Finite Group Databasebeta

← #29649359

Subgroups — #29649359

Order 15616.

592 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454949C211yesmaximal, characteristic
2#21454940C211yesmaximal, characteristic
3#21454780C211yesmaximal, characteristic
4#21444608C211yesmaximal, characteristic, complemented
5#21444598C212yesmaximal, complemented
6#21444588C212yesmaximal, complemented
7#21444585C212yesmaximal, complemented
8#21444574C211yesmaximal, characteristic, complemented
9#21444568C212yesmaximal, complemented
10#21444565C211yesmaximal, characteristic, complemented
11C61×#1574C211yesmaximal, characteristic, complemented
12C4×#11758222C2211yescharacteristic
13#16839892—21no—
14#16839889C411yescharacteristic
15#16839889C411yescharacteristic
16#16839886—21no—
17#16839839—21no—
18#16839839C411yescharacteristic
19#16839839C411yescharacteristic
20#16839838—22no—
21#16839836—22no—
22#16839830C412yes—
23#16839830C412yes—
24C8×D244—21no—
25#16839818C2212yes—
26#16839815C2211yescharacteristic
27#16839814C2212yes—
28#16839813C2212yes—
29#16839786C2211yescharacteristic
30#16839784C2212yes—
31C4⋊C8×D61C2212yes—
32C4×#11758189C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839515C2212yes—
35#16839514C2211yescharacteristic
36#16839514C2211yescharacteristic
37#16839513C2212yes—
38#16839512C2211yescharacteristic
39#16839512C2211yescharacteristic
40#16839512C2212yes—
41#16839512C2212yes—
42C61×#431C2211yescharacteristic
43C61×#431C2211yescharacteristic
44C61×#423C2211yescharacteristic
45C61×#335C2212yes—
46C61×#334C2212yes—
47C2×#11758222Q811yescharacteristic
48C2×#11758194—21no—
49C2×#11758189—22no—
50C2×#11758189C2×C412yes—
51C2×#11758189C2×C412yes—
52C22×#92418C2×C411yescharacteristic
53C22×#92418C2×C411yescharacteristic
54C2×C8×D61—22no—
55C4×Dic122C2×C411yescharacteristic
56C4×Dic122D411yescharacteristic
57C4×D244C2×C411yescharacteristic
58C4×D244D411yescharacteristic
59C4×C22⋊D61C2×C412yes—
60C4×C22⋊D61D412yes—
61#14519736Q811yescharacteristic
62#14519734Q812yes—
63C42×D61C2312yes—
64C4⋊#92418—21no—
65C4⋊#92418C2312yes—
66C4⋊#92418C2311yescharacteristic
67C4⋊#92418C2311yescharacteristic
68#14519706—22no—
69#14519706—22no—
70#14519706C2×C412yes—
71#14519706C2×C412yes—
72C22⋊#92418—21no—
73#14519689—22no—
74#14519689—22no—
75#14519689C2×C412yes—
76#14519689C2×C412yes—
77#14519687—22no—
78#14519687C2×C412yes—
79#14519687C2×C412yes—
80C4×#92418—21no—
81C4×#92418C2×C411yescharacteristic
82C4×#92418C2×C411yescharacteristic
83C4×#92418C2311yescharacteristic
84C4×#92418C2311yescharacteristic
85C8×Dic61—22no—
86C2×M16×C61—21no—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61—21no—
89C4⋊C8×C61C2×C411yescharacteristic
90C4⋊C8×C61C2×C411yescharacteristic
91C4⋊C8×C61C2311yescharacteristic
92C4⋊C8×C61C2312yes—
93C4⋊C8×C61C2311yescharacteristic
94C22⋊C8×C61—21no—
95C22⋊C8×C61C2×C412yes—
96C4×C8×C61—21no—
97C4×C8×C61C2312yes—
98C22×C4×C61C22×C411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×Q811yescharacteristic
101C2×C22⋊D61C4⋊C412yes—
102#11758222—42no—
103C2×D244C4⋊C411yescharacteristic
104C2×C4×D61C22×C412yes—
105C2×C4×D61C2×D412yes—
106C2×C4×D61C2×Q812yes—
107C2×Dic122C4⋊C411yescharacteristic
108M16×C61—81no—
109C2×C8×C61—21no—
110C2×C8×C61—21no—
111C2×C8×C61—22no—
112C2×C8×C61C22×C412yes—
