Finite Group Databasebeta

← #29836881

Subgroups — #29836881

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458163C211yesmaximal, characteristic
2Q16×C22⋊D61C211yesmaximal, characteristic, complemented
3#21448045C211yesmaximal, characteristic, complemented
4#21448017C211yesmaximal, characteristic, complemented
5#21448015C211yesmaximal, characteristic, complemented
6#21447959C211yesmaximal, characteristic, complemented
7#21447913C211yesmaximal, characteristic, complemented
8#21447912C211yesmaximal, characteristic, complemented
9#21447862C211yesmaximal, characteristic, complemented
10#21447734C211yesmaximal, characteristic, complemented
11SD32×Dic61C211yesmaximal, characteristic, complemented
12#21447636C211yesmaximal, characteristic, complemented
13C2×#16839566C211yesmaximal, characteristic, complemented
14#358×D61C211yesmaximal, characteristic, complemented
15C61×#2185C211yesmaximal, characteristic, complemented
16C2×Q16×D61C2211yescharacteristic
17Q8×C22⋊D61—21no—
18#16840412—21no—
19C2×#14519721C2211yescharacteristic
20C22×#11758185C2211yescharacteristic
21C2×C16×D61C2211yescharacteristic
22#16840166C2211yescharacteristic
23#16840140C2211yescharacteristic
24#16840074—21no—
25#16840073—21no—
26#16840040—21no—
27#16840003C2211yescharacteristic
28#16840000C2211yescharacteristic
29#16839995C2211yescharacteristic
30#16839970—21no—
31#16839918—21no—
32#16839916—21no—
33Q16×Dic61C2211yescharacteristic
34D8×Dic61C2211yescharacteristic
35C8×C22⋊D61C2211yescharacteristic
36C2×#14519711C2211yescharacteristic
37#107×D61C2211yescharacteristic
38#16839590C2211yescharacteristic
39#16839590C2211yescharacteristic
40#16839589C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839566C2211yescharacteristic
48#16839565C2211yescharacteristic
49#16839563C2211yescharacteristic
50#16839541C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C22×Q16×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×#11758224—41no—
60C2×Q8×D61—21no—
61C2×Dic244C2311yescharacteristic
62C2×#11758202—21no—
63C2×#11758202—21no—
64Q16×D61—22no—
65Q16×D61—41no—
66C2×C8×D61C2311yescharacteristic
67#14519776—21no—
68#14519775—21no—
69#14519775—41no—
70#14519759—41no—
71#14519758—41no—
72#14519756—21no—
73#14519756—41no—
74#14519754—41no—
75#14519753—21no—
76#14519753—41no—
77#14519748—21no—
78Q8×Dic61—21no—
79D4×Dic61—21no—
80C4×C22⋊D61—41no—
81C4×C22⋊D61D411yescharacteristic
82C2×C4⋊Dic61—21no—
83C4⋊C4×D61—21no—
84#14519721—41no—
85C2×#11758185—41no—
86C2×#11758185C2311yescharacteristic
87C2×#11758185C2311yescharacteristic
88C16×D61—41no—
89#14519711—21no—
90#14519711C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519710—21no—
94#14519710—21no—
95#14519706D411yescharacteristic
96#14519702—21no—
97#14519701—21no—
98#14519701—21no—
99#14519698—21no—
100#14519698—21no—
101C22⋊#92418D411yescharacteristic
102#14519689D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C22×Q8×C61—21no—
105C2×Q16×C61—22no—
106C2×Q16×C61—21no—
107C2×Q16×C61—41no—
108C2×Q16×C61C2311yescharacteristic
109C2×Q16×C61C2311yescharacteristic
110C2×D8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#121×C61—21no—
113SD32×C61—22no—
114SD32×C61—41no—
115C2×C16×C61C2311yescharacteristic
116C2×C16×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#102×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—42no—
122C2×Q8×C61—41no—
123C2×Q8×C61—81no—
124C2×D4×C61—21no—
125C2×D4×C61—41no—
126C22×C4×C61—41no—
127C22×C4×C61C2×D411yescharacteristic
128C2×C22⋊D61—41no—
129C2×C22⋊D61D811yescharacteristic
130C22×Dic61—41no—
131C22×Dic61—41no—
132Q8×D61—42no—
133Q8×D61—81no—
134#11758224—84no—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61C2×D411yescharacteristic
138C2×Dic122—21no—
139C2×Dic122—41no—
140C2×Dic122—41no—
141Q16×C61—21no—
142Q16×C61—21no—
143Q16×C61—21no—
144Q16×C61—42no—
145Q16×C61#4112yescomplemented
146D8×C61—21no—
