Finite Group Databasebeta

← #29663878

Subgroups — #29663878

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456396C211yesmaximal, characteristic
2#21455940C212yesmaximal, complemented
3C8×Q8×D61C211yesmaximal, characteristic, complemented
4#21445234C212yesmaximal, complemented
5#21445233C212yesmaximal, complemented
6#21445224C211yesmaximal, characteristic, complemented
7#21445223C211yesmaximal, characteristic, complemented
8#21445176C211yesmaximal, characteristic, complemented
9#21444581C211yesmaximal, characteristic, complemented
10#21444511C212yesmaximal, complemented
11C61×#1668C211yesmaximal, characteristic, complemented
12#16840310C2212yes—
13C4×Q8×D61C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840108C2212yes—
16#16840074—22no—
17#16840058C2212yes—
18#16840057C2212yes—
19#16839995—21no—
20#16839995—21no—
21#16839959C2212yes—
22#16839899—22no—
23Q8×#92418C2211yescharacteristic
24C8×Dic122—21no—
25C8×Dic122C2211yescharacteristic
26#16839886—21no—
27#16839886C2211yescharacteristic
28#16839863—22no—
29#16839859—22no—
30D8×Dic61—21no—
31D8×Dic61—21no—
32#16839818C2212yes—
33C4⋊C8×D61—21no—
34C4⋊C8×D61C2211yescharacteristic
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839510C2212yes—
40#16839510C2212yes—
41#16839506C2212yes—
42C61×#493C2211yescharacteristic
43C61×#460C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#331C2212yes—
47C2×#11758224—22no—
48C2×Q8×D61D411yescharacteristic
49C2×#11758200—22no—
50C2×C8×D61—21no—
51C2×C8×D61D411yescharacteristic
52C2×C8×D61D411yescharacteristic
53#14519777C2312yes—
54#14519764—22no—
55#14519748—22no—
56#14519748—22no—
57Q8×Dic61D411yescharacteristic
58C4×Dic122—21no—
59C4×Dic122C2311yescharacteristic
60#14519742—21no—
61#14519742D411yescharacteristic
62D4×Dic61—22no—
63D4×Dic61—22no—
64C4⋊C4×D61—21no—
65C4⋊C4×D61D411yescharacteristic
66C42×D61—21no—
67C42×D61C2311yescharacteristic
68#14519711—21no—
69#14519711—21no—
70#14519711—22no—
71#14519710—22no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75#14519706—21no—
76#14519706—21no—
77#14519706—21no—
78#14519702—22no—
79#14519701—22no—
80#14519698—22no—
81#14519698—22no—
82#14519698—22no—
83#14519689—22no—
84#14519687—22no—
85C4×#92418—21no—
86C4×#92418C2311yescharacteristic
87C8×Dic61—21no—
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×SD16×C61—22no—
92C2×D8×C61—21no—
93C2×D8×C61—21no—
94#127×C61C2312yes—
95C4×Q8×C61C2311yescharacteristic
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#102×C61—22no—
100C4×C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61C2×D411yescharacteristic
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×C22⋊D61—42no—
107C22×Dic61—42no—
108Q8×D61—22no—
109Q8×D61—22no—
110#11758224—84no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61C2×D411yescharacteristic
114C2×C4×D61C2×D411yescharacteristic
115C2×C4×D61#4111yescharacteristic
116C2×Dic122—21no—
117C2×Dic122—22no—
118C2×Dic122C2×D411yescharacteristic
119SD16×C61—44no—
120D8×C61—42no—
121D8×C61—42no—
122C2×C8×C61—21no—
123C2×C8×C61—22no—
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C4⋊C4×C61—21no—
127C4⋊C4×C61C2×D411yescharacteristic
128C42×C61—21no—
129C42×C61C2411yescharacteristic
130C22⋊Dic61—42no—
131C22⋊Dic61—42no—
132#11758200—44no—
