Finite Group Databasebeta

← #29722498

Subgroups — #29722498

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#21455474C211yesmaximal, characteristic, complemented
3#21455367C212yesmaximal
4#21454414C211yesmaximal, characteristic, complemented
5#21447181C212yesmaximal, complemented
6#21447035C212yesmaximal, complemented
7#21446690C211yesmaximal, characteristic, complemented
8#21446572C211yesmaximal, characteristic, complemented
9#21446186C212yesmaximal, complemented
10#21445878C211yesmaximal, characteristic, complemented
11C2×C61×#444C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C22×Dic244C2211yescharacteristic
14#16840333C2212yes—
15#16840245C2212yes—
16#16840037—24no—
17#16840020—21no—
18#16840020—21no—
19#16840003—22no—
20#16839982—22no—
21#16839975—22no—
22#16839970—22no—
23#16839965—22no—
24C2×#14519713C2211yescharacteristic
25#16839804—22no—
26#16839804C2212yes—
27#16839800—21no—
28#16839800C2211yescharacteristic
29C2×#14519709—21no—
30C2×#14519709C2211yescharacteristic
31#16839788C2212yes—
32#16839772C2212yes—
33#16839763C2212yes—
34C2×#14519702C2212yes—
35C2×#14519695C2212yes—
36#16839668C2212yes—
37#16839530C2212yes—
38#16839522C2212yes—
39C2×#135×C61C2211yescharacteristic
40C2×#131×C61C2211yescharacteristic
41C2×#131×C61C2211yescharacteristic
42C61×#444—24no—
43C2×#108×C61C2211yescharacteristic
44C2×#104×C61C2211yescharacteristic
45C2×#104×C61C2211yescharacteristic
46C2×#97×C61C2211yescharacteristic
47C22×Dic122C2312yes—
48#14519807—24no—
49#14519807—24no—
50C2×Dic244—21no—
51C2×Dic244—21no—
52C2×Dic244—22no—
53C2×Dic244—22no—
54C2×#11758202—22no—
55C2×#11758200—22no—
56C2×#11758194C2312yes—
57#14519775—22no—
58#14519764—22no—
59#14519758—42no—
60#14519756—22no—
61#14519756—22no—
62Q8×Dic61—22no—
63D4×Dic61—22no—
64D4×Dic61—42no—
65#14519736—22no—
66#14519736C2312yes—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61C2312yes—
69#14519713—22no—
70#14519711—22no—
71#14519711D412yes—
72#14519709—21no—
73#14519709—21no—
74#14519709—22no—
75#14519709D411yescharacteristic
76#14519709D411yescharacteristic
77#14519706—22no—
78#14519706—22no—
79#14519702—22no—
80#14519702—22no—
81#14519702—22no—
82#14519702—22no—
83#14519701—22no—
84#14519698—22no—
85#14519695—24no—
86#14519684—42no—
87C2×#41×C61C2311yescharacteristic
88C2×#41×C61C2311yescharacteristic
89#135×C61—22no—
90#135×C61—22no—
91C2×Q16×C61D411yescharacteristic
92C2×SD16×C61D411yescharacteristic
93C2×SD16×C61D411yescharacteristic
94C2×D8×C61D411yescharacteristic
95#131×C61—22no—
96#131×C61—22no—
97#131×C61—22no—
98#131×C61D414yes—
99C2×M16×C61—21no—
100C2×M16×C61—21no—
101C2×M16×C61C2311yescharacteristic
102C2×M16×C61C2311yescharacteristic
103C22×C8×C61—21no—
104C22×C8×C61—21no—
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#108×C61—22no—
109#104×C61—22no—
110#104×C61—22no—
111#97×C61—22no—
112#41×C61—21no—
113#41×C61—21no—
114#41×C61—21no—
115#41×C61—21no—
116#41×C61—21no—
117#41×C61—21no—
118#41×C61C2×D412yes—
119#41×C61#4112yes—
120C2×Q8×C61C2×D411yescharacteristic
121C2×Q8×C61C2×D411yescharacteristic
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61C2×D411yescharacteristic
125C2×D4×C61C2×D411yescharacteristic
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2411yescharacteristic
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—42no—
133C2×Dic122—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139Q16×C61—22no—
140SD16×C61—22no—
141SD16×C61—22no—
142D8×C61—22no—
143M16×C61—21no—
144M16×C61—24no—
145M16×C61—21no—
146M16×C61—21no—
147M16×C61—42no—
