Finite Group Databasebeta

← #30601660

Subgroups — #30601660

Order 15616.

920 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 122 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
1#21461673C211yesmaximal, characteristic, complemented
2#21461545C211yesmaximal, characteristic, complemented
3C2×#16840606C212yesmaximal, complemented
4C2×#16840605C212yesmaximal, complemented
5C2×#16839662C211yesmaximal, characteristic
6C2×C4⋊#92420—21no—
7C2×C22⋊#92420C2212yes—
8C2×C22⋊#92420C2212yes—
9C2×#14519832—21no—
10#16840628—21no—
11#16840628—21no—
12#16840606—24no—
13#16840605—24no—
14C23×C22⋊D61C411yescharacteristic
15C22×#11758196C411yescharacteristic
16C22×C22⋊Dic61C2211yescharacteristic
17C22×#11758198C411yescharacteristic
18C22×#11758198C2211yescharacteristic
19C22×C4×Dic61C2211yescharacteristic
20#16839662—21no—
21#16839662—21no—
22#16839662—21no—
23#16839662—21no—
24C22×C22⋊C4×C61C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27C4⋊#92420—42no—
28C22⋊#92420—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C22⋊#92420D414yescomplemented
32#14519832—42no—
33C24×D61C2×C411yescharacteristic
34C22×C22⋊D61—22no—
35C22×C22⋊D61—22no—
36C22×C22⋊D61—21no—
37C22×C22⋊D61—21no—
38C22×C22⋊D61—42no—
39C22×C22⋊D61—42no—
40C23×Dic61C2×C411yescharacteristic
41C23×Dic61C2×C411yescharacteristic
42C23×Dic61C2311yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196—42no—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—22no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×#11758198—22no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×#11758198—22no—
56C2×#11758198—42no—
57C2×C4×Dic61—42no—
58C2×C4×Dic61D411yescharacteristic
59C2×C4×Dic61D411yescharacteristic
60C2×C4×Dic61D411yescharacteristic
61C2×C4×Dic61D411yescharacteristic
62C25×C61C2×C411yescharacteristic
63C23×C4×C61C2×C411yescharacteristic
64C23×C4×C61C2×C411yescharacteristic
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61—42no—
68C24×C61—21no—
69C24×C61—21no—
70C24×C61—22no—
71C24×C61—22no—
72C24×C61—41no—
73C24×C61—41no—
74C24×C61—41no—
75C24×C61—41no—
76C24×C61M1612yes—
77C24×C61C22×C411yescharacteristic
78C23×D61—21no—
79C23×D61—21no—
80C23×D61—22no—
81C23×D61—22no—
82C23×D61—41no—
83C23×D61—41no—
84C23×D61—41no—
85C23×D61—41no—
86C23×D61M1612yes—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61C22⋊C411yescharacteristic
95C22×C4×C61C22⋊C411yescharacteristic
96C2×C22⋊D61—24no—
97C2×C22⋊D61—24no—
98C2×C22⋊D61—48no—
99C2×C22⋊D61—48no—
100C2×C22⋊D61—42no—
101C2×C22⋊D61—42no—
102C2×C22⋊D61—42no—
103C2×C22⋊D61—42no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—22no—
107C22×Dic61—22no—
108C22×Dic61—21no—
109C22×Dic61—22no—
110C22×Dic61—41no—
111C22×Dic61—41no—
112C22×Dic61—41no—
113C22×Dic61—41no—
114C22×Dic61C22⋊C411yescharacteristic
115C22×Dic61C22⋊C411yescharacteristic
116C22×Dic61C2×D411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C22×Dic61C2×D412yes—
119C22×Dic61#4112yes—
120C2×#92420—24no—
121C2×#92420—42no—
122C2×#92420—42no—
123C2×#92420—42no—
124C2×#92420—42no—
125C2×#92420—42no—
126C22⋊C4×C61—42no—
127C22⋊C4×C61—42no—
128C22⋊Dic61—42no—
129C22⋊Dic61—41no—
130C22⋊Dic61—41no—
131#11758198—42no—
132#11758198—42no—
133#11758198—42no—
134#11758198—41no—
135#11758198—41no—
136#11758196—42no—
137#11758196—42no—
138C4×Dic61—22no—
139C4×Dic61—22no—
140C4×Dic61—41no—
