268#107—1222no—
269#107—1222no—
270#107—1222no—
271#106—1224no—
272#106—1224no—
273C4⋊C8—21no—
274C4⋊C8—612no—
275C4⋊C8C22×D6112yes—
276C4⋊C8C22×D6112yes—
277#102—1224no—
278#102—1224no—
279C22⋊C8—21no—
280C22⋊C8—1222no—
281C22⋊C8—1222no—
282C22⋊C8—1221no—
283C4×C8—21no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—1221no—
289#41—2442no—
290#41—2442no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—82no—
294C2×Q8—1221no—
295C2×Q8—1222no—
296C2×Q8—4882no—
297C2×Q8—2441no—
298C2×D4—612no—
299C2×D4—1222no—
300C2×D4—1221no—
301C2×D4—1221no—
302C22×C4—41no—
303C22×C4—1222no—
304C22×C4—2441no—
305C22×C4#1175822211yescharacteristic
306C22×C4D4×D6111yescharacteristic
307C22×C4D4×D6111yescharacteristic
308M16—42no—
309M16—2444no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—611no—
313C2×C8—611no—
314C2×C8—614no—
315C2×C8—611no—
316C2×C8—611no—
317C2×C8—1222no—
318C2×C8—1222no—
319C2×C8—2444no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—612no—
325C4⋊C4—1221no—
326C4⋊C4—1222no—
327C4⋊C4—1222no—
328C4⋊C4—2441no—
329C4⋊C4—2441no—
330C22⋊C4—1222no—
331C22⋊C4—1222no—
332C42#1175822211yescharacteristic
333C42D4×D6111yescharacteristic
334C42D4×D6111yescharacteristic
335C42C23×D6111yescharacteristic
336C23—1222no—
337C23#1451975011yescharacteristic
338Q8—82no—
339Q8—82no—
340Q8—4882no—
341Q8—4882no—
342Q8—2442no—
343D4—1224no—
344D4—2444no—
345D4—2442no—
346D4—2442no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—1222no—
352C2×C4—4882no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4#1451975011yescharacteristic
357C2×C4#1451975011yescharacteristic
358C2×C4#1451975011yescharacteristic
359C2×C4SD16×D6112yes—
360C2×C4#1451980912yes—
361C2×C4C2×D4×D6111yescharacteristic
362C2×C4C2×D4×D6111yescharacteristic
363C2×C4C2×#1175822211yescharacteristic
364C8—42no—
365C8—42no—
366C8—612no—
367C8—612no—
368C8—1221no—
369C8—1221no—
370C8—2442no—
371C22—21no—
372C22—21no—
373C22—2442no—
374C22—2442no—
375C22—2442no—
376C22#1683996612yes—
377C22C2×#1451975011yescharacteristic
378C4—21no—
379C4—21no—
380C4—82no—
381C4—4882no—
382C4—2442no—
383C4#1683996612yes—
384C4#1684003112yes—
385C4C2×SD16×D6111yescharacteristic
386C4C2×#1451980911yescharacteristic
387C4#1684057812yes—
388C2—22no—
389C2—4882no—
390C2C2×#1683996611yescharacteristic
391C2#2145547311yescharacteristic
392C2C2×#1684003111yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1548C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1548D61
hypocenterC61#19193
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C2×#16840165C2semidirect
5. #21445341C2semidirect
6. #21445217C2semidirect
7. #21444565C2semidirect
8. #21444501C2semidirect
9. C61×#1548C2semidirect
200. #1548D61semidirect
237. C61#19193semidirect

Maximal subgroups

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

subgroupindex
#214569492
#214569472
C2×#168401652
#214545302
#214453412
#214452172
#214445652
#214445012
C61×#15482
#1919361