Finite Group Databasebeta

← #29722492

Subgroups — #29722492

Order 15616.

842 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840037C211yesmaximal, characteristic, complemented
2#21455475C211yesmaximal, characteristic, complemented
3#21455369C212yesmaximal
4C2×#16839882C211yesmaximal, characteristic
5#21447183C212yesmaximal, complemented
6#21447038C212yesmaximal, complemented
7#21446688C211yesmaximal, characteristic, complemented
8#21446573C211yesmaximal, characteristic, complemented
9#21446188C212yesmaximal, complemented
10#21445879C211yesmaximal, characteristic, complemented
11C61×#2912C211yesmaximal, characteristic, complemented
12C2×#14519807C2212yes—
13C2×#14519797C2211yescharacteristic
14C22×Dic244C2211yescharacteristic
15C2×#14519783C2211yescharacteristic
16#16840338C2212yes—
17#16840037—24no—
18#16840021—21no—
19#16840021—21no—
20#16840004—22no—
21#16839984—22no—
22#16839977—22no—
23#16839972—22no—
24#16839967—22no—
25#16839882—24no—
26C2×#14519713C2211yescharacteristic
27C2×#14519712C2211yescharacteristic
28#16839789C2212yes—
29C2×#14519704C2211yescharacteristic
30C2×#14519703C2211yescharacteristic
31#16839770C2212yes—
32#16839761C2212yes—
33C2×#14519700C2212yes—
34C2×#14519695C2212yes—
35C2×#14519687C2211yescharacteristic
36#16839667C2212yes—
37#16839531C2212yes—
38#16839521C2212yes—
39C2×#131×C61C2211yescharacteristic
40C61×#520C2212yes—
41C61×#428—22no—
42C61×#428C2212yes—
43C61×#427—21no—
44C61×#427C2211yescharacteristic
45C2×#106×C61—21no—
46C2×#106×C61C2211yescharacteristic
47C2×#11758222C2311yescharacteristic
48C22×Dic122C2312yes—
49#14519807—24no—
50#14519807—24no—
51#14519797—22no—
52#14519797—22no—
53C2×Dic244—21no—
54C2×Dic244—21no—
55C2×Dic244—22no—
56C2×Dic244—22no—
57C2×#11758202—22no—
58C2×#11758202D411yescharacteristic
59C2×#11758200—22no—
60C2×#11758200D411yescharacteristic
61C2×#11758201D411yescharacteristic
62C2×#11758199D411yescharacteristic
63#14519783—22no—
64#14519783—22no—
65C2×#11758194C2312yes—
66C2×#11758194C2311yescharacteristic
67C2×#11758189—21no—
68C22×#92418C2311yescharacteristic
69C2×C8×D61—21no—
70#14519777—22no—
71#14519765—22no—
72#14519761—42no—
73#14519757—22no—
74#14519757—22no—
75C2×C4×Dic61C2311yescharacteristic
76#14519713—22no—
77#14519712—22no—
78C4⋊#92418—22no—
79C4⋊#92418—22no—
80#14519704—22no—
81#14519703—22no—
82#14519700—22no—
83#14519700—22no—
84#14519700—22no—
85#14519700—22no—
86#14519700—22no—
87#14519697—22no—
88#14519695—24no—
89#14519687—22no—
90#14519684—42no—
91C2×#41×C61C2311yescharacteristic
92#131×C61—22no—
93#131×C61D414yes—
94C2×M16×C61—21no—
95C2×M16×C61C2311yescharacteristic
96C22×C8×C61—21no—
97C22×C8×C61C2311yescharacteristic
98C4×Q8×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—42no—
101#117×C61—22no—
102#117×C61C2312yes—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61C2312yes—
105#107×C61—22no—
106#107×C61D412yes—
107#106×C61—21no—
108#106×C61—21no—
109#106×C61—22no—
110#106×C61D411yescharacteristic
111#106×C61D411yescharacteristic
112#41×C61—21no—
113#41×C61—21no—
114#41×C61C2×D412yes—
115#41×C61#4112yes—
116C2×Q8×C61C2×D411yescharacteristic
117C2×D4×C61—21no—
118C2×D4×C61C2×D411yescharacteristic
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—42no—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—21no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126#11758222—21no—
127#11758222—21no—
128#11758222—21no—
129#11758222—21no—
130C2×D244C2×D411yescharacteristic
131C2×C4×D61—21no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122C2×D411yescharacteristic
139M16×C61—21no—
140M16×C61—24no—
141M16×C61C2×D412yes—
142C2×C8×C61—21no—
143C2×C8×C61—24no—
144C2×C8×C61C2×D412yes—