141C4×Dic61—41no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—22no—
147C23×C61—24no—
148C23×C61—42no—
149C23×C61—42no—
150C23×C61—42no—
151C23×C61—42no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—42no—
159C23×C61—82no—
160C23×C61—82no—
161C23×C61—82no—
162C23×C61—82no—
163C23×C61C2×C22⋊C411yescharacteristic
164C23×C61C4×D412yes—
165C23×C61C2×M1611yescharacteristic
166C23×C61C2×M1611yescharacteristic
167C23×C61#13112yes—
168C22×D61—22no—
169C22×D61—24no—
170C22×D61—42no—
171C22×D61—42no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61—82no—
182C22×D61—82no—
183C22×D61—82no—
184C22×D61—82no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—44no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—82no—
195C2×C4×C61#10012yes—
196C22⋊D61—48no—
197C22⋊D61—48no—
198C2×Dic61—24no—
199C2×Dic61—24no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—82no—
214C2×Dic61#9912yes—
215C2×Dic61#10012yes—
216C2×Dic61#12112yes—
217C2×Dic61#12411yescharacteristic
218C2×Dic61#12711yescharacteristic
219#92420—44no—
220#92420—82no—
221#9653—611nomaximal, Sylow
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—84no—
234C22×C61—84no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61#39011yescharacteristic
240C22×C61#40711yescharacteristic
241C22×C61#40811yescharacteristic
242C22×C61C2×#9911yescharacteristic
243C22×C61#41011yescharacteristic
244C22×C61C2×#10011yescharacteristic
245C22×C61C2×#10011yescharacteristic
246C22×C61#43512yes—
247C22×C61#43512yes—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—84no—
252C2×D61—81no—
253C2×D61—81no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—84no—
257C4×C61—22no—
258C4×C61—41no—
259C4×C61—81no—
260Dic61—22no—
261Dic61—21no—
262Dic61—21no—
263Dic61—22no—
264Dic61—22no—
265Dic61—81no—
266C2×#390—611no—
267#1797—611no—
268#1749—611no—
269C2×#323—612no—
270C2×#323—612no—
271C2×C61—22no—
272C2×C61—22no—
273C2×C61—82no—
274C2×C61#146211yescharacteristic
275C2×C61#146611yescharacteristic
276C2×C61#146911yescharacteristic
277C2×C61#147111yescharacteristic
278C2×C61#195511yescharacteristic
279C2×C61#196411yescharacteristic
280C2×C61#196511yescharacteristic
281D61—82no—
282C23×D4—611no—
283C22×C4⋊C4—611no—
284C22×C22⋊C4—611no—
285C22×C22⋊C4—611no—
286C22×C22⋊C4—611no—
287C22×C22⋊C4#692311yescharacteristic
288C22×C42—611no—
289C2×C4⋊C8—1221no—
290C2×C22⋊C8—612no—
291C2×C22⋊C8—612no—
292C2×#97—1221no—
293#390—1221no—
294#390—1221no—
295#390—1221no—
296#390—1221no—
297#336—1221no—
298#336—1221no—
299#323—1224no—
300#323—1224no—
301C61#965311yescharacteristic, complemented, Sylow
302C25—611no—
303C25C2×#692311yescharacteristic
304C22×D4—1221no—
305C22×D4—1221no—
306C22×D4—1222no—
307C22×D4—1222no—
308C22×D4—2442no—
309C22×D4—2442no—
310C23×C4—611no—
311C23×C4—611no—
312C23×C4—611no—
313C23×C4C2×#692311yescharacteristic
314C23×C4C2×#692311yescharacteristic
315C22×C8—1222no—
316C22×C8—1222no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—2442no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1222no—
328C2×C22⋊C4—1221no—
329C2×C22⋊C4—1221no—
330C2×C22⋊C4—1222no—
331C2×C22⋊C4—1221no—
332C2×C22⋊C4—1221no—
333C2×C22⋊C4—2442no—
334C2×C42—611no—
335C2×C42—611no—
336C2×C42—611no—
337C2×C42—611no—
338C2×C42—2442no—
339C4⋊C8—2442no—
340C22⋊C8—614no—
341C22⋊C8—1222no—
342C22⋊C8—1222no—
343C22⋊C8—1222no—
344#97—2442no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—41no—
