Finite Group Databasebeta

← #29580566

Subgroups — #29580566

Order 15616.

1020 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839913C211yesmaximal, characteristic
2#21451935C211yesmaximal, characteristic, complemented
3C2×#16839500C211yesmaximal, characteristic, complemented
4C2×#16839499C211yesmaximal, characteristic, complemented
5#21444060C211yesmaximal, characteristic
6#21443661C211yesmaximal, characteristic, complemented
7C2×C61×#450C211yesmaximal, characteristic
8C22×#11758200—21no—
9C2×#14519756C2211yescharacteristic
10C22×C22⋊Dic61—22no—
11#16839913—22no—
12#16839913—22no—
13C2×#14519698—21no—
14C2×C22⋊#92418C2211yescharacteristic
15#16839693—21no—
16#16839692—21no—
17#16839682—21no—
18#16839681—21no—
19#16839620—22no—
20C2×#14519684C2211yescharacteristic
21C2×#14519684C2211yescharacteristic
22#16839500—22no—
23#16839499—22no—
24C23×D4×C61C2211yescharacteristic
25C22×SD16×C61—21no—
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#450—22no—
28C61×#450—22no—
29C2×#102×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C23×Dic61—21no—
33C23×Dic61D412yes—
34C2×#11758200—24no—
35C2×#11758200—44no—
36C22×#92418—21no—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196—22no—
44C2×#11758196—22no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—44no—
47C2×C22⋊Dic61—44no—
48C2×C22⋊Dic61—44no—
49#14519698—24no—
50C22⋊#92418—21no—
51C22⋊#92418D412yes—
52#14519684—22no—
53#14519684—22no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C25×C61—21no—
59C22×Q8×C61D411yescharacteristic
60C22×D4×C61—21no—
61C22×D4×C61—21no—
62C22×D4×C61—21no—
63C22×D4×C61—21no—
64C22×D4×C61—21no—
65C22×D4×C61—44no—
66C22×D4×C61D412yes—
67C23×C4×C61C2311yescharacteristic
68C2×SD16×C61—24no—
69C2×SD16×C61—44no—
70C22×C8×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61D411yescharacteristic
77C2×C22⋊C4×C61—21no—
78#102×C61—24no—
79C22⋊C8×C61—21no—
80C22⋊C8×C61D412yes—
81C24×C61—24no—
82C24×C61—22no—
83C24×C61—42no—
84C24×C61—42no—
85C24×C61—42no—
86C24×C61—41no—
87C24×C61—41no—
88C24×C61—42no—
89C24×C61—42no—
90C24×C61C2×D411yescharacteristic
91C2×Q8×C61—21no—
92C2×Q8×C61—21no—
93C2×Q8×C61—42no—
94C2×D4×C61—24no—
95C2×D4×C61—24no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61—42no—
99C2×D4×C61—42no—
100C2×D4×C61—44no—
101C2×D4×C61—44no—
102C2×D4×C61—44no—
103C2×D4×C61—44no—
104C2×D4×C61—41no—
105C2×D4×C61—41no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—42no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×Dic61—22no—
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61—42no—
121C22×Dic61—42no—
122C22×Dic61—41no—
123C22×Dic61—42no—
124C22×Dic61—41no—
125C22×Dic61—42no—
126C22×Dic61SD1614yescomplemented
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×Dic122—42no—
130SD16×C61—48no—
131C2×C8×C61—22no—
132C2×C8×C61—42no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C22⋊C4×C61—41no—
139C22⋊Dic61—416no—
140C22⋊Dic61—41no—
141C22⋊Dic61—41no—
142#11758200—48no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145#11758196—41no—
146#11758196—41no—
147#11758196—82no—
148#11758196—82no—
149C2×#92418—22no—
150C2×#92418—42no—
151C23×C61—21no—
152C23×C61—21no—
153C23×C61—22no—
154C23×C61—24no—
155C23×C61—44no—
156C23×C61—44no—
157C23×C61—42no—
158C23×C61—44no—
159C23×C61—44no—