147D8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61C2411yescharacteristic
152C4⋊C4×C61—21no—
153C22⋊C4×C61—41no—
154C22⋊Dic61—41no—
155C22⋊Dic61—41no—
156C22⋊Dic61—41no—
157C22⋊Dic61D811yescharacteristic
158#11758202—42no—
159#11758202—81no—
160#11758198—41no—
161#11758198D811yescharacteristic
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—41no—
166C4⋊Dic61—41no—
167#11758196—41no—
168#11758196—41no—
169#11758196—41no—
170#11758196—41no—
171#11758196—41no—
172#11758196—41no—
173#11758196D811yescharacteristic
174C4×Dic61—41no—
175C4×Dic61C2×D411yescharacteristic
176C2×#92418C2×D411yescharacteristic
177C2×#92418C2×D411yescharacteristic
178Dic244—21no—
179Dic244—21no—
180C8×D61—21no—
181C8×D61—21no—
182C16×C61—21no—
183C16×C61—21no—
184#11758185—21no—
185#11758185—21no—
186C23×C61—41no—
187C23×C61C2×D811yescharacteristic
188C22×D61C2×D811yescharacteristic
189Q8×C61—22no—
190Q8×C61—41no—
191Q8×C61—41no—
192Q8×C61—41no—
193Q8×C61—82no—
194D4×C61—41no—
195D4×C61—41no—
196D4×C61—82no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—82no—
203C2×C4×C61C2×D811yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C22⋊D61—82no—
206C22⋊D61SD3214yescomplemented
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—82no—
212C2×Dic61—82no—
213C2×Dic61C2×D811yescharacteristic
214C2×Dic61C2×D811yescharacteristic
215C2×Dic61C2×D811yescharacteristic
216C4×D61—21no—
217C4×D61—21no—
218C4×D61—82no—
219C4×D61—82no—
220Dic122—41no—
221Dic122—41no—
222Dic122—82no—
223Dic122—82no—
224C8×C61—21no—
225C8×C61C2×#4111yescharacteristic
226C8×C61H(5,𝔽2)11yescharacteristic
227#92418—21no—
228#92418—21no—
229#33865—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—81no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61C2×SD3212yes—
236C22×C61C22×D811yescharacteristic
237C2×D61—21no—
238C2×D61—21no—
239C2×D61C2×SD3212yes—
240C4×C61—21no—
241C4×C61—42no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61#54611yescharacteristic
245C4×C61#57711yescharacteristic
246Dic61—21no—
247Dic61—21no—
248Dic61—81no—
249Dic61—81no—
250Dic61—81no—
251Dic61—81no—
252Dic61C2×SD3212yes—
253#3334—611no—
254D4×Q16—611no—
255#2241—611no—
256#2235—611no—
257#2235—611no—
258#2225—611no—
259#2217—611no—
260#2217—611no—
261#2208—611no—
262#2185—611no—
263#2185D6111yescharacteristic, complemented
264C4×SD32—611no—
265D4×C16—611no—
266C2×#358—611no—
267C2×#358—611no—
268C2×C61—22no—
269C2×C61—161no—
270C2×C61#329411yescharacteristic
271C2×C61C22×SD3211yescharacteristic
272C2×C61#341711yescharacteristic
273D61—22no—
274C22×Q16—611no—
275C22×Q16C2×D6111yescharacteristic
276D4×Q8—1221no—
277#547—1221no—
278C2×SD32—611no—
279C2×SD32C2×D6111yescharacteristic
280C2×SD32C2×D6111yescharacteristic
281C22×C16—611no—
282C22×C16—611no—
283#500—611no—
284#494—611no—
285#480—1221no—
286#480—1221no—
287#474—1221no—
288#467—611no—
289#467—611no—
290#466—611no—
291#466C2×D6111yescharacteristic
292#462—1221no—
293#451—1221no—
294#451—1221no—
295C4×Q16—611no—
296C4×D8—611no—
297C8×D4—611no—
298C2×#107—611no—
299C2×#107—611no—
300#367—611no—
301#367—611no—
302#367—611no—
303C4⋊C16—611no—
304#358—611no—
305#358—611no—
306#358—611no—
307#358—611no—
308#358—611no—
309#358—611no—
310#358—611no—
311#358C2×D6111yescharacteristic
312#358C2×D6111yescharacteristic
313#357—611no—
314C22⋊C16—611no—
315C22⋊C16C2×D6111yescharacteristic
316C4×C16—611no—
317C61#3386511yescharacteristic, complemented, Sylow
318C2×#41—2441no—
319C22×Q8—21no—
320C22×Q8—1221no—
321C2×Q16—22no—
322C2×Q16—21no—
323C2×Q16—41no—
324C2×Q16—611no—
325C2×Q16—1221no—
326C2×Q16—1221no—
327C2×Q16—1222no—
328C2×Q16—2441no—
329C2×Q16C22×D6111yescharacteristic
330C2×Q16C22×D6111yescharacteristic
331C2×D8C22×D6111yescharacteristic
332C22×C8—611no—
333C22×C8C22×D6111yescharacteristic
334C4⋊Q8—1221no—
335C4⋊Q8—1221no—
336C4⋊Q8—2441no—
337#123—2441no—
338#123—2441no—
339C22⋊Q8—1221no—
340C22⋊Q8—1221no—
341C22⋊Q8—2441no—