350C24—41no—
351C24—41no—
352C24—41no—
353C24—612no—
354C24—1222no—
355C24—1222no—
356C24—1221no—
357C24—1221no—
358C24—2441no—
359C24—2441no—
360C24—2441no—
361C24—2441no—
362C24#1175821312yes—
363C24C22×#692311yescharacteristic
364C2×D4—1224no—
365C2×D4—1224no—
366C2×D4—2442no—
367C2×D4—2442no—
368C2×D4—2448no—
369C2×D4—2448no—
370C2×D4—2442no—
371C2×D4—2442no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—612no—
384C22×C4—612no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1222no—
390C22×C4—1221no—
391C22×C4—2441no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4—2441no—
395C22×C4C22⋊#692311yescharacteristic
396C22×C4C22⋊#692311yescharacteristic
397C2×C8—1224no—
398C2×C8—2442no—
399C2×C8—2442no—
400C2×C8—2442no—
401C2×C8—2442no—
402C2×C8—2442no—
403C4⋊C4—2442no—
404C4⋊C4—2442no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—2442no—
408C22⋊C4—2442no—
409C22⋊C4—2441no—
410C22⋊C4—2441no—
411C22⋊C4—2442no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C22⋊C4—2442no—
415C42—1222no—
416C42—1222no—
417C42—2441no—
418C42—2441no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—1224no—
441C23—1222no—
442C23—4882no—
443C23—4882no—
444C23—4882no—
445C23—4882no—
446C23—2442no—
447C23—2442no—
448C23—2442no—
449C23—2442no—
450C23—2441no—
451C23—2441no—
452C23—2441no—
453C23—2441no—
454C23—2441no—
455C23—2441no—
456C23—2442no—
457C23C2×C22⋊#692311yescharacteristic
458C23D4×#692312yes—
459C23D61⋊M1611yescharacteristic
460C23C2×#1175821311yescharacteristic
461C23#1451986412yes—
462D4—2448no—
463D4—2448no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—612no—
475C2×C4—611no—
476C2×C4—611no—
477C2×C4—612no—
478C2×C4—612no—
479C2×C4—1224no—
480C2×C4—1224no—
481C2×C4—1221no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—4882no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4#1451983712yes—
496C8—4882no—
497C8—2444no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—84no—
510C22—84no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—4884no—
516C22—4884no—
517C22—4881no—
518C22—4881no—
519C22—4881no—
520C22—4881no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22#1684071411yescharacteristic
525C22#1684073311yescharacteristic
526C22#1684073511yescharacteristic
527C22D61⋊#9911yescharacteristic
528C22#1684074111yescharacteristic
529C22C2×#1451983711yescharacteristic
530C22D61⋊#10011yescharacteristic
531C22#1684079112yes—
532C22#1684079412yes—
533C4—22no—
534C4—41no—
535C4—81no—
536C4—1222no—
537C4—1222no—
538C4—1221no—
539C4—1222no—
540C4—1221no—
541C4—4881no—
542C2—22no—
543C2—22no—
544C2—82no—
545C2—4882no—
546C2#2146106511yescharacteristic
547C2#2146107811yescharacteristic
548C2#2146108811yescharacteristic
549C2#2146109511yescharacteristic
550C2#2146194411yescharacteristic
551C2#2146196011yescharacteristic
552C2#2146196111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×C22⋊D61C4
cosocleC23×Dic61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#9653
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #21461673C2semidirect
2. #21461545C2semidirect
3. C2×#16840606C2semidirect
4. C2×#16840605C2semidirect
31. C22⋊#92420D4semidirect
301. C61#9653semidirect

Maximal subgroups

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

subgroupindex
#214616732
#214615452
C2×#168406062
C2×#168406052
C2×#168396622
#965361