Finite Group Databasebeta

← #29722494

Subgroups — #29722494

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×#16840034C211yesmaximal, characteristic, complemented
2#21455365C212yesmaximal
3C2×#16839882C211yesmaximal, characteristic
4#21454411C211yesmaximal, characteristic, complemented
5#21447186C212yesmaximal, complemented
6#21447033C212yesmaximal, complemented
7#21446691C211yesmaximal, characteristic, complemented
8#21446573C211yesmaximal, characteristic, complemented
9#21446185C212yesmaximal, complemented
10#21445878C211yesmaximal, characteristic, complemented
11C61×#3059C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519797C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840242C2212yes—
16#16840034—24no—
17#16839981—22no—
18#16839974—22no—
19#16839969—22no—
20#16839964—22no—
21#16839882—24no—
22C2×#14519713C2211yescharacteristic
23C2×#14519712C2211yescharacteristic
24#16839805C2212yes—
25#16839803—22no—
26#16839797—21no—
27C2×#14519709C2211yescharacteristic
28#106×D61—21no—
29#16839789C2212yes—
30C2×#14519704C2211yescharacteristic
31C2×#14519703C2211yescharacteristic
32#16839763C2212yes—
33C2×#14519701C2212yes—
34C2×#14519696C2212yes—
35C2×#14519687C2211yescharacteristic
36#16839669C2212yes—
37#16839530C2212yes—
38#16839521C2212yes—
39C22×Q16×C61C2211yescharacteristic
40C2×#131×C61C2211yescharacteristic
41C61×#536C2212yes—
42C61×#470—21no—
43C61×#470—21no—
44C61×#467—22no—
45C61×#427C2211yescharacteristic
46C61×#419C2212yes—
47C2×#11758222C2311yescharacteristic
48#14519808—24no—
49#14519808—24no—
50#14519797—22no—
51#14519797—22no—
52C2×#11758202—22no—
53C2×#11758202D411yescharacteristic
54C2×#11758200D411yescharacteristic
55C2×#11758201—22no—
56C2×#11758201D411yescharacteristic
57C2×#11758199D411yescharacteristic
58#14519783—22no—
59#14519783D414yes—
60C2×#11758194C2312yes—
61C2×#11758194C2311yescharacteristic
62C2×#11758189—21no—
63C22×#92418C2311yescharacteristic
64C2×C8×D61—21no—
65C4×Dic122—22no—
66C4×D244—22no—
67#14519737—42no—
68#14519735—22no—
69C2×C4⋊Dic61C2312yes—
70C4⋊C4×D61—22no—
71C2×C4×Dic61C2311yescharacteristic
72#14519713—22no—
73#14519712—22no—
74#14519710—22no—
75#14519709—21no—
76#14519709—21no—
77#14519708—22no—
78C4⋊#92418—22no—
79C4⋊#92418—22no—
80#14519704—22no—
81#14519703—22no—
82#14519701—22no—
83#14519701—22no—
84#14519701—22no—
85#14519698—22no—
86#14519696—24no—
87#14519687—22no—
88#14519684—42no—
89C2×#41×C61C2311yescharacteristic
90C22×Q8×C61C2312yes—
91C2×Q16×C61—22no—
92C2×Q16×C61—21no—
93C2×Q16×C61—21no—
94C2×Q16×C61—22no—
95#131×C61—22no—
96#131×C61—22no—
97C2×M16×C61—21no—
98C2×M16×C61C2311yescharacteristic
99C22×C8×C61—21no—
100C22×C8×C61C2311yescharacteristic
101C4⋊Q8×C61—22no—
102#124×C61—22no—
103#123×C61—42no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—22no—
106#117×C61C2312yes—
107#107×C61D412yes—
108#106×C61D411yescharacteristic
109#106×C61D411yescharacteristic
110#103×C61—22no—
111#103×C61—22no—
112#41×C61—21no—
113#41×C61—21no—
114#41×C61—21no—
115#41×C61—21no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61C2×D411yescharacteristic
123C2×D4×C61—21no—
124C2×D4×C61C2×D411yescharacteristic
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—21no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131#11758222—21no—
132#11758222—21no—
133#11758222C2×D412yes—
134#11758222#4112yes—
135C2×D244C2×D411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—42no—
138C2×Dic122C2×D411yescharacteristic
139Q16×C61—42no—
140Q16×C61—42no—