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61#4112yes—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61C2×D412yes—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—42no—
155C42×C61—22no—
156C42×C61—42no—
157#11758202—22no—
158#11758202—44no—
159#11758201—22no—
160#11758200—22no—
161#11758200—44no—
162#11758199—22no—
163#11758198—42no—
164#11758198—42no—
165C4⋊Dic61—42no—
166C4⋊Dic61—42no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169#11758194—22no—
170#11758194—21no—
171#11758194—21no—
172#11758194—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—21no—
177C2×#92418C2×D411yescharacteristic
178C2×#92418C2×D411yescharacteristic
179Dic244—42no—
180Dic244—42no—
181#11758189—42no—
182C8×D61—42no—
183C23×C61—21no—
184C23×C61H(5,𝔽2)11yescharacteristic
185C22×D61—21no—
186Q8×C61—21no—
187Q8×C61#12112yes—
188D4×C61—22no—
189D4×C61—22no—
190D4×C61—21no—
191D4×C61#12112yes—
192C2×C4×C61—21no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—42no—
198C2×C4×C61—82no—
199C2×C4×C61—162no—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C612−1+411yescharacteristic
206C22⋊D61—41no—
207C22⋊D61—41no—
208C2×Dic61—21no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214D244—21no—
215D244—21no—
216C4×D61—41no—
217C4×D61—41no—
218Dic122—21no—
219Dic122—21no—
220Dic122—42no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224C8×C61—22no—
225C8×C61#12112yes—
226C8×C61#12112yes—
227#92418—21no—
228#92418—21no—
229#92418—41no—
230#92418—42no—
231#24800—611nomaximal, Sylow
232C22×C61—22no—
233C22×C61—22no—
234C22×C61—42no—
235C22×C61#53711yescharacteristic
236C22×C61D4211yescharacteristic
237C22×C61#55311yescharacteristic
238C2×D61—41no—
239C2×D61—42no—
240C4×C61—22no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—82no—
244C4×C61C2×#12111yescharacteristic
245C4×C61C2×#12111yescharacteristic
246C4×C61#54711yescharacteristic
247C4×C61#54811yescharacteristic
248Dic61—41no—
249Dic61—81no—
250Dic61—82no—
251C2×#473—611no—
252#3059—611no—
253#3046—612no—
254C2×#444—611no—
255#2912D6111yescharacteristic, complemented
256#2064—612no—
257#2038—612no—
258#1971—611no—
259#1946—611no—
260#1882—612no—
261#1815—611no—
262C2×C61—21no—
263C2×C61—21no—
264C2×C61—42no—
265C2×C61#241611yescharacteristic
266C2×C61#328912yes—
267D61—81no—
268C2×#137—612no—
269C2×#135—611no—
270C22×Q16—611no—
271C2×#131—611no—
272C2×#131C2×D6111yescharacteristic
273#536—612no—
274#520C2×D6112yes—
275#473—1224no—
276#470—1221no—
277#470—1221no—
278#467—1222no—
279#464—1222no—
280#463—1222no—
281#462—1222no—
282#461—1222no—
283#444—1224no—
284C2×#108—611no—
285C2×#108—611no—
286#428—22no—
287#428C2×D6112yes—
288#427—21no—
289#427C2×D6111yescharacteristic
290C2×#106—21no—
291C2×#106C2×D6111yescharacteristic
292#424—612no—
293C2×#104—611no—
294C2×#104—611no—
295#419—612no—
296#417—612no—
297C2×#103—612no—
298C2×#101—612no—
299C2×#97—611no—
300#391—612no—
301#343—612no—
302#339—612no—
303C61#2480011yescharacteristic, complemented, Sylow
304C2×#41—611no—
305C2×#41C22×D6111yescharacteristic
306C22×Q8—612no—
307#137—1224no—
308#137—1224no—
309#135—1222no—
310#135—1222no—
311C2×Q16—611no—
312C2×Q16—1222no—
313C2×Q16—1222no—
314C2×Q16—1222no—
315C2×Q16—1221no—
316C2×Q16—1221no—
317C2×SD16—611no—
318C2×SD16—611no—
319C2×SD16—1222no—
320C2×D8—611no—
321#131—22no—
322#131—1222no—
323#131—1222no—
324#131D24414yes—
325C2×M16—21no—
326C2×M16—611no—
327C2×M16—612no—
328C2×M16—1221no—
329C2×M16C22×D6111yescharacteristic
330C22×C8—21no—
331C22×C8—611no—
332C22×C8—1221no—
333C22×C8C22×D6111yescharacteristic
334C4⋊Q8—1222no—
335#124—1222no—
336#123—2442no—
337C22⋊Q8—1222no—
338C22⋊Q8—1222no—
339C4×Q8—22no—
340C4×D4—22no—
341C4×D4—42no—
342#117—22no—
343#117C22×D6112yes—
344C2×C4⋊C4—22no—
345C2×C4⋊C4C22×D6112yes—
