Finite Group Databasebeta

← #29656291

Subgroups — #29656291

Order 15616.

1156 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456272C211yesmaximal, characteristic, complemented
2#21455299C212yesmaximal, complemented
3#21444902C212yesmaximal, complemented
4#21444897C212yesmaximal, complemented
5#21444884C214yesmaximal, complemented
6#21444554C211yesmaximal, characteristic, complemented
7#21444258C212yesmaximal, complemented
8C61×#3483C211yesmaximal, characteristic, complemented
9C2×#14519802—22no—
10C2×#14519802—22no—
11C2×#14519774C2211yescharacteristic
12#16840152—22no—
13#16840138C2212yes—
14#16840136C2212yes—
15#16839962—22no—
16#16839962C2214yes—
17#16839957—22no—
18#16839957C2214yes—
19#16839786C2212yes—
20#16839766—22no—
21C4×#11758194C2211yescharacteristic
22#16839511C2212yescomplemented
23#16839506C2212yescomplemented
24Q8⋊#92418C2214yescomplemented
25D4⋊#92418C2214yescomplemented
26Q82×C61C2211yescharacteristic, complemented
27C61×#555—22no—
28C61×#555—22no—
29C61×#555C2211yescharacteristic, complemented
30C61×#550—22no—
31C61×#550C2212yescomplemented
32C61×#532—22no—
33C61×#532—22no—
34C61×#532C2211yescharacteristic
35C4×#41×C61—22no—
36C4×#41×C61C2212yes—
37C22×D244—22no—
38C22×D244—21no—
39#14519802—48no—
40#14519802—48no—
41C2×#11758201—24no—
42C2×#11758201—24no—
43C2×#11758199—24no—
44C2×#11758199—24no—
45C2×#11758194—21no—
46C2×#11758194—22no—
47#14519774—22no—
48#14519774—21no—
49#14519774C2312yes—
50C4⋊#92418—22no—
51C4⋊#92418C2314yes—
52#14519697—24no—
53#14519697—24no—
54#14519688—21no—
55C4×#92418C2312yes—
56C2×#41×C61—22no—
57C2×#41×C61—22no—
58C2×#41×C61D412yes—
59C2×#41×C61D412yes—
60C4⋊Q8×C61—22no—
61C4⋊Q8×C61—22no—
62C4⋊Q8×C61C2311yescharacteristic
63#127×C61—22no—
64#127×C61—22no—
65#127×C61C2311yescharacteristic
66#124×C61—22no—
67#124×C61—24no—
68#124×C61—22no—
69#124×C61D412yes—
70#121×C61—24no—
71#121×C61—24no—
72#121×C61—22no—
73#121×C61—24no—
74#121×C61—22no—
75#121×C61—22no—
76C4×Q8×C61—22no—
77C4×Q8×C61D412yes—
78C4×Q8×C61C2312yes—
79C4×D4×C61—24no—
80C4×D4×C61—24no—
81C4×D4×C61—22no—
82C4×D4×C61—22no—
83C4×D4×C61—22no—
84C4×D4×C61—22no—
85C4×D4×C61D412yes—
86C4×D4×C61C2312yes—
87#117×C61—24no—
88#117×C61—22no—
89#117×C61—22no—
90#117×C61D412yes—
91C2×C42×C61—22no—
92C2×C42×C61—22no—
93C2×C42×C61C2311yescharacteristic
94C23×D61—41no—
95#41×C61—24no—
96#41×C61—42no—
97#41×C61—44no—
98#41×C61—44no—
99#41×C61—44no—
100#41×C61—84no—
101C2×Q8×C61—22no—
102C2×Q8×C61C2×D412yes—
103C2×Q8×C61C2×D412yes—
104C2×D4×C61—24no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—24no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61C2×D412yes—
111C2×D4×C61C2×D412yes—
112C22×C4×C61—24no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C2×D244—24no—
120C2×D244—22no—
121C2×D244—44no—
122C2×D244—44no—
123C2×D244—44no—
124C2×D244—42no—
125C2×D244—41no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61C2×D412yes—
131C4⋊C4×C61C2×D412yes—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—24no—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—24no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C42×C61—22no—