148M16×C61C2×D412yes—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—24no—
153C2×C8×C61—21no—
154C2×C8×C61—42no—
155C2×C8×C61C2×D411yescharacteristic
156C2×C8×C61C2×D411yescharacteristic
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61C2×D411yescharacteristic
159C2×C8×C61C2×D412yes—
160C2×C8×C61#4112yes—
161C42×C61—21no—
162C42×C61—21no—
163C22⋊Dic61—22no—
164C22⋊Dic61—42no—
165C22⋊Dic61—42no—
166C22⋊Dic61—42no—
167#11758202—44no—
168#11758200—44no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61C2×D412yes—
173C4⋊Dic61C2×D412yes—
174#11758196—42no—
175#11758196—42no—
176C4×Dic61—22no—
177C4×Dic61—42no—
178#11758194—22no—
179#11758194—22no—
180C2×#92418—22no—
181Dic244—42no—
182Dic244—42no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61H(5,𝔽2)11yescharacteristic
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—21no—
189Q8×C61#12112yes—
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—21no—
193D4×C61—21no—
194D4×C61—21no—
195D4×C61—41no—
196D4×C61—41no—
197D4×C61#12112yes—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—82no—
219C2×Dic61—162no—
220Dic122—42no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224C8×C61—22no—
225C8×C61—21no—
226C8×C61—21no—
227C8×C61—41no—
228C8×C61#12112yes—
229C8×C61#12112yes—
230#92418—42no—
231#24800—611nomaximal, Sylow
232C22×C61—22no—
233C22×C61—22no—
234C22×C61—42no—
235C22×C61—41no—
236C22×C61—42no—
237C22×C61#53711yescharacteristic
238C22×C61D4211yescharacteristic
239C22×C61#55311yescharacteristic
240C4×C61—22no—
241C4×C61—41no—
242C4×C61—81no—
243C4×C61C2×#12111yescharacteristic
244C4×C61C2×#12111yescharacteristic
245C4×C61#54711yescharacteristic
246C4×C61#54811yescharacteristic
247Dic61—42no—
248Dic61—42no—
249Dic61—82no—
250Dic61—82no—
251C2×#473—611no—
252#3059—611no—
253#3046—612no—
254C2×#444D6111yescharacteristic
255#2912—611no—
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—81no—
266C2×C61#241611yescharacteristic
267C2×C61#328912yes—
268C2×#137—612no—
269C2×#135C2×D6111yescharacteristic
270C22×Q16—611no—
271C2×#131C2×D6111yescharacteristic
272C2×#131C2×D6111yescharacteristic
273#536—612no—
274#520—612no—
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—24no—
284C2×#108—611no—
285C2×#108C2×D6111yescharacteristic
286#428—612no—
287#428—1222no—
288#427—611no—
289#427—1221no—
290C2×#106—611no—
291C2×#106—1221no—
292#424—612no—
293C2×#104C2×D6111yescharacteristic
294C2×#104C2×D6111yescharacteristic
295#419—612no—
296#417—612no—
297C2×#103—612no—
298C2×#101—612no—
299C2×#97C2×D6111yescharacteristic
300#391—612no—
301#343—612no—
302#339—612no—
303C61#2480011yescharacteristic, complemented, Sylow
304C2×#41C22×D6111yescharacteristic
305C2×#41C22×D6111yescharacteristic
306C22×Q8—612no—
307#137—1224no—
308#137—1224no—
309#135—22no—
310#135—22no—
311C2×Q16—1222no—
312C2×Q16—1221no—
313C2×Q16—1221no—
314C2×Q16—1222no—
315C2×Q16—1222no—
316C2×Q16D24411yescharacteristic
317C2×SD16—1222no—
318C2×SD16D24411yescharacteristic
319C2×SD16D24411yescharacteristic
320C2×D8D24411yescharacteristic
321#131—22no—
322#131—22no—
323#131—22no—
324#131C22⋊D6114yes—
325C2×M16—21no—
326C2×M16—21no—
327C2×M16—612no—
328C2×M16C22×D6111yescharacteristic
329C2×M16C22×D6111yescharacteristic
330C22×C8—21no—
331C22×C8—21no—
332C22×C8C22×D6111yescharacteristic
333C22×C8C22×D6111yescharacteristic
334C4⋊Q8—1222no—
335#124—1222no—
336#123—2442no—
337C22⋊Q8—1222no—
338C22⋊Q8—1222no—
339C4×Q8—1222no—
340C4×D4—1222no—
341C4×D4—2442no—
342#117—612no—
343#117—1222no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—1222no—
