Finite Group Databasebeta

← #29576261

Subgroups — #29576261

Order 15616.

686 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#21455823C211yesmaximal, characteristic
2C4×#14519739C211yesmaximal, characteristic
3#21446797C211yesmaximal, characteristic, complemented
4#21444245C211yesmaximal, characteristic
5#21443722C211yesmaximal, characteristic, complemented
6#21443651C211yesmaximal, characteristic, complemented
7C61×#1503C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9C2×#14519739—21no—
10C2×C42×D61—21no—
11#16840090—21no—
12#16840049C412yescomplemented
13#16839964C412yescomplemented
14#16839912—21no—
15C4×#11758201—22no—
16#16839815C2211yescharacteristic
17#16839786C2211yescharacteristic
18C4⋊C4×Dic61—21no—
19C4×#11758198—21no—
20#16839609—22no—
21C42×Dic61C2211yescharacteristic
22#16839521—21no—
23#16839521—21no—
24#16839491—22no—
25C61×#533C2211yescharacteristic
26C61×#431C2211yescharacteristic
27C4×C4⋊C4×C61C2211yescharacteristic
28C61×#328—22no—
29C22×D244D411yescharacteristic
30C22×C4×D61—22no—
31C22×C4×D61—21no—
32C2×#11758201—22no—
33C2×#11758194—21no—
34C4×D244—21no—
35C4×D244—21no—
36C4×D244C2×C412yes—
37C4×D244D412yes—
38#14519739—28no—
39C2×C4⋊Dic61D411yescharacteristic
40C2×#11758198—22no—
41C2×#11758198—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—22no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47C42×D61—42no—
48C42×D61—42no—
49#14519708—22no—
50C4⋊#92418—21no—
51C4⋊#92418—21no—
52C4⋊#92418C2×C412yescomplemented
53#14519701—22no—
54#14519697—22no—
55C22⋊#92418—21no—
56C4×#92418—21no—
57#14519684—22no—
58C4⋊Q8×C61C2×C411yescharacteristic, complemented
59C4⋊Q8×C61C2×C411yescharacteristic, complemented
60#125×C61—21no—
61C22⋊Q8×C61—21no—
62C22⋊Q8×C61—21no—
63C4×Q8×C61—21no—
64C4×Q8×C61—21no—
65#117×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C42×C61—21no—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71C4×C8×C61—21no—
72#95×C61D412yes—
73C23×D61—21no—
74C2×Q8×C61—21no—
75C2×Q8×C61—21no—
76C22×C4×C61—22no—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—21no—
83C2×D244—21no—
84C2×D244—24no—
85C2×D244C22⋊C412yes—
86C2×C4×D61—28no—
87C2×C4×D61—22no—
88C2×C4×D61—22no—
89C2×C4×D61—44no—
90C2×C4×D61—44no—
91C2×C4×D61—41no—
92C2×C4×D61—41no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C8×C61—41no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C42×C61—41no—
105C42×C61—41no—
106C42×C61—41no—
107C42×C61C22×C411yescharacteristic
108C42×C61C2×D411yescharacteristic
109C42×C61#4111yescharacteristic
110C42×C61#4111yescharacteristic
111#11758201—44no—
112#11758198—28no—
113#11758198—22no—
114#11758198—22no—
115C4⋊Dic61—21no—
116C4⋊Dic61C22⋊C412yes—
117C4×Dic61—24no—
118C4×Dic61—21no—
119C4×Dic61—21no—
120C4×Dic61—21no—
121C4×Dic61—44no—
122C4×Dic61—42no—
123C4×Dic61SD1612yescomplemented
124#11758194—41no—
125#11758194—41no—
126C2×#92418—22no—
127C2×#92418—41no—
128C23×C61#12011yescharacteristic
129C22×D61—24no—
130C22×D61—22no—
131C22×D61—42no—
132C22×D61—41no—
133C22×D61—41no—
134Q8×C61—42no—
135Q8×C61—81no—
136C2×C4×C61—22no—
137C2×C4×C61—24no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61C2×C22⋊C411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C2×C4×C61C4×D411yescharacteristic
146C2×C4×C61#12111yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12311yescharacteristic
149C2×C4×C61C2×SD1611yescharacteristic
150C2×C4×C61#13511yescharacteristic
151C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C61#13711yescharacteristic
153C2×Dic61—22no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162D244—24no—
163D244—22no—
164D244#10414yes—
165C4×D61—416no—
166C4×D61—42no—
167C4×D61—42no—
168C4×D61—42no—
169C4×D61—42no—
170C8×C61—81no—
171#92418—42no—
172#92418—81no—
173#9231—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61#38611yescharacteristic
177C22×C61#45011yescharacteristic
178C22×C61#45211yescharacteristic
179C2×D61—24no—
180C2×D61—44no—
181C2×D61—44no—
182C2×D61—42no—
183C2×D61—42no—
184C4×C61—21no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—44no—
188C4×C61—81no—
189C4×C61—81no—
190C4×C61C2×#10412yes—
191C4×C61C4×SD1611yescharacteristic
192C4×C61#44011yescharacteristic
193C4×C61#46111yescharacteristic
194C4×C61#46311yescharacteristic
195Dic61—24no—
196Dic61—42no—
197Dic61—42no—
198#3101—611no—
199C42×D4—611no—
200#1994—611no—
201#1503D6111yescharacteristic, complemented
202#1486—611no—
203#1303—611no—
204#1276—611no—
205C2×C61—21no—
206C2×C61—21no—
207C2×C61#187711yescharacteristic