139C42×C61—22no—
140C42×C61—22no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61C2×D412yes—
144C42×C61C2×D411yescharacteristic
145C42×C61C2411yescharacteristic
146#11758201—48no—
147#11758201—48no—
148#11758199—48no—
149#11758199—48no—
150#11758194—24no—
151#11758194—44no—
152C2×#92418—22no—
153C2×#92418—24no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61#12011yescharacteristic
157C22×D61—42no—
158C22×D61—84no—
159C22×D61—82no—
160Q8×C61—22no—
161Q8×C61—42no—
162Q8×C61—82no—
163Q8×C61AGL(1,ℤ/8)14yescomplemented
164D4×C61—22no—
165D4×C61—22no—
166D4×C61—44no—
167D4×C61—42no—
168D4×C61—48no—
169D4×C61—42no—
170D4×C61—42no—
171D4×C61—82no—
172D4×C61—82no—
173D4×C61—84no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—44no—
182C2×C4×C61—48no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—44no—
186C2×C4×C61—44no—
187C2×C4×C61#12012yes—
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D412yes—
191D244—44no—
192D244—44no—
193D244—44no—
194D244—84no—
195D244—82no—
196#92418—24no—
197#92418—44no—
198#18548—611nomaximal, Sylow
199C22×C61—22no—
200C22×C61—21no—
201C22×C61—42no—
202C22×C61—44no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61C2×#12011yescharacteristic
207C2×D61—84no—
208C2×D61—82no—
209C2×D61—164no—
210C4×C61—22no—
211C4×C61—24no—
212C4×C61—21no—
213C4×C61—42no—
214C4×C61—42no—
215C4×C61—42no—
216C4×C61#45312yes—
217C4×C61C2×AGL(1,ℤ/8)12yes—
218C4×C61#57512yes—
219#3483D6111yescharacteristic, complemented
220#3151—611no—
221#3038—612no—
222#1624—612no—
223#1623—612no—
224#1621—614no—
225#1560—611no—
226#1489—612no—
227C2×C61—41no—
228C2×C61—82no—
229C2×C61—82no—
230C2×C61#300612yes—
231C2×C61C2×#45311yescharacteristic
232D61—162no—
233C2×AGL(1,ℤ/8)—1222no—
234C2×AGL(1,ℤ/8)—1222no—
235Q82C2×D6111yescharacteristic, complemented
236#555—22no—
237#555—22no—
238#555C2×D6111yescharacteristic, complemented
239#550—22no—
240#550C2×D6112yescomplemented
241#532—22no—
242#532—22no—
243#532C2×D6111yescharacteristic
244C2×#127—611no—
245C4×#41—22no—
246C4×#41C2×D6112yes—
247#496—1222no—
248#493—612no—
249#492—612no—
250#460—614no—
251#460—1222no—
252#459—614no—
253#459—1222no—
254#423—612no—
255#418—1222no—
256C4×M16—611no—
257#333—612no—
258#331—612no—
259Q8⋊C8—614no—
260D4⋊C8—614no—
261C61#1854811yescharacteristic, complemented, Sylow
262C2×#41—22no—
263C2×#41—22no—
264C2×#41C22⋊D6112yes—
265C2×#41C22⋊D6112yes—
266C22×D4—1221no—
267C22×D4—1222no—
268AGL(1,ℤ/8)—2448no—
269AGL(1,ℤ/8)—2448no—
270C2×SD16—1224no—
271C2×SD16—1224no—
272C2×D8—1224no—
273C2×D8—1224no—
274C2×M16—1221no—
275C2×M16—1222no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8C22×D6111yescharacteristic
279#127—22no—
280#127—22no—
281#127—612no—
282#127—1222no—
283#127—1221no—
284#127C22×D6111yescharacteristic
285#124—22no—
286#124—24no—
287#124—22no—
288#124C22⋊D6112yes—
289#121—24no—
290#121—24no—
291#121—22no—
292#121—24no—
293#121—22no—
294#121—22no—
295C4×Q8—22no—
296C4×Q8C22⋊D6112yes—
297C4×Q8C22×D6112yes—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4C22⋊D6112yes—
305C4×D4C22×D6112yes—
306#117—24no—
307#117—22no—