346C2×C42C22×D6111yescharacteristic
347#108—22no—
348#108—1222no—
349#107—612no—
350#107—1222no—
351#106—611no—
352#106—611no—
353#106—1222no—
354#106—1221no—
355#106—1221no—
356C4⋊C8—1222no—
357C4⋊C8—1222no—
358#104—22no—
359#104—22no—
360#103—1222no—
361#103—1222no—
362#103—1222no—
363#103—1222no—
364#103—1222no—
365#102—1222no—
366#101—1224no—
367#97—22no—
368#95—2442no—
369#41—21no—
370#41—21no—
371#41—21no—
372#41—21no—
373#41—21no—
374#41—21no—
375#41#1175822212yes—
376#41C2×C22⋊D6112yes—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—1222no—
381C2×Q8—1222no—
382C2×Q8—1222no—
383C2×Q8C2×D24411yescharacteristic
384C2×Q8C2×D24411yescharacteristic
385C2×D4—21no—
386C2×D4—21no—
387C2×D4C2×D24411yescharacteristic
388C2×D4C2×D24411yescharacteristic
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—1222no—
393C22×C4—1222no—
394C22×C4—2442no—
395C22×C4C23×D6111yescharacteristic
396Q16—22no—
397Q16—2444no—
398Q16—2442no—
399Q16—2442no—
400SD16—22no—
401SD16—22no—
402SD16—2444no—
403D8—22no—
404M16—21no—
405M16—24no—
406M16—21no—
407M16—21no—
408M16—42no—
409M16—1222no—
410M16—1222no—
411M16C2×C22⋊D6112yes—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—24no—
416C2×C8—21no—
417C2×C8—42no—
418C2×C8—1222no—
419C2×C8C2×D24411yescharacteristic
420C2×C8C2×D24411yescharacteristic
421C2×C8D4×D6111yescharacteristic
422C2×C8D4×D6111yescharacteristic
423C2×C8#1175822412yes—
424C2×C8C2×C22⋊D6112yes—
425C4⋊C4—612no—
426C4⋊C4—612no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4—2442no—
430C4⋊C4—2442no—
431C4⋊C4—2442no—
432C22⋊C4—1222no—
433C22⋊C4—2442no—
434C22⋊C4—2442no—
435C22⋊C4—2442no—
436C42—21no—
437C42—21no—
438C42—1222no—
439C42—2442no—
440C23—21no—
441C23—21no—
442C23#1451982311yescharacteristic
443Q8—21no—
444Q8—21no—
445Q8—21no—
446Q8—2442no—
447Q8—2442no—
448Q8—2442no—
449Q8—2442no—
450Q8#1451975112yes—
451D4—22no—
452D4—22no—
453D4—21no—
454D4—21no—
455D4—21no—
456D4—41no—
457D4—41no—
458D4#1451975112yes—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—9762no—
470C2×C4—4882no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C22×D24411yescharacteristic
478C2×C4C2×#1175822411yescharacteristic
479C2×C4C2×#1175822211yescharacteristic
480C2×C4#1451982411yescharacteristic
481C8—22no—
482C8—21no—
483C8—21no—
484C8—41no—
485C8—2442no—
486C8#1451974912yes—
487C8#1451974912yes—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—41no—
492C22—42no—
493C22#1684034011yescharacteristic
494C22D4×D24411yescharacteristic
495C22#1684045811yescharacteristic
496C4—22no—
497C4—41no—
498C4—81no—
499C4—4882no—
500C4—4882no—
501C4—2442no—
502C4—2442no—
503C4C2×#1451974911yescharacteristic
504C4C2×#1451975111yescharacteristic
505C4#1684041611yescharacteristic
506C4#1684042111yescharacteristic
507C2—21no—
508C2—21no—
509C2—42no—
510C2—81no—
511C2#2144925911yescharacteristic
512C2#2145759512yes—
subgroupG/H
centerC22D4×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#444C2
socleC22×C61D42
prime radicalC2×#135×C61C22
hypercenterC2×#444D61
hypocenterC61#24800
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
1. C2×#16840037C2semidirect
2. #21455474C2semidirect
4. #21454414C2semidirect
5. #21447181C2semidirect
6. #21447035C2semidirect
7. #21446690C2semidirect
8. #21446572C2semidirect
9. #21446186C2semidirect
10. #21445878C2semidirect
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
#214554742
#214553672
#214544142
#214471812
#214470352
#214466902
#214465722
#214461862
#214458782
C2×C61×#4442
#2480061