Finite Group Databasebeta

← #30600411

Subgroups — #30600411

Order 15616.

1010 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 112 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×C4×C22⋊#6923C211yesmaximal, characteristic
2#21461642C211yesmaximal, characteristic
3C2×#16840616C212yesmaximal, complemented
4#21460692C212yesmaximal, complemented
5#21454103C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7C22×C4×#6923—21no—
8C2×#14519834—21no—
9C2×#14519834C2212yes—
10C2×C22⋊#92420—21no—
11C4×C22⋊#6923—24no—
12C2×(C2×#6923)⋊C4C2212yes—
13#16840616—24no—
14C23×C4×D61C2211yescharacteristic
15C2×#14519753C411yescharacteristic, complemented
16C2×#14519753C2211yescharacteristic
17#16839634—21no—
18#16839633—21no—
19#16839633—21no—
20#16839622C411yescharacteristic, complemented
21#16839619C411yescharacteristic, complemented
22#16839619C2211yescharacteristic
23#16839617—21no—
24C2×C22⋊Q8×C61C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C2×C22⋊#6923—24no—
28C2×C22⋊#6923—24no—
29C2×C22⋊#6923—24no—
30C2×C4×#6923—24no—
31C2×C4×#6923—42no—
32C2×C4×#6923—42no—
33#14519834—22no—
34#14519834—22no—
35#14519834—42no—
36C22⋊#92420—42no—
37(C2×#6923)⋊C4—22no—
38(C2×#6923)⋊C4D414yes—
39C24×D61D411yescharacteristic
40C23×Dic61D411yescharacteristic
41C22×C4×D61—22no—
42C22×C4×D61—22no—
43C22×C4×D61—42no—
44C22×C4×D61—42no—
45C22×C4×D61C2×C411yescharacteristic
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49#14519753—22no—
50#14519753—22no—
51#14519753—42no—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×C4⋊Dic61C2×C411yescharacteristic
54C2×#11758196—21no—
55C2×#11758196—21no—
56C2×C22⋊Dic61—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64C22×Q8×C61C2×C411yescharacteristic
65C23×C4×C61C2×C411yescharacteristic
66C22⋊Q8×C61—42no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61C2×C411yescharacteristic
69C2×C22⋊C4×C61—21no—
70C24×C61C22⋊C411yescharacteristic
71C23×D61—22no—
72C23×D61—22no—
73C23×D61—21no—
74C23×D61—21no—
75C23×D61—22no—
76C23×D61—22no—
77C23×D61—41no—
78C23×D61—41no—
79C23×D61C22⋊C411yescharacteristic
80C23×D61C2×D411yescharacteristic
81C22×#6923—24no—
82C22×#6923—24no—
83C22×#6923—24no—
84C22×#6923—42no—
85C22×#6923—42no—
86C22×#6923—42no—
87C22×#6923—42no—
88C2×Q8×C61—22no—
89C2×Q8×C61—41no—
90C2×Q8×C61—41no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—41no—
96C22×C4×C61C22⋊C411yescharacteristic
97C22×C4×C61C22×C411yescharacteristic
98C22×Dic61—22no—
99C22×Dic61—22no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—41no—
103C22×Dic61C22⋊C411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C2×C4×D61—21no—
106C2×C4×D61—21no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—22no—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—48no—
114C2×C4×D61—48no—
115C2×C4×D61—41no—
116C2×C4×D61—41no—
117C2×C4×D61—41no—
118C2×C4×D61—41no—
119C2×C4×D61C2×D412yes—
120C2×C4×D61#4112yes—
121C22⋊#6923—216no—
122C2×#92420—42no—
123C2×#92420—42no—
124C2×#92420—42no—
125C4×#6923—48no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61—42no—
128C22⋊C4×C61—42no—
129C22⋊Dic61—42no—
130#11758198—42no—
131#11758198—42no—
132#11758198—41no—
133#11758198—41no—
134#11758198—42no—
135C4⋊Dic61—22no—
136C4⋊Dic61—41no—
137#11758196—42no—
138#11758196—42no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61#10412yes—
146C23×C61C2×C22⋊C411yescharacteristic
