Finite Group Databasebeta

← #29665310

Subgroups — #29665310

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456734C211yesmaximal, characteristic
2#21455287C211yesmaximal, characteristic, complemented
3C2×#16839963C211yesmaximal, characteristic, complemented
4#21454899C211yesmaximal, characteristic, complemented
5#21445266C214yesmaximal, complemented
6#21445178C214yesmaximal, complemented
7#21444593C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#1540C211yesmaximal, characteristic, complemented
10C2×#14519807—21no—
11C22×#11758190—21no—
12C2×#14519774C2211yescharacteristic
13C2×C4×Dic122C2211yescharacteristic
14#16840138C2212yes—
15#16840110C2212yes—
16#16839981—22no—
17#16839969C2212yescomplemented
18#16839963—22no—
19#16839963C2214yes—
20#16839959C2212yescomplemented
21#16839957—22no—
22#16839895—22no—
23C8×Dic122—22no—
24#16839886C2214yescomplemented
25#16839815C2211yescharacteristic
26#16839812—22no—
27#16839786C2211yescharacteristic
28#16839763—21no—
29C2×#14519699—21no—
30#16839514C2214yescomplemented
31#16839505C2214yes—
32C61×#532C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#332C2212yescomplemented
36C61×#331C2212yescomplemented
37C22×Dic122D411yescharacteristic
38C22×D244—21no—
39C22×D244—21no—
40C22×D244—21no—
41#14519807—44no—
42C2×#11758202—22no—
43C2×#11758190—24no—
44C2×#11758190—44no—
45C2×#11758200—22no—
46C2×#11758199—24no—
47C2×#11758194—21no—
48#14519774—21no—
49#14519774—21no—
50#14519774—21no—
51#14519774C2312yes—
52C4×Dic122—21no—
53C4×Dic122—21no—
54C4×Dic122D412yes—
55C4×Dic122C2312yes—
56C2×C4⋊Dic61D411yescharacteristic
57C2×#11758196—21no—
58C2×C4×Dic61—21no—
59C4⋊#92418—21no—
60C4⋊#92418D412yes—
61C4⋊#92418C2312yes—
62#14519706—24no—
63#14519701—22no—
64#14519699—24no—
65#14519698—22no—
66#14519697—24no—
67C22⋊#92418D412yes—
68C4×#92418C2312yes—
69C8×Dic61—24no—
70C2×#41×C61—21no—
71C22×C8×C61—21no—
72C4⋊Q8×C61C2311yescharacteristic
73#127×C61C2311yescharacteristic
74#124×C61—21no—
75#121×C61—21no—
76#121×C61—21no—
77C4×D4×C61—21no—
78C4×D4×C61—21no—
79C2×C42×C61C2311yescharacteristic
80C4⋊C8×C61—21no—
81C4⋊C8×C61C2312yes—
82C4⋊C8×C61C2312yes—
83C22⋊C8×C61—21no—
84C4×C8×C61—21no—
85C23×D61—41no—
86#41×C61—42no—
87#41×C61—42no—
88C2×Q8×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C22×Dic61—21no—
98C22×Dic61—21no—
99C2×D244—22no—
100C2×D244—22no—
101C2×D244—22no—
102C2×D244—42no—
103C2×D244—42no—
104C2×D244—41no—
105C2×D244—41no—
106C2×D244—41no—
107C2×D244—82no—
108C2×D244—82no—
109C2×Dic122—21no—
110C2×Dic122—22no—
111C2×Dic122D814yescomplemented
112C2×Dic122C2×D412yes—
113C2×C8×C61—22no—
114C2×C8×C61—22no—
115C2×C8×C61—42no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C42×C61C2411yescharacteristic
124#11758202—44no—
125#11758200—44no—
126#11758199—84no—
127C4⋊Dic61—21no—
128C4⋊Dic61C2×D412yes—
129#11758196—22no—
130#11758196—22no—
131C4×Dic61—22no—
132C4×Dic61—22no—
133#11758194—42no—
134C2×#92418—22no—
135C2×#92418C2×D414yes—
136#11758190—48no—
137C23×C61—41no—
138C23×C61#12111yescharacteristic
139C22×D61—42no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143Q8×C61—41no—
144Q8×C61—41no—
145D4×C61—41no—
146D4×C61—41no—
147D4×C61—42no—
148D4×C61—81no—
149D4×C61—82no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—82no—
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61C2×D812yes—
158C2×C4×C61#13512yes—
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C2×#4111yescharacteristic
162C2×Dic61—22no—
163C2×Dic61—24no—
164C2×Dic61—22no—
165C2×Dic61—81no—
166D244—42no—
167D244—42no—
168D244—44no—
169D244—82no—
170D244—82no—
171D244—82no—
172D244—82no—
173Dic122—22no—
174Dic122—24no—
175Dic122C2×D814yes—
176C8×C61—42no—
177C8×C61—42no—
178#92418—24no—
179#92418—42no—
180#19398—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61#46612yes—
187C22×C61C2×#12111yescharacteristic
188C2×D61—82no—
189C2×D61—82no—
190C2×D61—82no—
191C2×D61—164no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—81no—
196C4×C61#46612yes—
197C4×C61C22×D811yescharacteristic
198C4×C61C2×#13511yescharacteristic
199C4×C61#57712yes—
200Dic61—24no—
201Dic61—42no—
202#3205—611no—
203#3037—611no—
204C2×#460—611no—
205#2982—611no—
206#1682—614no—
207#1668—614no—
208#1571—611no—
209#1545—611no—
210#1540D6111yescharacteristic, complemented
211C2×C61—22no—
212C2×C61—161no—
213C2×C61#304311yescharacteristic
214C2×C61C2×#46611yescharacteristic
215C2×C61#306811yescharacteristic
216D61—162no—
217C2×#137—1221no—
218C22×SD16—1221no—
219#532C2×D6111yescharacteristic
220C2×#127—611no—