160C23×C61—44no—
161C23×C61—44no—
162C23×C61—42no—
163C23×C61—44no—
164C23×C61—44no—
165C23×C61—82no—
166C23×C61—82no—
167C23×C61—82no—
168C23×C61—82no—
169C23×C61—82no—
170C23×C61—82no—
171C23×C61—81no—
172C23×C61—81no—
173C23×C61—81no—
174C23×C61—81no—
175C23×C61#12011yescharacteristic
176C23×C61#12111yescharacteristic
177C23×C61#12711yescharacteristic
178C23×C61C2×SD1612yes—
179C23×C61#13712yes—
180Q8×C61—42no—
181Q8×C61—81no—
182D4×C61—24no—
183D4×C61—24no—
184D4×C61—42no—
185D4×C61—42no—
186D4×C61—44no—
187D4×C61—44no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61#12111yescharacteristic
203C2×Dic61—24no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—44no—
207C2×Dic61—44no—
208C2×Dic61—42no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—42no—
214C2×Dic61—82no—
215C2×Dic61—82no—
216C2×Dic61—81no—
217C2×Dic61—81no—
218Dic122—42no—
219Dic122—81no—
220C8×C61—42no—
221#92418—42no—
222#9966—611nomaximal, Sylow
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—48no—
232C22×C61—44no—
233C22×C61—84no—
234C22×C61—84no—
235C22×C61—84no—
236C22×C61—84no—
237C22×C61—84no—
238C22×C61—82no—
239C22×C61—82no—
240C22×C61#39211yescharacteristic
241C22×C61#44812yes—
242C22×C61#45311yescharacteristic
243C22×C61#45411yescharacteristic
244C22×C61B(4,𝔽2)12yes—
245C22×C61#46112yes—
246C22×C61#49211yescharacteristic
247C22×C61#49711yescharacteristic
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—81no—
253C4×C61—81no—
254Dic61—44no—
255Dic61—82no—
256Dic61—81no—
257Dic61—81no—
258C2×#450—611no—
259C2×#450D6111yescharacteristic
260#2659—611no—
261C2×#328—611no—
262C2×#328—611no—
263#1438—611no—
264#1280—611no—
265C2×C61—21no—
266C2×C61—22no—
267C2×C61—21no—
268C2×C61—84no—
269C2×C61#159612yes—
270C2×C61#159912yes—
271C2×C61#200611yescharacteristic
272C2×C61#201311yescharacteristic
273C2×C61#202111yescharacteristic
274C23×D4C2×D6111yescharacteristic
275C22×SD16—21no—
276C22×SD16—1221no—
277C2×C22⋊Q8—611no—
278C2×C22⋊Q8C2×D6111yescharacteristic
279C22×C22⋊C4—1222no—
280#450—22no—
281#450—22no—
282#450—1222no—
283#450—1222no—
284C2×#102—21no—
285C2×#102—1221no—
286C2×C22⋊C8—611no—
287C2×C22⋊C8C2×D6111yescharacteristic
288#398—1221no—
289#398—1221no—
290#395—1221no—
291#395—1221no—
292#380—1222no—
293C2×#95—611no—
294C2×#95—611no—
295#328—1222no—
296#328—1222no—
297C61#996611yescharacteristic, complemented, Sylow
298C25—21no—
299C22×Q8—611no—
300C22×Q8C22⋊D6111yescharacteristic
301C22×D4—21no—
302C22×D4—21no—
303C22×D4—21no—
304C22×D4—21no—
305C22×D4—21no—
306C22×D4—44no—
307C22×D4C22⋊D6112yes—
308C23×C4—612no—
309C23×C4—1221no—
310C23×C4C22×D6111yescharacteristic
311C2×SD16—24no—
312C2×SD16—44no—
313C2×SD16—1224no—
314C2×SD16—2444no—
315C22×C8—21no—
316C22×C8—1221no—
317C22⋊Q8—21no—
318C22⋊Q8—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—1221no—
322C22⋊Q8—1221no—
323C22⋊Q8—1221no—
324C22⋊Q8—1221no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—1222no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4C22⋊D6111yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1221no—
333C2×C22⋊C4—2444no—
334C2×C22⋊C4—2444no—
335C2×C22⋊C4—2444no—
336#102—24no—
337#102—1224no—
338C22⋊C8—21no—
339C22⋊C8—612no—
340C22⋊C8—1221no—
341C22⋊C8C22⋊D6112yes—
342#95—1222no—
343#95—1222no—
344#95—1221no—
345#95—1221no—