147C22×D61—21no—
148C22×D61—21no—
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—24no—
152C22×D61—24no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—21no—
158C22×D61—21no—
159C22×D61—44no—
160C22×D61—44no—
161C22×D61—44no—
162C22×D61—44no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—42no—
166C22×D61—42no—
167C22×D61#10412yes—
168C22×D61#12011yescharacteristic
169C22×D61#12311yescharacteristic
170C22×D61#13512yes—
171C22×D61#13712yes—
172C2×#6923—28no—
173C2×#6923—44no—
174C2×#6923—44no—
175C2×#6923—44no—
176C2×#6923—44no—
177C2×#6923—44no—
178C2×#6923—44no—
179Q8×C61—82no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—44no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61#9912yes—
192C2×C4×C61C4×D412yes—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—44no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61#9912yes—
205C2×Dic61#12112yes—
206C4×D61—24no—
207C4×D61—24no—
208C4×D61—42no—
209C4×D61—42no—
210C4×D61—44no—
211C4×D61—44no—
212#92420—82no—
213#9146—611nomaximal, Sylow
214C22×C61—22no—
215C22×C61—22no—
216C22×C61—22no—
217C22×C61—41no—
218C22×C61—41no—
219C22×C61—41no—
220C22×C61—41no—
221C22×C61#38611yescharacteristic
222C22×C61C2×#9911yescharacteristic
223C22×C61C2×#9911yescharacteristic
224C22×C61#41611yescharacteristic
225C22×C61#41911yescharacteristic
226C22×C61C2×#10411yescharacteristic
227C22×C61C2×#10411yescharacteristic
228C2×D61—21no—
229C2×D61—21no—
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—22no—
239C2×D61—48no—
240C2×D61—48no—
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—41no—
245C2×D61#45212yes—
246C2×D61#48512yes—
247#6923—48no—
248C4×C61—22no—
249C4×C61—22no—
250C4×C61—81no—
251C4×C61—81no—
252Dic61—22no—
253Dic61—22no—
254Dic61—81no—
255Dic61—81no—
256#2848—611no—
257C2×C4×C22⋊C4—611no—
258#1779—611no—
259C2×#328—612no—
260#1294—612no—
261C2×C61—42no—
262C2×C61#150111yescharacteristic
263C2×C61#150111yescharacteristic
264C2×C61#150811yescharacteristic
265C2×C61#150811yescharacteristic
266C2×C61#187311yescharacteristic
267C2×C61#189611yescharacteristic
268C2×C61#189811yescharacteristic
269D61—22no—
270D61—22no—
271D61—42no—
272C24×C4—611no—
273C2×C22⋊Q8—611no—
274C2×C22⋊Q8—611no—
275C2×C22⋊Q8#692311yescharacteristic, complemented
276C22×C22⋊C4—1221no—
277C22×C42—1221no—
278C2×C22⋊C8—612no—
279C2×C22⋊C8—1221no—
280C2×C22⋊C8—1221no—
281#385—1221no—
282#385—1221no—
283#385—1221no—
284#380—611no—
285#380—611no—
286#380—611no—
287#379—1221no—
288C4×C22⋊C4—1224no—
289C2×#95—612no—
290#328—1224no—
291C61#914611yescharacteristic, complemented, Sylow
292C25—611no—
293C22×Q8C2×#692311yescharacteristic
294C23×C4—611no—
295C23×C4—611no—
296C23×C4—611no—
297C23×C4—611no—
298C23×C4—611no—
299C23×C4—1222no—
300C23×C4—1222no—
301C23×C4—1222no—
302C23×C4—2442no—
303C23×C4—2442no—
304C23×C4C2×#692311yescharacteristic
305C22×C8—1222no—
306C22⋊Q8—42no—
307C22⋊Q8—1222no—
308C22⋊Q8—1222no—
309C22⋊Q8—2442no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4C2×#692311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—1224no—
318C2×C22⋊C4—1224no—
319C2×C22⋊C4—1224no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4—1221no—
323C2×C22⋊C4—1221no—
324C2×C22⋊C4—1221no—
325C2×C42—1224no—
326C2×C42—2442no—
327C2×C42—2442no—
328C22⋊C8—1222no—
329C22⋊C8—1222no—
330C22⋊C8—2442no—
331C22⋊C8—2442no—
332#95—614no—
333#95—1221no—
334#95—1221no—