208C2×C61#189611yescharacteristic
209C2×C61#190611yescharacteristic
210D61—44no—
211#533C2×D6111yescharacteristic
212C2×C4×D4—611no—
213C2×C4×D4—1221no—
214C22×C42—1221no—
215#483—1221no—
216#476—612no—
217#461—612no—
218#450—1221no—
219C4×SD16—1222no—
220#431—611no—
221#431C2×D6111yescharacteristic
222#423—611no—
223C4×C4⋊C4—1221no—
224C4×C4⋊C4C2×D6111yescharacteristic
225C4×C22⋊C4—1221no—
226C4×C22⋊C4—1222no—
227C43—611no—
228#339—1221no—
229#339—1221no—
230#328—22no—
231D4⋊C8—1222no—
232C61#923111yescharacteristic, complemented, Sylow
233C22×D4—611no—
234C23×C4—1222no—
235C23×C4—1221no—
236C2×SD16—1222no—
237C2×M16—1221no—
238C4⋊Q8C4×D6111yescharacteristic, complemented
239C4⋊Q8C4×D6111yescharacteristic, complemented
240#125—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×D4—612no—
246C4×D4—612no—
247C4×D4—1228no—
248C4×D4—1221no—
249C4×D4—1221no—
250#117—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—611no—
253C2×C22⋊C4—1222no—
254C2×C22⋊C4—1221no—
255C2×C42—21no—
256C2×C42—611no—
257C2×C42—611no—
258C2×C42—1221no—
259C2×C42—1222no—
260C2×C42—1221no—
261C2×C42—2442no—
262C2×C42—2442no—
263C2×C42C22×D6111yescharacteristic
264#106—1222no—
265C4⋊C8—21no—
266C4⋊C8—612no—
267C4⋊C8—1221no—
268C4⋊C8—1221no—
269#103—1222no—
270#102—1222no—
271C22⋊C8—21no—
272C22⋊C8—1221no—
273C4×C8—21no—
274C4×C8—1221no—
275#95—1222no—
276#95C22⋊D6112yes—
277C24—1221no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×D4—612no—
281C2×D4—1224no—
282C2×D4—1221no—
283C22×C4—22no—
284C22×C4—1228no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—1222no—
289C22×C4—1222no—
290C22×C4—2444no—
291C22×C4—2444no—
292C22×C4—2441no—
293C22×C4—2441no—
294C22×C4—2441no—
295C22×C4—2441no—
296C22×C4D4×D6111yescharacteristic
297C22×C4D4×D6111yescharacteristic
298C22×C4C2×C22⋊D6111yescharacteristic
299SD16—2444no—
300M16—2441no—
301M16—2441no—
302C2×C8—41no—
303C2×C8—1222no—
304C2×C8—2441no—
305C4⋊C4—24no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—612no—
313C4⋊C4—1221no—
314C22⋊C4—41no—
315C22⋊C4—1228no—
316C22⋊C4—1222no—
317C22⋊C4—1222no—
318C42—41no—
319C42—41no—
320C42—41no—
321C42—612no—
322C42—1224no—
323C42—1221no—
324C42—1221no—
325C42—1221no—
326C42—2444no—
327C42—2442no—
328C42C2×C4×D6111yescharacteristic
329C42#1175822211yescharacteristic
330C42D4×D6111yescharacteristic
331C42#1175822411yescharacteristic
332C23—1224no—
333C23—1222no—
334C23—2442no—
335C23—2441no—
336C23—2441no—
337C23#1451974511yescharacteristic
338Q8—42no—
339Q8—81no—
340D4—614no—
341D4—1224no—
342D4—1222no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—1222no—
351C2×C4—1222no—
352C2×C4—1222no—
353C2×C4—1222no—
354C2×C4—1222no—
355C2×C4—24416no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4C22⋊C4×D6111yescharacteristic
365C2×C4#1451973711yescharacteristic
366C2×C4C4×C22⋊D6111yescharacteristic
367C2×C4#1451974811yescharacteristic
368C2×C4#1451975011yescharacteristic
369C2×C4#1451975911yescharacteristic
370C2×C4SD16×D6111yescharacteristic
371C2×C4#1451980011yescharacteristic
372C2×C4#1451980411yescharacteristic
373C2×C4#1451980911yescharacteristic
374C8—81no—
375C8—4881no—
376C8—2442no—
377C22—21no—
378C22—21no—
379C22—1224no—
380C22—2444no—
381C22—2444no—
382C22—2442no—
383C22—2442no—
384C22#1683963811yescharacteristic
385C22#1683991111yescharacteristic
386C22#1683992311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—44no—
391C4—81no—
392C4—81no—
393C4—1224no—
394C4—2442no—
395C4—2442no—
396C4#104×D6112yes—
397C4#1683984711yescharacteristic
398C4#1683986511yescharacteristic
399C4#1683996611yescharacteristic
400C4#1683998011yescharacteristic
401C2—21no—
402C2—21no—
403C2—2444no—
404C2#2144615011yescharacteristic
405C2#2144625911yescharacteristic
406C2#2144631711yescharacteristic
subgroupG/H
centerC2×C4#14519750
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1503C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1503D61
hypocenterC61#9231
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
3. #21446797C2semidirect
5. #21443722C2semidirect
6. #21443651C2semidirect
7. C61×#1503C2semidirect
12. #16840049C4semidirect
13. #16839964C4semidirect
52. C4⋊#92418C2×C4semidirect
58. C4⋊Q8×C61C2×C4semidirect
59. C4⋊Q8×C61C2×C4semidirect
123. C4×Dic61SD16semidirect
201. #1503D61semidirect
232. C61#9231semidirect
238. C4⋊Q8C4×D61semidirect
239. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214558232
C4×#145197392
#214467972
#214442452
#214437222
#214436512
C61×#15032
#923161