Finite Group Databasebeta

← #29595000

Subgroups — #29595000

Order 15616.

794 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839746C211yesmaximal, characteristic
2#21448411C211yesmaximal, characteristic
3#21443945C214yesmaximal, complemented
4C2×C61×#413C211yesmaximal, characteristic
5C22×#11758194—21no—
6C2×#14519736C411yescharacteristic
7C22×C4⋊Dic61C411yescharacteristic
8#16839746—22no—
9#16839746C414yescomplemented
10C2×#14519696C2212yes—
11C2×#14519694—21no—
12#16839669C2212yes—
13#16839665C412yes—
14#16839642—21no—
15#16839642—21no—
16#16839622—21no—
17#16839622—21no—
18C22×#41×C61C2211yescharacteristic
19C22×M16×C61—21no—
20C61×#413—22no—
21C61×#413C414yescomplemented
22C2×#101×C61C2212yes—
23C2×#100×C61—21no—
24C23×Dic61—21no—
25C2×#11758194—24no—
26C2×#11758194—22no—
27C2×#11758194—41no—
28C2×#11758194—42no—
29C22×#92418—41no—
30#14519736—42no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61—41no—
34C2×C4⋊Dic61—41no—
35C2×C4⋊Dic61C2×C412yes—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61—22no—
39C2×C4×Dic61C2×C412yes—
40#14519696—22no—
41#14519696C2×C414yescomplemented
42#14519694—24no—
43#14519684—22no—
44#14519684—22no—
45C2×#41×C61—41no—
46C2×#41×C61—41no—
47C2×#41×C61—42no—
48C2×#41×C61C2×C412yes—
49C2×#41×C61C2×C412yes—
50C2×#41×C61D412yes—
51C2×#41×C61Q812yes—
52C22×Q8×C61C2311yescharacteristic
53C22×D4×C61C2×C411yescharacteristic
54C22×D4×C61D411yescharacteristic
55C22×D4×C61D411yescharacteristic
56C23×C4×C61C2×C411yescharacteristic
57C23×C4×C61D411yescharacteristic
58C23×C4×C61D411yescharacteristic
59C2×M16×C61—24no—
60C2×M16×C61—22no—
61C2×M16×C61—41no—
62C2×M16×C61—42no—
63C22×C8×C61—41no—
64#101×C61—22no—
65#101×C61C2×C414yescomplemented
66#100×C61—24no—
67C24×C61C22⋊C411yescharacteristic
68C24×C61C22⋊C411yescharacteristic
69C24×C61C22⋊C411yescharacteristic
70#41×C61—42no—
71#41×C61—42no—
72#41×C61—42no—
73#41×C61—42no—
74#41×C61—42no—
75#41×C61—42no—
76#41×C61—42no—
77#41×C61—42no—
78C2×Q8×C61—41no—
79C2×Q8×C61—41no—
80C2×Q8×C61C22×C411yescharacteristic
81C2×Q8×C61C22×C411yescharacteristic
82C2×Q8×C61C2×D411yescharacteristic
83C2×Q8×C61C2×Q811yescharacteristic
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C2×D4×C61—41no—
90C2×D4×C61C4212yes—
91C2×D4×C61C22⋊C412yes—
92C2×D4×C61C22⋊C412yes—
93C2×D4×C61C4⋊C412yes—
94C2×D4×C61C4⋊C412yes—
95C2×D4×C61C4⋊C412yes—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61C4212yes—
106C22×C4×C61C22⋊C412yes—
107C22×C4×C61C22⋊C412yes—
108C22×C4×C61C22⋊C411yescharacteristic
109C22×C4×C61C4⋊C412yes—
110C22×C4×C61C4⋊C412yes—
111C22×C4×C61C4⋊C412yes—
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×Dic61—22no—
116C22×Dic61—22no—
117C22×Dic61—42no—
118C22×Dic61—82no—
119M16×C61—24no—
120M16×C61—42no—
121M16×C61—44no—
122C2×C8×C61—42no—
123C2×C8×C61—44no—
124C22⋊Dic61—42no—
125C22⋊Dic61—82no—
126C22⋊Dic61—82no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129C4⋊Dic61—42no—
130C4⋊Dic61—42no—
131C4×Dic61—42no—
132#11758194—24no—