113C2×C8×C61C22×C412yes—
114C42×C61C22×C411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61C2×Q811yescharacteristic
117C42×C61C2411yescharacteristic
118C22⋊Dic61C4⋊C412yes—
119#11758198C4⋊C412yes—
120#11758198C4⋊C412yes—
121C4⋊Dic61C4⋊C411yescharacteristic
122C4⋊Dic61C4⋊C411yescharacteristic
123#11758196C4⋊C412yes—
124#11758196C4⋊C412yes—
125C4×Dic61C22×C412yes—
126C4×Dic61C2×D412yes—
127C4×Dic61C2×Q812yes—
128#11758194—81no—
129C2×#92418—22no—
130C2×#92418—21no—
131C2×#92418—21no—
132C2×#92418—22no—
133C2×#92418—41no—
134C2×#92418C22×C411yescharacteristic
135C2×#92418C22×C411yescharacteristic
136C2×#92418C22×C411yescharacteristic
137C2×#92418C22×C411yescharacteristic
138#11758189—24no—
139#11758189—42no—
140#11758189—82no—
141C8×D61—82no—
142C23×C61C2×C4⋊C411yescharacteristic
143C22×D61C2×C4⋊C412yes—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—22no—
149C2×C4×C61C2×C4⋊C411yescharacteristic
150C2×C4×C61C2×C4⋊C411yescharacteristic
151C2×C4×C61C2×C4⋊C411yescharacteristic
152C2×C4×C61C23×C411yescharacteristic
153C2×C4×C61C22×D411yescharacteristic
154C2×C4×C61C22×Q811yescharacteristic
155C22⋊D61—42no—
156C2×Dic61C2×C4⋊C412yes—
157C2×Dic61C2×C4⋊C412yes—
158C2×Dic61C2×C4⋊C412yes—
159D244—22no—
160C4×D61—22no—
161C4×D61—22no—
162C4×D61—22no—
163C4×D61—22no—
164C4×D61—42no—
165Dic122—22no—
166C8×C61—22no—
167C8×C61—22no—
168C8×C61—41no—
169C8×C61—42no—
170C8×C61—41no—
171#92418—21no—
172#92418—21no—
173#92418—21no—
174#92418—42no—
175#92418—41no—
176#92418—41no—
177#92418#13112yes—
178#17841—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61C22×C4⋊C411yescharacteristic
183C2×D61—22no—
184C2×D61—22no—
185C2×D61—22no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61#43012yes—
190C4×C61C2×#13111yescharacteristic
191C4×C61#56811yescharacteristic
192C4×C61#57711yescharacteristic
193C4×C61#57711yescharacteristic
194Dic61—22no—
195Dic61—22no—
196Dic61—22no—
197#2991—611no—
198#2990—611no—
199#2965—611no—
200#1574—611no—
201#1574D6111yescharacteristic, complemented
202#1572—612no—
203#1570—612no—
204#1569—612no—
205#1566—611no—
206#1565—612no—
207#1564—611no—
208C2×C61—41no—
209C2×C61#290611yescharacteristic
210C2×C61#291211yescharacteristic
211C2×C61C2×#43011yescharacteristic
212D61—42no—
213C4×#41—611no—
214#446—611no—
215#446—611no—
216#446—1221no—
217C8×Q8—1221no—
218#436—611no—
219#436—611no—
220#436—1221no—
221#436—1222no—
222#435—612no—
223#435—612no—
224#435—1222no—
225C8×D4—1221no—
226#432—612no—
227#431—612no—
228#431—611no—
229#431—612no—
230#431C2×D6111yescharacteristic
231#431C2×D6111yescharacteristic
232#423—611no—
233#423—612no—
234#423C2×D6111yescharacteristic
235C2×C4⋊C8—612no—
236C4×M16—612no—
237C2×C4×C8—611no—
238#335—611no—
239#335—611no—
240#335—612no—
241#335C2×D6112yes—
242#334—611no—
243#334—611no—
244#334—612no—
245#334—612no—
246#334—612no—
247#334C2×D6112yes—
248C61#1784111yescharacteristic, complemented, Sylow
249C2×#41—611no—
250C2×M16—21no—
251C2×M16—612no—
252C2×M16—612no—
253C2×M16—1222no—
254C2×M16—1221no—
255C22×C8—611no—
256C22×C8—611no—
257C22×C8—1222no—
258C4×Q8—611no—
259C4×Q8—611no—
260C4×D4—612no—
261C4×D4—612no—
262C4×D4—611no—
263C4×D4—611no—
264#117—611no—
265#117—612no—
266C2×C42—612no—
267C2×C42C22×D6111yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—612no—