346#95—1221no—
347#95—1221no—
348C24—24no—
349C24—22no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—41no—
354C24—41no—
355C24—42no—
356C24—42no—
357C24D4×D6111yescharacteristic
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8—1221no—
362C2×Q8—1221no—
363C2×Q8—2442no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—41no—
375C2×D4—41no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—42no—
379C22×C4—614no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—2442no—
386C22×C4—2441no—
387C22×C4—2441no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4—2442no—
391C22×C4D4×D6111yescharacteristic
392C22×C4D4×D6111yescharacteristic
393C22×C4#1175822411yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C2×C22⋊D6111yescharacteristic
397SD16—48no—
398SD16—2448no—
399C2×C8—22no—
400C2×C8—42no—
401C2×C8—1222no—
402C2×C8—2442no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—4882no—
410C4⋊C4—4882no—
411C4⋊C4—2441no—
412C4⋊C4—2441no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—24416no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C23—22no—
419C23—21no—
420C23—21no—
421C23—24no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—42no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—81no—
438C23—81no—
439C23—82no—
440C23—81no—
441C23—81no—
442C23#1451974511yescharacteristic
443C23#1451974911yescharacteristic
444C23#1451977311yescharacteristic
445C23SD16×D6112yes—
446C23#1451980612yes—
447Q8—42no—
448Q8—81no—
449Q8—4881no—
450Q8—2442no—
451D4—24no—
452D4—24no—
453D4—42no—
454D4—42no—
455D4—44no—
456D4—44no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—1224no—
469C2×C4—4881no—
470C2×C4—4882no—
471C2×C4—4881no—
472C2×C4—4882no—
473C2×C4—2444no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2444no—
483C2×C4#1451974511yescharacteristic
484C2×C4#1451974611yescharacteristic
485C2×C4#1451974811yescharacteristic
486C2×C4#1451974911yescharacteristic
487C8—42no—
488C8—2442no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—21no—
496C22—21no—
497C22—48no—
498C22—44no—
499C22—84no—
500C22—84no—
501C22—84no—
502C22—84no—
503C22—84no—
504C22—82no—
505C22—82no—
506C22#1683967011yescharacteristic
507C22#1683989912yes—
508C22#1683992811yescharacteristic
509C22#1683993311yescharacteristic
510C22#1683994812yes—
511C22#1683996512yes—
512C22#1684013311yescharacteristic
513C22#1684015311yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—81no—
519C4—81no—
520C4—4882no—
521C4—4881no—
522C4—4881no—
523C4—2444no—
524C2—21no—
525C2—22no—
526C2—21no—
527C2—84no—
528C2#2144474012yes—
529C2#2144475712yes—
530C2#2144685911yescharacteristic
531C2#2144690011yescharacteristic
532C2#2144694211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#450C2
socleC23×C61#120
prime radicalC23×D4×C61C22
hypercenterC2×#450D61
hypocenterC61#9966
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

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

NH ≅ G/Ntype
2. #21451935C2semidirect
3. C2×#16839500C2semidirect
4. C2×#16839499C2semidirect
6. #21443661C2semidirect
126. C22×Dic61SD16semidirect
297. C61#9966semidirect

Maximal subgroups

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

subgroupindex
C2×#168399132
#214519352
C2×#168395002
C2×#168394992
#214440602
#214436612
C2×C61×#4502
#996661