221C2×C4×Q8—611no—
222#493—612no—
223#486—612no—
224#464—1222no—
225#462—612no—
226#460—612no—
227#460—614no—
228#460—1222no—
229#459—1222no—
230#447—1222no—
231C8×Q8—614no—
232C8×Q8—1222no—
233#431—611no—
234#431—1222no—
235#431C2×D6111yescharacteristic
236#423—611no—
237C2×C4⋊C8C2×D6111yescharacteristic
238#417—1221no—
239C2×#102—1221no—
240#335—614no—
241#332C2×D6112yescomplemented
242#331—614no—
243#331C2×D6112yescomplemented
244C61#1939811yescharacteristic, complemented, Sylow
245C2×#41—21no—
246C22×Q8—611no—
247C22×D4—1221no—
248C22×D4—1221no—
249C22×D4—1221no—
250#137—2444no—
251C2×Q16—1222no—
252C2×SD16—1224no—
253C2×SD16—1222no—
254C2×SD16—2444no—
255C2×D8—1224no—
256C2×M16—1221no—
257C22×C8—21no—
258C4⋊Q8C22×D6111yescharacteristic
259#127—612no—
260#127—1221no—
261#127—1221no—
262#127—1221no—
263#127C22×D6111yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—612no—
268C4×Q8—612no—
269C4×Q8—1221no—
270C4×Q8—1221no—
271C4×D4—21no—
272C4×D4—21no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—1221no—
275C2×C42—1221no—
276C2×C42C22×D6111yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—612no—
279C4⋊C8—612no—
280C4⋊C8—1224no—
281C4⋊C8—1221no—
282C4⋊C8C22×D6112yes—
283C4⋊C8C22×D6112yes—
284#103—1222no—
285#102—1224no—
286#102—1224no—
287#102—1222no—
288C22⋊C8—21no—
289C22⋊C8—612no—
290C4×C8—21no—
291C4×C8—612no—
292C4×C8—1224no—
293C24—2441no—
294#41—42no—
295#41—42no—
296C2×Q8—21no—
297C2×Q8—612no—
298C2×Q8—614no—
299C2×Q8—1221no—
300C2×Q8—1222no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—1222no—
306C2×D4—1222no—
307C2×D4—1222no—
308C2×D4—4882no—
309C2×D4—4882no—
310C2×D4—2441no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2442no—
314C2×D4—2442no—
315C22×C4—41no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4#1175822211yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4D4×D6111yescharacteristic
321Q16—2444no—
322SD16—2448no—
323SD16—2444no—
324D8—4884no—
325M16—2442no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—42no—
329C2×C8—614no—
330C2×C8—1222no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—612no—
334C4⋊C4—1222no—
335C4⋊C4—1221no—
336C4⋊C4—1222no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C42—1222no—
340C42—1222no—
341C42#1175822211yescharacteristic
342C42D4×D6111yescharacteristic
343C42D4×D6111yescharacteristic
344C42C23×D6111yescharacteristic
345C23—41no—
346C23—4882no—
347C23—4882no—
348C23—4882no—
349C23—2442no—
350C23#1451975011yescharacteristic
351Q8—41no—
352Q8—41no—
353Q8—614no—
354Q8—1224no—
355Q8—1222no—
356D4—41no—
357D4—41no—
358D4—42no—
359D4—81no—
360D4—82no—
361D4—4882no—
362D4—4882no—
363D4—4882no—
364D4—4882no—
365D4—2442no—
366D4—2442no—
367D4—2444no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—82no—
372C2×C4—1224no—
373C2×C4—1222no—
374C2×C4—1222no—
375C2×C4—4881no—
376C2×C4#1451975011yescharacteristic
377C2×C4#1451975011yescharacteristic
378C2×C4#1451975011yescharacteristic
379C2×C4D8×D6112yes—
380C2×C4#1451979812yes—
381C2×C4C2×D4×D6111yescharacteristic
382C2×C4C2×D4×D6111yescharacteristic
383C2×C4C2×#1175822211yescharacteristic
384C8—42no—
385C8—42no—
386C8—1224no—
387C8—2442no—
388C22—21no—
389C22—21no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—9764no—
394C22—4882no—
395C22—4882no—
396C22—4882no—
397C22#1683999712yes—
398C22C2×#1451975011yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—1224no—
404C4—2442no—
405C4#1683999712yes—
406C4C2×D8×D6111yescharacteristic
407C4C2×#1451979811yescharacteristic
408C4#1684056912yes—
409C2—22no—
410C2—161no—
411C2—9762no—
412C2#2145534311yescharacteristic
413C2C2×#1683999711yescharacteristic
414C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1540C2
socleC22×C61C2×#121
prime radical#21456734C2
hypercenter#1540D61
hypocenterC61#19398
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
2. #21455287C2semidirect
3. C2×#16839963C2semidirect
4. #21454899C2semidirect
5. #21445266C2semidirect
6. #21445178C2semidirect
7. #21444593C2semidirect
8. #21444486C2semidirect
9. C61×#1540C2semidirect
17. #16839969C22semidirect
20. #16839959C22semidirect
24. #16839886C22semidirect
30. #16839514C22semidirect
35. C61×#332C22semidirect
36. C61×#331C22semidirect
111. C2×Dic122D8semidirect
210. #1540D61semidirect
241. #332C2×D61semidirect
243. #331C2×D61semidirect
244. C61#19398semidirect

Maximal subgroups

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

subgroupindex
#214567342
#214552872
C2×#168399632
#214548992
#214452662
#214451782
#214445932
#214444862
C61×#15402
#1939861