133#11758194—42no—
134#11758194—44no—
135C2×#92418—42no—
136C2×#92418—44no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61#9512yes—
150C23×C61#9512yes—
151C23×C61#9512yes—
152C23×C61C2×C22⋊C411yescharacteristic
153Q8×C61—41no—
154Q8×C61—41no—
155Q8×C61—41no—
156Q8×C61—41no—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—42no—
160D4×C61—42no—
161D4×C61—42no—
162D4×C61—42no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61#9512yes—
182C2×C4×C61#9912yes—
183C2×C4×C61#9911yescharacteristic
184C2×C4×C61#9911yescharacteristic
185C2×C4×C61C2×C4211yescharacteristic
186C2×C4×C61C2×C22⋊C411yescharacteristic
187C2×C4×C61C2×C22⋊C411yescharacteristic
188C2×C4×C61C2×C4⋊C411yescharacteristic
189C2×C4×C61C2×C4⋊C411yescharacteristic
190C2×C4×C61C2×C4⋊C411yescharacteristic
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—84no—
198C8×C61—44no—
199#92418—44no—
200#12587—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—22no—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61—41no—
213C22×C61—41no—
214C22×C61—41no—
215C22×C61—41no—
216C22×C61—81no—
217C22×C61—81no—
218C22×C61C2×#9511yescharacteristic
219C22×C61C2×#9911yescharacteristic
220C22×C61C2×#9911yescharacteristic
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61#34212yes—
228C4×C61#34212yes—
229Dic61—82no—
230Dic61—82no—
231C2×#413—611no—
232C2×#413D6111yescharacteristic
233#2311—611no—
234#1394—614no—
235C2×C61—21no—
236C2×C61—21no—
237C2×C61—42no—
238C2×C61—42no—
239C2×C61—42no—
240C2×C61C2×#34211yescharacteristic
241C2×C61#213411yescharacteristic
242C2×C61#213511yescharacteristic
243C22×#41C2×D6111yescharacteristic
244C22×M16—21no—
245C22×M16—1221no—
246C2×#117—611no—
247C22×C4⋊C4—611no—
248#413—22no—
249#413—614no—
250#413—1222no—
251#413Dic6114yescomplemented
252C2×#101—612no—
253C2×#101C2×D6112yes—
254C2×#100—21no—
255C2×#100—1221no—
256#391—612no—
257#390—612no—
258#386—1221no—
259#386—1221no—
260#380—1221no—
261#380—1221no—
262C61#1258711yescharacteristic, complemented, Sylow
263C2×#41—41no—
264C2×#41—41no—
265C2×#41—42no—
266C2×#41Dic12212yes—
267C2×#41C4×D6112yes—
268C2×#41C2×Dic6112yes—
269C2×#41C22⋊D6112yes—
270C22×Q8C22×D6111yescharacteristic
271C22×D4C4×D6111yescharacteristic
272C22×D4D24411yescharacteristic
273C22×D4C22⋊D6111yescharacteristic
274C23×C4—1221no—
275C23×C4C4×D6111yescharacteristic
276C23×C4D24411yescharacteristic
277C23×C4C22⋊D6111yescharacteristic
278C2×M16—24no—
279C2×M16—22no—
280C2×M16—41no—
281C2×M16—42no—
282C2×M16—1224no—
283C2×M16—1222no—
284C2×M16—2441no—
285C2×M16—2442no—
286C22×C8—41no—
287C22×C8—2441no—
288#117—2442no—
289C2×C4⋊C4—612no—
290C2×C4⋊C4—1221no—
291C2×C4⋊C4—1221no—
292C2×C4⋊C4—2441no—
293C2×C4⋊C4—2441no—
294C2×C22⋊C4—1222no—
295C2×C22⋊C4—1222no—
296C2×C22⋊C4—1221no—
297C2×C42—612no—
298#101—22no—
299#101—614no—
300#101—1222no—
301#101C2×Dic6114yescomplemented
302#100—24no—
303#100—1224no—
304#95—1222no—
305#95—1222no—
306C24#1175819811yescharacteristic
307C24#1175819811yescharacteristic