342C22⋊Q8—2441no—
343C22⋊Q8—2441no—
344#121—21no—
345#121—1221no—
346C4×Q8—1221no—
347C4×D4—611no—
348C4×D4—1221no—
349C4×D4—2441no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352SD32—22no—
353SD32—41no—
354SD32—2441no—
355C2×C16—611no—
356C2×C16—611no—
357C2×C16—2441no—
358C2×C16—2441no—
359C2×C16C22×D6111yescharacteristic
360C2×C16C22×D6111yescharacteristic
361#107—611no—
362#107—611no—
363#107—611no—
364#107—1221no—
365#107—1221no—
366#107—1221no—
367#107C22×D6111yescharacteristic
368C4⋊C8—611no—
369#103—1221no—
370#103—1221no—
371#103—1221no—
372#102—21no—
373#102—1221no—
374#102—1221no—
375C22⋊C8—611no—
376C22⋊C8—611no—
377C4×C8—611no—
378#41—4884no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—42no—
382C2×Q8—41no—
383C2×Q8—81no—
384C2×Q8—1221no—
385C2×Q8—4881no—
386C2×Q8—2441no—
387C2×Q8—2441no—
388C2×Q8—2442no—
389C2×D4—21no—
390C2×D4—41no—
391C2×D4—611no—
392C2×D4—2441no—
393C22×C4—41no—
394C22×C4—611no—
395C22×C4—2441no—
396C22×C4—2441no—
397C22×C4—2441no—
398C22×C4—2441no—
399C22×C4D4×D6111yescharacteristic
400Q16—21no—
401Q16—21no—
402Q16—21no—
403Q16—42no—
404Q16—1221no—
405Q16—1221no—
406Q16—4881no—
407Q16—2442no—
408Q16#1175822412yescomplemented
409D8—21no—
410D8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—611no—
414C2×C8—611no—
415C2×C8—1221no—
416C2×C8—1221no—
417C2×C8D4×D6111yescharacteristic
418C2×C8C23×D6111yescharacteristic
419C4⋊C4—21no—
420C4⋊C4—611no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C4⋊C4—2441no—
429C4⋊C4—2441no—
430C4⋊C4—2441no—
431C4⋊C4—2441no—
432C22⋊C4—41no—
433C22⋊C4—611no—
434C22⋊C4—611no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C42—611no—
440C42—2441no—
441C16—21no—
442C16—21no—
443C16—1221no—
444C16—1221no—
445C23—41no—
446C23—611no—
447C23D8×D6111yescharacteristic
448Q8—22no—
449Q8—41no—
450Q8—41no—
451Q8—41no—
452Q8—82no—
453Q8—4882no—
454Q8—4882no—
455Q8—2441no—
456Q8—2441no—
457D4—41no—
458D4—41no—
459D4—82no—
460D4—614no—
461D4—4882no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—82no—
468C2×C4—611no—
469C2×C4—611no—
470C2×C4—611no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—4882no—
474C2×C4—4882no—
475C2×C4—4882no—
476C2×C4—4882no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4D8×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C8—21no—
484C8—1221no—
485C8—1221no—
486C8C2×#1175822411yescharacteristic
487C8#1451982211yescharacteristic
488C22—21no—
489C22—21no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—612no—
494C22—1221no—
495C22—1221no—
496C22SD32×D6112yes—
497C22C2×D8×D6111yescharacteristic
498C4—21no—
499C4—42no—
500C4—81no—
501C4—81no—
502C4—612no—
503C4—1221no—
504C4—1221no—
505C4—4881no—
506C4—4881no—
507C4—4881no—
508C4—4881no—
509C4#1684039911yescharacteristic
510C4#1684057211yescharacteristic
511C2—22no—
512C2—161no—
513C2—1222no—
514C2#2145762511yescharacteristic
515C2C2×SD32×D6111yescharacteristic
516C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2185C2
socleC22×C61C22×D8
prime radical#21458163C2
hypercenter#2185D61
hypocenterC61#33865
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2185

Complements

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

NH ≅ G/Ntype
2. Q16×C22⋊D61C2semidirect
3. #21448045C2semidirect
4. #21448017C2semidirect
5. #21448015C2semidirect
6. #21447959C2semidirect
7. #21447913C2semidirect
8. #21447912C2semidirect
9. #21447862C2semidirect
10. #21447734C2semidirect
11. SD32×Dic61C2semidirect
12. #21447636C2semidirect
13. C2×#16839566C2semidirect
14. #358×D61C2semidirect
15. C61×#2185C2semidirect
145. Q16×C61#41semidirect
206. C22⋊D61SD32semidirect
263. #2185D61semidirect
317. C61#33865semidirect
408. Q16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214581632
Q16×C22⋊D612
#214480452
#214480172
#214480152
#214479592
#214479132
#214479122
#214478622
#214477342
SD32×Dic612
#214476362
C2×#168395662
#358×D612
C61×#21852
#3386561