346C2×C42—611no—
347#108—1222no—
348#108—1222no—
349#107—22no—
350#107C22⋊D6112yes—
351#106—21no—
352#106—21no—
353#106—22no—
354#106C22⋊D6111yescharacteristic
355#106C22⋊D6111yescharacteristic
356C4⋊C8—1222no—
357C4⋊C8—1222no—
358#104—1222no—
359#104—1222no—
360#103—1222no—
361#103—1222no—
362#103—1222no—
363#103—1222no—
364#103—1222no—
365#102—1222no—
366#101—1224no—
367#97—1222no—
368#95—2442no—
369#41—21no—
370#41—21no—
371#41—1221no—
372#41—1221no—
373#41—1221no—
374#41—1221no—
375#41C2×D24412yes—
376#41#1175822612yes—
377C2×Q8—611no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—1222no—
381C2×Q8—1222no—
382C2×Q8—1222no—
383C2×Q8—1222no—
384C2×Q8D4×D6111yescharacteristic
385C2×D4—21no—
386C2×D4—611no—
387C2×D4—1221no—
388C2×D4D4×D6111yescharacteristic
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—42no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4C23×D6111yescharacteristic
396Q16—1222no—
397Q16—2444no—
398Q16—2442no—
399Q16—2442no—
400SD16—1222no—
401SD16—1222no—
402SD16—2444no—
403D8—1222no—
404M16—21no—
405M16—24no—
406M16—1222no—
407M16—1221no—
408M16—1221no—
409M16—1222no—
410M16—2442no—
411M16C2×D24412yes—
412C2×C8—21no—
413C2×C8—24no—
414C2×C8—611no—
415C2×C8—611no—
416C2×C8—1221no—
417C2×C8—1221no—
418C2×C8—1222no—
419C2×C8—1221no—
420C2×C8—2442no—
421C2×C8C2×D24412yes—
422C2×C8#1175822212yes—
423C2×C8C2×C22⋊D6111yescharacteristic
424C2×C8C2×C22⋊D6111yescharacteristic
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—42no—
428C4⋊C4—2442no—
429C4⋊C4—2442no—
430C4⋊C4C2×C22⋊D6112yes—
431C4⋊C4C2×C22⋊D6112yes—
432C22⋊C4—22no—
433C22⋊C4—42no—
434C22⋊C4—2442no—
435C22⋊C4—2442no—
436C42—22no—
437C42—42no—
438C42—1221no—
439C42—1221no—
440C23—21no—
441C23—1221no—
442C23#1451982211yescharacteristic
443Q8—21no—
444Q8—1221no—
445Q8—1221no—
446Q8—2442no—
447Q8—2442no—
448Q8—2442no—
449Q8—2442no—
450Q8#1451975212yes—
451D4—22no—
452D4—22no—
453D4—21no—
454D4—1221no—
455D4—1221no—
456D4—2441no—
457D4—2441no—
458D4#1451975212yes—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—82no—
466C2×C4—162no—
467C2×C4—1221no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C22×D24411yescharacteristic
478C2×C4C2×#1175822211yescharacteristic
479C2×C4C2×#1175822611yescharacteristic
480C2×C4#1451982511yescharacteristic
481C8—22no—
482C8—1221no—
483C8—1221no—
484C8—2441no—
485C8—2442no—
486C8#1451975112yes—
487C8#1451975112yes—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—2441no—
492C22—2442no—
493C22#1684034311yescharacteristic
494C22D4×C22⋊D6111yescharacteristic
495C22#1684046011yescharacteristic
496C4—22no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4—4881no—
501C4—4882no—
502C4—2441no—
503C4C2×#1451975111yescharacteristic
504C4C2×#1451975211yescharacteristic
505C4#1684041311yescharacteristic
506C4#1684042311yescharacteristic
507C2—21no—
508C2—21no—
509C2—42no—
510C2—4881no—
511C2#2144925311yescharacteristic
512C2#2145759212yes—
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2912C2
socleC22×C61D42
prime radicalC2×#14519797C22
hypercenter#2912D61
hypocenterC61#24800
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2912

Complements

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

NH ≅ G/Ntype
1. C2×#16840037C2semidirect
2. #21455475C2semidirect
5. #21447183C2semidirect
6. #21447038C2semidirect
7. #21446688C2semidirect
8. #21446573C2semidirect
9. #21446188C2semidirect
10. #21445879C2semidirect
11. C61×#2912C2semidirect
255. #2912D61semidirect
303. C61#24800semidirect

Maximal subgroups

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

subgroupindex
C2×#168400372
#214554752
#214553692
C2×#168398822
#214471832
#214470382
#214466882
#214465732
#214461882
#214458792
C61×#29122
#2480061