Finite Group Databasebeta

← #29835015

Subgroups — #29835015

Order 15616.

552 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456387C211yesmaximal, characteristic
2C4×Q16×D61C211yesmaximal, characteristic, complemented
3#21448018C211yesmaximal, characteristic, complemented
4#21447998C211yesmaximal, characteristic
5#21447986C211yesmaximal, characteristic, complemented
6#21447985C211yesmaximal, characteristic, complemented
7#21447880C211yesmaximal, characteristic, complemented
8#21447879C211yesmaximal, characteristic, complemented
9#21447754C211yesmaximal, characteristic, complemented
10#21447734C211yesmaximal, characteristic
11C4×Dic488C211yesmaximal, characteristic, complemented
12C4×#14519723C211yesmaximal, characteristic, complemented
13C4×#14519721C211yesmaximal, characteristic, complemented
14#21447616C211yesmaximal, characteristic, complemented
15C4×SD32×C61C211yesmaximal, characteristic, complemented
16C2×Q16×D61C2211yescharacteristic
17#16840294—21no—
18C4×Q8×D61—21no—
19C2×Dic488C2211yescharacteristic
20C2×#14519723C2211yescharacteristic
21C2×#14519721C2211yescharacteristic
22#16840163C2211yescharacteristic
23#16840145C2211yescharacteristic
24#16840115—21no—
25#16840004C2211yescharacteristic
26#16839995C2211yescharacteristic
27#16839990—21no—
28C4×Dic244C2211yescharacteristic
29C4×Dic244C2211yescharacteristic
30Q16×Dic61C2211yescharacteristic
31#16839856C2211yescharacteristic
32C4×#11758202—21no—
33#16839854—21no—
34C4×#11758200—21no—
35#107×D61C2211yescharacteristic
36#103×D61—21no—
37C4×C8×D61C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839565C2211yescharacteristic
47#16839565C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839541C2211yescharacteristic
51C4×#11758185C2211yescharacteristic
52C2×SD32×C61C2211yescharacteristic
53C4×Q16×C61C2211yescharacteristic
54C4×D8×C61C2211yescharacteristic
55C61×#367C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C4×C16×C61C2211yescharacteristic
59C2×Q8×D61—21no—
60C2×Dic244C2311yescharacteristic
61C2×Dic244C2311yescharacteristic
62C2×#11758202—21no—
63Q16×D61—22no—
64Q16×D61—22no—
65C2×#11758200—21no—
66C2×C8×D61D411yescharacteristic
67C2×C8×D61C2311yescharacteristic
68#14519770—41no—
69#14519768—21no—
70#14519764—21no—
71#14519759—41no—
72#14519757—21no—
73#14519748—21no—
74Q8×Dic61—21no—
75C4×Dic122—21no—
76C4×Dic122—21no—
77C4×Dic122—41no—
78#14519742—21no—
79#14519742—41no—
80C4×C22⋊D61—41no—
81#14519736—41no—
82C4⋊C4×D61—21no—
83C4⋊C4×D61—41no—
84C42×D61—41no—
85C42×D61D411yescharacteristic
86Dic488—22no—
87#14519723—22no—
88#14519721—22no—
89C2×#11758185C2311yescharacteristic
90C2×#11758185C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519710—21no—
94#14519708—21no—
95#14519702—21no—
96#14519702—21no—
97#14519701—21no—
98#14519700—21no—
99#14519700—21no—
100#14519698—21no—
101C4×#92418D411yescharacteristic
102C8×Dic61D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C2×Q16×C61C2311yescharacteristic
105C2×D8×C61C2311yescharacteristic
106C4×Q8×C61—21no—
107C4×D4×C61—21no—
108SD32×C61—22no—
109C2×C16×C61C2311yescharacteristic
110C2×C16×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#103×C61—21no—
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C2×Q8×C61—21no—
117C2×D4×C61—21no—
118C22×C4×C61—41no—
119C2×C22⋊D61—41no—
120Q8×D61—24no—
121Q8×D61—42no—
122C2×C4×D61—41no—
123C2×C4×D61—41no—
124C2×C4×D61D811yescharacteristic
125C2×C4×D61D811yescharacteristic
126C2×C4×D61C2×D411yescharacteristic
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×Dic122—41no—
130Q16×C61—21no—
131Q16×C61#4112yes—
132D8×C61—21no—
133D8×C61—21no—
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2411yescharacteristic