141M16×C61—21no—
142M16×C61—21no—
143M16×C61—42no—
144C2×C8×C61—21no—
145C2×C8×C61—42no—
146C2×C8×C61C2×D412yes—
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2×D411yescharacteristic
149C2×C8×C61#4112yes—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—42no—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—42no—
156C22⋊C4×C61—42no—
157C42×C61—22no—
158#11758202—22no—
159#11758202—44no—
160#11758201—22no—
161#11758201—44no—
162#11758200—22no—
163#11758199—22no—
164#11758198—42no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167#11758196—42no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61—42no—
171#11758194—21no—
172#11758194—22no—
173#11758194—22no—
174#11758194C2×D412yes—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418C2×D411yescharacteristic
180C2×#92418C2×D411yescharacteristic
181#11758189—24no—
182C8×D61—24no—
183C23×C61—21no—
184C23×C61H(5,𝔽2)11yescharacteristic
185C22×D61—21no—
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—42no—
189Q8×C61—42no—
190Q8×C61—42no—
191Q8×C61—42no—
192D4×C61—21no—
193D4×C61—21no—
194D4×C61—41no—
195D4×C61—41no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C22⋊D61—22no—
211C22⋊D61—22no—
212C2×Dic61—21no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—82no—
217D244—21no—
218D244#12112yes—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—162no—
222Dic122—21no—
223Dic122#12112yes—
224C8×C61—41no—
225C8×C61#12112yes—
226C8×C61#12112yes—
227#92418—22no—
228#92418—21no—
229#92418—21no—
230#92418—42no—
231#24800—611nomaximal, Sylow
232C22×C61—22no—
233C22×C61—41no—
234C22×C61—42no—
235C22×C61#53711yescharacteristic
236C22×C61D4211yescharacteristic
237C22×C61#55311yescharacteristic
238C2×D61—22no—
239C2×D61—42no—
240C4×C61—41no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—82no—
244C4×C61C2×#12111yescharacteristic
245C4×C61C2×#12111yescharacteristic
246C4×C61#54711yescharacteristic
247C4×C61#54811yescharacteristic
248Dic61—22no—
249Dic61—81no—
250Dic61—82no—
251C2×#473—611no—
252#3059D6111yescharacteristic, complemented
253#3046—612no—
254C2×#444—611no—
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—81no—
265C2×C61#241611yescharacteristic
266C2×C61#328912yes—
267D61—42no—
268C2×#137—612no—
269C2×#135—611no—
270C22×Q16C2×D6111yescharacteristic
271C2×#131—611no—
272C2×#131C2×D6111yescharacteristic
273#536C2×D6112yes—
274#520—612no—
275#473—1224no—
276#470—21no—
277#470—21no—
278#467—22no—
279#464—1222no—
280#463—1222no—
281#462—1222no—
282#461—1222no—
283#444—1224no—
284C2×#108—611no—
285C2×#108—611no—
286#428—612no—
287#428—1222no—
288#427—1221no—
289#427C2×D6111yescharacteristic
290C2×#106—611no—
291C2×#106—1221no—
292#424—612no—
293C2×#104—611no—
294C2×#104—611no—
295#419C2×D6112yes—
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×Q8C22×D6112yes—
307#137—1224no—
308#137—1224no—
309#135—1222no—
310#135—1222no—
311C2×Q16—22no—
312C2×Q16—22no—
313C2×Q16—21no—
314C2×Q16—21no—
315C2×Q16—611no—
316C2×Q16—1222no—
317C2×SD16—611no—
318C2×SD16—611no—
319C2×SD16—1222no—
320C2×D8—611no—
321#131—22no—
322#131—22no—
323#131—614no—
324#131—1222no—
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—22no—
335#124—22no—
336#123—42no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C4×Q8—1222no—
340C4×D4—1222no—
341C4×D4—2442no—
342#117—1222no—
343#117C22×D6112yes—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—1222no—
346C2×C42—611no—
347#108—1222no—
348#108—1222no—
349#107—1222no—
350#107C22⋊D6112yes—