308#117—22no—
309#117C22⋊D6112yes—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42C22×D6111yescharacteristic
313C4⋊C8—614no—
314C4⋊C8—1222no—
315#102—1224no—
316#102—1224no—
317#97—1221no—
318C4×C8—612no—
319C24—2441no—
320#41—24no—
321#41—42no—
322#41—44no—
323#41—44no—
324#41—44no—
325#41—84no—
326C2×Q8—22no—
327C2×Q8C2×C22⋊D6112yes—
328C2×Q8C2×C22⋊D6112yes—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—1224no—
336C2×D4—1222no—
337C2×D4—2444no—
338C2×D4—2444no—
339C2×D4—2444no—
340C2×D4—2442no—
341C2×D4—2441no—
342C2×D4C2×C22⋊D6112yes—
343C2×D4C2×C22⋊D6112yes—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4C2×C22⋊D6112yes—
350C22×C4C2×C22⋊D6111yescharacteristic
351SD16—2448no—
352SD16—2448no—
353D8—2448no—
354D8—2448no—
355M16—1224no—
356M16—2444no—
357C2×C8—1222no—
358C2×C8—1224no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4C2×C22⋊D6112yes—
364C4⋊C4C2×C22⋊D6112yes—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—24no—
370C22⋊C4—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42C2×C22⋊D6111yescharacteristic
377C42C2×C22⋊D6112yes—
378C42C23×D6111yescharacteristic
379C23—22no—
380C23—22no—
381C23—4882no—
382C23—4884no—
383C23—2442no—
384C23#1451974611yescharacteristic
385Q8—22no—
386Q8—42no—
387Q8—82no—
388Q8#1451980214yescomplemented
389D4—22no—
390D4—22no—
391D4—44no—
392D4—42no—
393D4—48no—
394D4—42no—
395D4—42no—
396D4—82no—
397D4—82no—
398D4—84no—
399D4—4884no—
400D4—4882no—
401D4—2444no—
402D4—2444no—
403D4—2444no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4#1451974612yes—
418C2×C4#1451974611yescharacteristic
419C2×C4C22×C22⋊D6111yescharacteristic
420C2×C4C22×C22⋊D6112yes—
421C8—1224no—
422C8—2444no—
423C22—21no—
424C22—22no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—44no—
429C22—42no—
430C22—9764no—
431C22—4882no—
432C22—4884no—
433C22C2×#1451974611yescharacteristic
434C4—21no—
435C4—22no—
436C4—24no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4#1683993012yes—
441C4C2×#1451980212yes—
442C4#1684056012yes—
443C2—41no—
444C2—82no—
445C2—82no—
446C2—9762no—
447C2#2145507112yes—
448C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3483C2
socleC22×C61C2×#120
hypercenter#3483D61
hypocenterC61#18548
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3483

Complements

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

NH ≅ G/Ntype
1. #21456272C2semidirect
2. #21455299C2semidirect
3. #21444902C2semidirect
4. #21444897C2semidirect
5. #21444884C2semidirect
6. #21444554C2semidirect
7. #21444258C2semidirect
8. C61×#3483C2semidirect
22. #16839511C22semidirect
23. #16839506C22semidirect
24. Q8⋊#92418C22semidirect
25. D4⋊#92418C22semidirect
26. Q82×C61C22semidirect
29. C61×#555C22semidirect
31. C61×#550C22semidirect
163. Q8×C61AGL(1,ℤ/8)semidirect
219. #3483D61semidirect
235. Q82C2×D61semidirect
238. #555C2×D61semidirect
240. #550C2×D61semidirect
261. C61#18548semidirect
388. Q8#14519802semidirect

Maximal subgroups

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

subgroupindex
#214562722
#214552992
#214449022
#214448972
#214448842
#214445542
#214442582
C61×#34832
#1854861