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—41no—
139C22⋊C4×C61—41no—
140C42×C61—41no—
141C42×C61C2×D411yescharacteristic
142C22⋊Dic61—41no—
143C22⋊Dic61—41no—
144C22⋊Dic61—41no—
145#11758202—24no—
146#11758200—42no—
147#11758198—41no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—41no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153#11758196—41no—
154#11758196—41no—
155#11758196—41no—
156#11758196—41no—
157#11758196—41no—
158C4×Dic61—41no—
159C4×Dic61—41no—
160C4×Dic61—41no—
161C4×Dic61—41no—
162C4×Dic61D811yescharacteristic
163C4×Dic61D811yescharacteristic
164C4×Dic61C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167Dic244—21no—
168Dic244—21no—
169Dic244—21no—
170Dic244#4112yes—
171C8×D61—21no—
172C8×D61—21no—
173C8×D61—41no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C22×D61C2×D811yescharacteristic
180Q8×C61—22no—
181Q8×C61—41no—
182D4×C61—41no—
183D4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—82no—
188C2×C4×C61C2×D811yescharacteristic
189C2×C4×C61C2×D811yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C22⋊D61—82no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61C2×D811yescharacteristic
197C2×Dic61C2×D811yescharacteristic
198C2×Dic61C2×D811yescharacteristic
199C4×D61—22no—
200C4×D61—21no—
201C4×D61—21no—
202C4×D61—22no—
203C4×D61—44no—
204C4×D61—82no—
205Dic122—22no—
206Dic122—44no—
207Dic122—41no—
208Dic122—41no—
209Dic122—41no—
210C8×C61—21no—
211C8×C61C2×#4111yescharacteristic
212C8×C61C2×#4111yescharacteristic
213#92418—21no—
214#92418—21no—
215#33668—611nomaximal, Sylow
216C22×C61—81no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61C22×D811yescharacteristic
220C2×D61—21no—
221C2×D61—21no—
222C2×D61#50812yes—
223C4×C61—21no—
224C4×C61—42no—
225C4×C61—81no—
226C4×C61—81no—
227C4×C61#51012yes—
228C4×C61#52511yescharacteristic
229C4×C61#57511yescharacteristic
230Dic61—21no—
231Dic61—21no—
232Dic61—21no—
233Dic61—42no—
234Dic61—81no—
235Dic61—81no—
236Dic61—81no—
237#3168—611no—
238C2×C4×Q16—611no—
239#2235—611no—
240#2232—611no—
241#2229—611no—
242#2229—611no—
243#2211—611no—
244#2211—611no—
245#2188—611no—
246#2185—611no—
247C4×Q32—611no—
248C4×Q32—611no—
249C4×SD32—611no—
250C4×SD32D6111yescharacteristic, complemented
251#2162—611no—
252C2×C61—161no—
253C2×C61#310011yescharacteristic
254C2×C61C2×#50811yescharacteristic
255C2×C61C2×#51011yescharacteristic
256D61—22no—
257C22×Q16—611no—
258#529—1221no—
259C2×C4×Q8—1221no—
260C2×Q32—611no—
261C2×Q32—611no—
262C2×SD32—611no—
263C2×SD32C2×D6111yescharacteristic
264#499—611no—
265#495—611no—
266#488—1221no—
267#467—611no—
268#466—611no—
269#465—1221no—
270C4×Q16—611no—
271C4×Q16—611no—
272C4×Q16—611no—
273C4×Q16—611no—
274C4×Q16—1221no—
275C4×Q16—1221no—
276C4×Q16C2×D6111yescharacteristic
277C4×SD16—1221no—
278C4×D8C2×D6111yescharacteristic
279C2×#107—611no—
280C2×#103—1221no—
281C2×C4×C8—611no—
282#367—611no—
283#367C2×D6111yescharacteristic
284#366—611no—
285#366—611no—
286C4⋊C16—611no—
287C4⋊C16—611no—
288#358—611no—
289#358—611no—
290#358—611no—
291#358—611no—
292#358—611no—
293#358C2×D6111yescharacteristic
294#357—611no—
295#357C2×D6111yescharacteristic
296C22⋊C16—611no—
297C22⋊C16—611no—
298C4×C16—611no—
299C4×C16C2×D6111yescharacteristic
300C61#3366811yescharacteristic, complemented, Sylow
301C22×Q8—1221no—
302C2×Q16—611no—
303C2×Q16—611no—
304C2×Q16—1221no—
305C2×Q16—1222no—
306C2×Q16—1222no—
307C2×Q16C22×D6111yescharacteristic
308C2×SD16—1221no—
309C2×D8C22×D6111yescharacteristic
310C22×C8—611no—
311C22×C8—611no—
312#126—2441no—
313#125—1221no—
314#124—1221no—
315#123—2441no—
316C22⋊Q8—1221no—
317#121—1221no—
318C4×Q8—21no—
319C4×Q8—1221no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×Q8—1221no—