308C24#1175819811yescharacteristic
309#41—42no—
310#41—42no—
311#41—42no—
312#41—42no—
313#41—42no—
314#41—42no—
315#41—42no—
316#41—42no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8C2×Dic12211yescharacteristic
320C2×Q8C2×C4×D6111yescharacteristic
321C2×Q8C22×Dic6111yescharacteristic
322C2×Q8C2×C22⋊D6111yescharacteristic
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4C4×Dic6112yes—
330C2×D4#1175819612yes—
331C2×D4#1175819612yes—
332C2×D4C4⋊Dic6112yes—
333C2×D4#1175819812yes—
334C2×D4C22⋊Dic6112yes—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—1222no—
345C22×C4—1222no—
346C22×C4—4882no—
347C22×C4—2442no—
348C22×C4C4×Dic6112yes—
349C22×C4#1175819612yes—
350C22×C4#1175819612yes—
351C22×C4C4⋊Dic6112yes—
352C22×C4#1175819812yes—
353C22×C4#1175819811yescharacteristic
354C22×C4C22⋊Dic6112yes—
355C22×C4C2×C4×D6111yescharacteristic
356C22×C4C2×D24411yescharacteristic
357C22×C4C2×C22⋊D6111yescharacteristic
358M16—24no—
359M16—42no—
360M16—44no—
361M16—1224no—
362M16—2442no—
363M16—2444no—
364C2×C8—42no—
365C2×C8—44no—
366C2×C8—2442no—
367C2×C8—2444no—
368C4⋊C4—2442no—
369C4⋊C4—2441no—
370C4⋊C4—2441no—
371C4⋊C4—2442no—
372C22⋊C4—4882no—
373C22⋊C4—4882no—
374C22⋊C4—2442no—
375C42—2442no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23#1451968412yes—
389C23#1451968412yes—
390C23#1451968412yes—
391C23C2×#1175819811yescharacteristic
392Q8—41no—
393Q8—41no—
394Q8—41no—
395Q8—41no—
396D4—42no—
397D4—42no—
398D4—42no—
399D4—42no—
400D4—42no—
401D4—42no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—4884no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4#1451968412yes—
428C2×C4#1451969112yes—
429C2×C4#1451969111yescharacteristic
430C2×C4#1451969111yescharacteristic
431C2×C4C2×C4×Dic6111yescharacteristic
432C2×C4C2×#1175819811yescharacteristic
433C2×C4C2×C22⋊Dic6111yescharacteristic
434C2×C4C2×#1175819611yescharacteristic
435C2×C4C2×#1175819611yescharacteristic
436C2×C4C2×C4⋊Dic6111yescharacteristic
437C8—44no—
438C8—2444no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—81no—
455C22—81no—
456C22C2×#1451968411yescharacteristic
457C22C2×#1451969111yescharacteristic
458C22C2×#1451969111yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—4882no—
466C4—4882no—
467C4#1683952812yes—
468C4#1683952812yes—
469C2—21no—
470C2—21no—
471C2—42no—
472C2—42no—
473C2—42no—
474C2C2×#1683952811yescharacteristic
475C2#2144751011yescharacteristic
476C2#2144751311yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC2×C4×C61C2×C42
FrattiniC22×Q8C22×D61
FittingC2×C61×#413C2
socleC22×C61C2×#99
prime radicalC22×#41×C61C22
cosocleC22×Q8×C61C23
hypercenterC2×#413D61
hypocenterC61#12587
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
3. #21443945C2semidirect
9. #16839746C4semidirect
21. C61×#413C4semidirect
41. #14519696C2×C4semidirect
65. #101×C61C2×C4semidirect
251. #413Dic61semidirect
262. C61#12587semidirect
301. #101C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168397462
#214484112
#214439452
C2×C61×#4132
#1258761