133C4⋊Dic61—21no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136C4⋊Dic61C2×D411yescharacteristic
137#11758196—21no—
138#11758196—21no—
139#11758196—21no—
140C4×Dic61—21no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143C4×Dic61C2×D411yescharacteristic
144C4×Dic61C2×D411yescharacteristic
145C4×Dic61#4111yescharacteristic
146C2×#92418—21no—
147C2×#92418—22no—
148C2×#92418C2×D411yescharacteristic
149C2×#92418C2×D411yescharacteristic
150C8×D61—22no—
151C8×D61—41no—
152C8×D61—42no—
153C23×C61—42no—
154C22×D61#12111yescharacteristic
155Q8×C61—21no—
156Q8×C61#13512yes—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—42no—
160D4×C61—42no—
161D4×C61—82no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C22⋊D61—84no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—42no—
171C2×Dic61—84no—
172C2×Dic61#12111yescharacteristic
173C2×Dic61#12111yescharacteristic
174C2×Dic61#12111yescharacteristic
175C4×D61—24no—
176C4×D61—22no—
177C4×D61—21no—
178C4×D61—21no—
179C4×D61—21no—
180C4×D61—21no—
181C4×D61—42no—
182Dic122—24no—
183Dic122—21no—
184Dic122—82no—
185Dic122#13512yes—
186C8×C61—21no—
187C8×C61—42no—
188C8×C61—41no—
189C8×C61#13712yes—
190#92418—21no—
191#92418—21no—
192#92418—42no—
193#92418—41no—
194#19264—611nomaximal, Sylow
195C22×C61—82no—
196C22×C61—82no—
197C22×C61—82no—
198C22×C61C2×#12111yescharacteristic
199C2×D61—21no—
200C2×D61—21no—
201C2×D61#47212yes—
202C4×C61—22no—
203C4×C61—21no—
204C4×C61—41no—
205C4×C61C2×#13511yescharacteristic
206C4×C61C2×#13511yescharacteristic
207C4×C61C2×#13711yescharacteristic
208C4×C61#57511yescharacteristic
209Dic61—21no—
210Dic61—21no—
211Dic61—22no—
212Dic61—21no—
213Dic61—41no—
214Dic61—82no—
215#3169—611no—
216#3112—612no—
217C2×C8×Q8—611no—
218#1678—612no—
219#1678—612no—
220#1676—611no—
221#1676—611no—
222#1668—611no—
223#1668D6111yescharacteristic, complemented
224#1568—611no—
225#1551—612no—
226C2×C61—162no—
227C2×C61#303711yescharacteristic
228C2×C61#305011yescharacteristic
229C2×C61C2×#47211yescharacteristic
230D61—22no—
231#532—612no—
232C2×C4×Q8—611no—
233#500—611no—
234#493C2×D6111yescharacteristic
235#486—612no—
236#480—1222no—
237#477—612no—
238#477—612no—
239#466—1221no—
240#466—1221no—
241#460—612no—
242#460C2×D6112yes—
243#448—1222no—
244C8×Q8—611no—
245C8×Q8—611no—
246C8×Q8—611no—
247C8×Q8—1221no—
248C8×Q8—1221no—
249C8×Q8C2×D6111yescharacteristic
250#440—1222no—
251#440—1222no—
252C4×D8—1221no—
253C4×D8—1221no—
254#432—612no—
255C2×C4⋊C8—611no—
256C2×C4⋊C8—1221no—
257C2×C4×C8—611no—
258C2×C4×C8—1221no—
259#335—611no—
260#335C2×D6111yescharacteristic
261#334—612no—
262#333—612no—
263#333—612no—
264#331—612no—
265#331C2×D6112yes—
266C61#1926411yescharacteristic, complemented, Sylow
267C2×#41—1222no—
268C22×Q8—611no—
269C2×SD16—22no—
270C2×SD16—1222no—
271C2×D8—21no—
272C2×D8—21no—
273C22×C8—611no—
274C22×C8—611no—
275C22×C8—1221no—
276C4⋊Q8—612no—
277#127C22×D6112yes—
278#124—1222no—
279#121—1222no—
280#121—1222no—
281C4×Q8—611no—
282C4×Q8—611no—
283C4×Q8—611no—
284C4×Q8—1221no—
285C4×Q8—1221no—
286C4×Q8C22×D6111yescharacteristic
287C4×D4—1222no—
288C4×D4—1222no—
289C2×C4⋊C4—611no—
290C2×C4⋊C4—1221no—
291C2×C42—611no—
292C2×C42—1221no—
293#107—1222no—
294#107—1221no—
295#107—1221no—
296#107—1222no—
297C4⋊C8—21no—
298C4⋊C8—611no—
299C4⋊C8—612no—
300C4⋊C8—1221no—