323C4×Q8—2441no—
324C4×Q8—2441no—
325C4×D4—21no—
326C4×D4—2441no—
327#117—2441no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—2441no—
330C2×C42—611no—
331C2×C42—2441no—
332Q32—1222no—
333Q32—1222no—
334SD32—22no—
335SD32—1222no—
336C2×C16—611no—
337C2×C16—611no—
338C2×C16C22×D6111yescharacteristic
339C2×C16C22×D6111yescharacteristic
340#107—611no—
341#107—611no—
342#107—1221no—
343#107C22×D6111yescharacteristic
344#107C22×D6111yescharacteristic
345#106—1221no—
346#103—21no—
347#103—1221no—
348#103—1221no—
349#103—1221no—
350#103—1221no—
351#103—1221no—
352#102—21no—
353#102—1221no—
354C4×C8—611no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8C22×D6111yescharacteristic
358C2×Q8—21no—
359C2×Q8—1224no—
360C2×Q8—1221no—
361C2×Q8—1221no—
362C2×Q8—2441no—
363C2×Q8—2442no—
364C2×D4—21no—
365C2×D4—2441no—
366C22×C4—41no—
367C22×C4—611no—
368C22×C4—611no—
369C22×C4—611no—
370C22×C4—2441no—
371C22×C4—2441no—
372Q16—21no—
373Q16—612no—
374Q16—1224no—
375Q16—1221no—
376Q16—1221no—
377Q16—1221no—
378Q16#1175822212yes—
379SD16—2442no—
380D8—21no—
381D8—21no—
382C2×C8—611no—
383C2×C8—611no—
384C2×C8—1221no—
385C2×C8—1221no—
386C2×C8—2441no—
387C2×C8D4×D6111yescharacteristic
388C2×C8C23×D6111yescharacteristic
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—41no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C4⋊C4—2441no—
398C4⋊C4—2441no—
399C4⋊C4—2441no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C22⋊C4—41no—
403C22⋊C4—2441no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C22⋊C4—2441no—
407C42—41no—
408C42—611no—
409C42—611no—
410C42—611no—
411C42—2441no—
412C42—2441no—
413C42—2441no—
414C42—2441no—
415C42D4×D6111yescharacteristic
416C16—21no—
417C16—21no—
418C16—1221no—
419C16—1221no—
420C23—41no—
421C23—611no—
422Q8—22no—
423Q8—41no—
424Q8—1222no—
425Q8—2444no—
426Q8—2441no—
427Q8—2441no—
428Q8—2441no—
429D4—41no—
430D4—41no—
431D4—4882no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—82no—
436C2×C4—611no—
437C2×C4—611no—
438C2×C4—611no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—4882no—
444C2×C4—2444no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4D8×D6111yescharacteristic
450C2×C4D8×D6111yescharacteristic
451C2×C4C2×D4×D6111yescharacteristic
452C8—21no—
453C8—1221no—
454C8—1221no—
455C8#41×D6111yescharacteristic
456C8C2×#1175822211yescharacteristic
457C22—81no—
458C22—81no—
459C22—81no—
460C22—612no—
461C22—1221no—
462C22—1221no—
463C22C2×D8×D6111yescharacteristic
464C4—21no—
465C4—42no—
466C4—81no—
467C4—81no—
468C4—1221no—
469C4—1221no—
470C4—1221no—
471C4—4881no—
472C4—4881no—
473C4—4881no—
474C4—2442no—
475C4#1684020112yes—
476C4#1684027111yescharacteristic
477C4#1684055911yescharacteristic
478C2—161no—
479C2—1222no—
480C2#2145581711yescharacteristic
481C2#508×D6111yescharacteristic
482C2C2×#1684020111yescharacteristic
subgroupG/H
centerC2×C4D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC4×SD32×C61C2
socleC22×C61C22×D8
prime radical#16839995C22
hypercenterC4×SD32D61
hypocenterC61#33668
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
2. C4×Q16×D61C2semidirect
3. #21448018C2semidirect
5. #21447986C2semidirect
6. #21447985C2semidirect
7. #21447880C2semidirect
8. #21447879C2semidirect
9. #21447754C2semidirect
11. C4×Dic488C2semidirect
12. C4×#14519723C2semidirect
13. C4×#14519721C2semidirect
14. #21447616C2semidirect
15. C4×SD32×C61C2semidirect
250. C4×SD32D61semidirect
300. C61#33668semidirect

Maximal subgroups

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

subgroupindex
#214563872
C4×Q16×D612
#214480182
#214479982
#214479862
#214479852
#214478802
#214478792
#214477542
#214477342
C4×Dic4882
C4×#145197232
C4×#145197212
#214476162
C4×SD32×C612
#3366861