270C4⋊C8—612no—
271C4⋊C8—612no—
272C4⋊C8—611no—
273C4⋊C8—611no—
274C4⋊C8—1222no—
275C4⋊C8—1222no—
276C4⋊C8—1221no—
277C4⋊C8C4×D6111yescharacteristic
278C4⋊C8C4×D6111yescharacteristic
279C4⋊C8C22×D6112yes—
280C4⋊C8C22×D6111yescharacteristic
281C4⋊C8C22×D6111yescharacteristic
282C22⋊C8—21no—
283C22⋊C8—612no—
284C22⋊C8—612no—
285C22⋊C8—1222no—
286C22⋊C8—1222no—
287C22⋊C8—1221no—
288C22⋊C8C4×D6112yes—
289#97—612no—
290#97—612no—
291#97—1222no—
292C4×C8—21no—
293C4×C8—611no—
294C4×C8—611no—
295C4×C8—611no—
296C4×C8—611no—
297C4×C8—1222no—
298C4×C8—1221no—
299C4×C8C22×D6112yes—
300#41—2442no—
301C2×Q8—611no—
302C2×D4—612no—
303C2×D4—611no—
304C22×C4—612no—
305C22×C4—612no—
306C22×C4—612no—
307C22×C4C2×C4×D6111yescharacteristic
308C22×C4D4×D6111yescharacteristic
309C22×C4Q8×D6111yescharacteristic
310M16—81no—
311M16—1224no—
312M16—4882no—
313M16—4881no—
314M16—2442no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—611no—
319C2×C8—611no—
320C2×C8—611no—
321C2×C8—611no—
322C2×C8—1222no—
323C2×C8—1221no—
324C2×C8—1221no—
325C2×C8—1222no—
326C2×C8—4882no—
327C2×C8—2441no—
328C2×C8C2×C4×D6112yes—
329C2×C8C2×C4×D6112yes—
330C4⋊C4—612no—
331C4⋊C4—612no—
332C4⋊C4—611no—
333C4⋊C4—611no—
334C22⋊C4—612no—
335C22⋊C4—612no—
336C22⋊C4—612no—
337C42—612no—
338C42—612no—
339C42—612no—
340C42C2×C4×D6111yescharacteristic
341C42D4×D6111yescharacteristic
342C42Q8×D6111yescharacteristic
343C42C23×D6111yescharacteristic
344C23—612no—
345C23C4⋊C4×D6111yescharacteristic
346Q8—1222no—
347D4—1222no—
348D4—2442no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—612no—
355C2×C4—612no—
356C2×C4—612no—
357C2×C4—1222no—
358C2×C4—1222no—
359C2×C4—1222no—
360C2×C4—1222no—
361C2×C4—2442no—
362C2×C4C4⋊C4×D6111yescharacteristic
363C2×C4C4⋊C4×D6111yescharacteristic
364C2×C4C4⋊C4×D6111yescharacteristic
365C2×C4C22×C4×D6111yescharacteristic
366C2×C4C2×D4×D6111yescharacteristic
367C2×C4C2×Q8×D6111yescharacteristic
368C8—22no—
369C8—22no—
370C8—41no—
371C8—41no—
372C8—42no—
373C8—612no—
374C8—1221no—
375C8—1221no—
376C8—1221no—
377C8—2441no—
378C8—2442no—
379C8—2441no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—1222no—
384C22—1222no—
385C22—1222no—
386C22C2×C4⋊C4×D6111yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—1222no—
391C4—1222no—
392C4—1222no—
393C4#1683980912yes—
394C4#131×D6111yescharacteristic
395C4#1684052711yescharacteristic
396C4#1684057111yescharacteristic
397C4#1684057211yescharacteristic
398C2—41no—
399C2—2442no—
400C2#2145438611yescharacteristic
401C2#2145441311yescharacteristic
402C2C2×#1683980911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1574C2
socleC22×C61C22×C4⋊C4
prime radicalC2×#11758222Q8
cosocleC42×C61C24
hypercenter#1574D61
hypocenterC61#17841
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
4. #21444608C2semidirect
5. #21444598C2semidirect
6. #21444588C2semidirect
7. #21444585C2semidirect
8. #21444574C2semidirect
9. #21444568C2semidirect
10. #21444565C2semidirect
11. C61×#1574C2semidirect
201. #1574D61semidirect
248. C61#17841semidirect

Maximal subgroups

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

subgroupindex
#214549492
#214549402
#214547802
#214446082
#214445982
#214445882
#214445852
#214445742
#214445682
#214445652
C61×#15742
#1784161