301C4⋊C8—1221no—
302C4⋊C8—1221no—
303C4⋊C8—1221no—
304C4⋊C8C22×D6111yescharacteristic
305C4⋊C8C22×D6112yes—
306#103—1222no—
307#103—1222no—
308#102—22no—
309#102—1222no—
310#102—1222no—
311#102—1222no—
312C22⋊C8—1222no—
313#97—1222no—
314C4×C8—21no—
315C4×C8—611no—
316C4×C8—611no—
317C4×C8—611no—
318C4×C8—1221no—
319C4×C8—1221no—
320C4×C8—1221no—
321C4×C8C22×D6111yescharacteristic
322#41—4884no—
323C2×Q8—611no—
324C2×Q8—1221no—
325C2×Q8—1222no—
326C2×Q8—1222no—
327C2×Q8—1222no—
328C2×Q8D4×D6111yescharacteristic
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—2442no—
333C22×C4—611no—
334C22×C4—611no—
335C22×C4—611no—
336C22×C4—1221no—
337C22×C4—1221no—
338C22×C4—2442no—
339SD16—44no—
340SD16—2444no—
341D8—42no—
342D8—42no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—611no—
346C2×C8—611no—
347C2×C8—1222no—
348C2×C8—1221no—
349C2×C8—1222no—
350C2×C8—2441no—
351C2×C8—2442no—
352C2×C8D4×D6111yescharacteristic
353C2×C8D4×D6111yescharacteristic
354C4⋊C4—21no—
355C4⋊C4—611no—
356C4⋊C4—1221no—
357C4⋊C4—1221no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C4⋊C4—1222no—
361C4⋊C4—1222no—
362C4⋊C4D4×D6111yescharacteristic
363C22⋊C4—2442no—
364C22⋊C4—2442no—
365C42—21no—
366C42—611no—
367C42—611no—
368C42—611no—
369C42—1221no—
370C42—1221no—
371C42—1221no—
372C42C23×D6111yescharacteristic
373C23—42no—
374C23—611no—
375Q8—21no—
376Q8—612no—
377Q8—1224no—
378Q8—1221no—
379Q8—4882no—
380Q8#1451979612yes—
381D4—42no—
382D4—42no—
383D4—42no—
384D4—42no—
385D4—82no—
386D4—4884no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—611no—
390C2×C4—611no—
391C2×C4—611no—
392C2×C4—1222no—
393C2×C4—1224no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—1221no—
399C2×C4—1221no—
400C2×C4—4884no—
401C2×C4—2442no—
402C2×C4—2442no—
403C2×C4C2×D4×D6111yescharacteristic
404C2×C4C2×D4×D6111yescharacteristic
405C2×C4#41×D6111yescharacteristic
406C8—21no—
407C8—42no—
408C8—41no—
409C8—1221no—
410C8—1221no—
411C8—2442no—
412C8—2441no—
413C8#1451980612yes—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—612no—
418C22—1221no—
419C22—1221no—
420C22#121×D6111yescharacteristic
421C4—22no—
422C4—21no—
423C4—41no—
424C4—1221no—
425C4—1222no—
426C4—1221no—
427C4—1221no—
428C4—4882no—
429C4—2441no—
430C4#135×D6111yescharacteristic
431C4C2×#1451979611yescharacteristic
432C4C2×#1451980611yescharacteristic
433C4#1684055811yescharacteristic
434C2—162no—
435C2—1222no—
436C2#2145528611yescharacteristic
437C2#2145539811yescharacteristic
438C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1668C2
socleC22×C61C2×#121
prime radical#21456396C2
hypercenter#1668D61
hypocenterC61#19264
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1668

Complements

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

NH ≅ G/Ntype
2. #21455940C2semidirect
3. C8×Q8×D61C2semidirect
4. #21445234C2semidirect
5. #21445233C2semidirect
6. #21445224C2semidirect
7. #21445223C2semidirect
8. #21445176C2semidirect
9. #21444581C2semidirect
10. #21444511C2semidirect
11. C61×#1668C2semidirect
223. #1668D61semidirect
266. C61#19264semidirect

Maximal subgroups

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

subgroupindex
#214563962
#214559402
C8×Q8×D612
#214452342
#214452332
#214452242
#214452232
#214451762
#214445812
#214445112
C61×#16682
#1926461