351#106—1221no—
352#106—1221no—
353#106—1222no—
354#106C22⋊D6111yescharacteristic
355#106C22⋊D6111yescharacteristic
356C4⋊C8—1222no—
357C4⋊C8—1222no—
358#104—1222no—
359#104—1222no—
360#103—22no—
361#103—22no—
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—21no—
372#41—21no—
373#41—612no—
374#41—612no—
375#41—1221no—
376#41—1221no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—611no—
384C2×Q8D4×D6111yescharacteristic
385C2×D4—21no—
386C2×D4—611no—
387C2×D4—1221no—
388C2×D4D4×D6111yescharacteristic
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—1221no—
392C22×C4—1222no—
393C22×C4—1221no—
394C22×C4—2442no—
395C22×C4C23×D6111yescharacteristic
396Q16—42no—
397Q16—42no—
398Q16—1222no—
399Q16—2444no—
400SD16—1222no—
401SD16—1222no—
402SD16—2444no—
403D8—1222no—
404M16—21no—
405M16—21no—
406M16—42no—
407M16—612no—
408M16—1221no—
409M16—1224no—
410M16—1222no—
411M16—1222no—
412C2×C8—21no—
413C2×C8—42no—
414C2×C8—611no—
415C2×C8—611no—
416C2×C8—1221no—
417C2×C8—1221no—
418C2×C8—1221no—
419C2×C8—1224no—
420C2×C8—1222no—
421C2×C8D4×D6112yes—
422C2×C8#1175822412yes—
423C2×C8C2×C22⋊D6111yescharacteristic
424C2×C8C2×C22⋊D6111yescharacteristic
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4—2442no—
430C4⋊C4C2×C22⋊D6112yes—
431C4⋊C4C2×C22⋊D6112yes—
432C22⋊C4—22no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—2442no—
436C42—22no—
437C42—1221no—
438C42—1221no—
439C42—2442no—
440C23—21no—
441C23—1221no—
442C23#1451982211yescharacteristic
443Q8—21no—
444Q8—21no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—612no—
450Q8—1221no—
451D4—21no—
452D4—21no—
453D4—41no—
454D4—41no—
455D4—612no—
456D4—1222no—
457D4—1222no—
458D4—1221no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—1221no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—9762no—
471C2×C4—4882no—
472C2×C4—2442no—
473C2×C4—2441no—
474C2×C4—2442no—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C2×D4×D6111yescharacteristic
477C2×C4C22×C22⋊D6111yescharacteristic
478C2×C4#41×D6111yescharacteristic
479C2×C4C2×#1175822411yescharacteristic
480C2×C4#1451982411yescharacteristic
481C8—41no—
482C8—1221no—
483C8—1221no—
484C8—1222no—
485C8—2442no—
486C8#1451974812yes—
487C8#1451974812yes—
488C22—22no—
489C22—41no—
490C22—42no—
491C22—1222no—
492C22—2442no—
493C22#1684034111yescharacteristic
494C22D4×C22⋊D6111yescharacteristic
495C22#1684045911yescharacteristic
496C4—41no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4—1222no—
501C4—4881no—
502C4—4882no—
503C4#121×D6111yescharacteristic
504C4C2×#1451974811yescharacteristic
505C4#1684041211yescharacteristic
506C4#1684041911yescharacteristic
507C2—21no—
508C2—21no—
509C2—81no—
510C2—2442no—
511C2#2144925511yescharacteristic
512C2#2145759012yes—
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3059C2
socleC22×C61D42
prime radicalC2×#14519797C22
hypercenter#3059D61
hypocenterC61#24800
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3059

Complements

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

NH ≅ G/Ntype
1. C2×#16840034C2semidirect
4. #21454411C2semidirect
5. #21447186C2semidirect
6. #21447033C2semidirect
7. #21446691C2semidirect
8. #21446573C2semidirect
9. #21446185C2semidirect
10. #21445878C2semidirect
11. C61×#3059C2semidirect
252. #3059D61semidirect
303. C61#24800semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
#214553652
C2×#168398822
#214544112
#214471862
#214470332
#214466912
#214465732
#214461852
#214458782
C61×#30592
#2480061