335#95—1222no—
336#95—1221no—
337C24—611no—
338C24—611no—
339C24—1221no—
340C24—1222no—
341C24—1221no—
342C24—1222no—
343C24—1222no—
344C24—1222no—
345C24—2441no—
346C24—2441no—
347C24C22⋊#692311yescharacteristic
348C2×Q8—22no—
349C2×Q8—41no—
350C2×Q8—41no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—612no—
359C22×C4—612no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4—1222no—
366C22×C4—1222no—
367C22×C4—1222no—
368C22×C4—1224no—
369C22×C4—1224no—
370C22×C4—1224no—
371C22×C4—1222no—
372C22×C4—1222no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—2441no—
376C22×C4—2441no—
377C22×C4—2441no—
378C22×C4—2441no—
379C22×C4—2448no—
380C22×C4—2448no—
381C22×C4—2441no—
382C22×C4—2442no—
383C22×C4—2442no—
384C22×C4—2442no—
385C22×C4—2442no—
386C22×C4C22⋊#692311yescharacteristic
387C22×C4C22×#692311yescharacteristic
388C2×C8—2442no—
389C2×C8—2442no—
390C2×C8—2442no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—1222no—
394C4⋊C4—2441no—
395C4⋊C4—2442no—
396C4⋊C4—2442no—
397C22⋊C4—42no—
398C22⋊C4—12216no—
399C22⋊C4—2442no—
400C22⋊C4—2442no—
401C22⋊C4—2441no—
402C22⋊C4—2441no—
403C22⋊C4—2442no—
404C22⋊C4—2442no—
405C42—2448no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—611no—
413C23—611no—
414C23—612no—
415C23—612no—
416C23—612no—
417C23—1224no—
418C23—1224no—
419C23—1221no—
420C23—1221no—
421C23—1221no—
422C23—1221no—
423C23—1221no—
424C23—1221no—
425C23—1221no—
426C23—1221no—
427C23—1221no—
428C23—1221no—
429C23—2444no—
430C23—2444no—
431C23—2442no—
432C23—2442no—
433C23—2444no—
434C23—2444no—
435C23—2442no—
436C23—2442no—
437C23#1451984212yes—
438C23C2×C22⋊#692311yescharacteristic
439Q8—82no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—612no—
452C2×C4—612no—
453C2×C4—1224no—
454C2×C4—1224no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1228no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2444no—
463C2×C4—2444no—
464C2×C4—2444no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2444no—
472C2×C4—2444no—
473C2×C4—2444no—
474C2×C4—2444no—
475C2×C4—2444no—
476C2×C4—2444no—
477C2×C4#1451983512yes—
478C2×C4D4×#692312yes—
479C8—4882no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—612no—
488C22—612no—
489C22—1221no—
490C22—1221no—
491C22—1221no—
492C22—1221no—
493C22—1222no—
494C22—1222no—
495C22—1222no—
496C22—1222no—
497C22—1222no—
498C22—1222no—
499C22—1222no—
500C22—2448no—
501C22—2448no—
502C22—2441no—
503C22—2441no—
504C22—2441no—
505C22—2441no—
506C22#1684071011yescharacteristic
507C22C2×#1451983511yescharacteristic
508C22D61⋊#9911yescharacteristic
509C22#1684075211yescharacteristic
510C22#1684075511yescharacteristic
511C22D61⋊#10411yescharacteristic
512C22C2×#1451984211yescharacteristic
513C4—22no—
514C4—22no—
515C4—81no—
516C4—81no—
517C4—1222no—
518C4—1222no—
519C4—4881no—
520C4—4881no—
521C4—2448no—
522C2—42no—
523C2—1222no—
524C2—1222no—
525C2—2442no—
526C2#2146119411yescharacteristic
527C2#2146119711yescharacteristic
528C2#2146121711yescharacteristic
529C2#2146121911yescharacteristic
530C2#2146180511yescharacteristic
531C2#2146183611yescharacteristic
532C2#2146184011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#9146
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16840616C2semidirect
4. #21460692C2semidirect
15. C2×#14519753C4semidirect
20. #16839622C4semidirect
21. #16839619C4semidirect
24. C2×C22⋊Q8×C61C4semidirect
275. C2×C22⋊Q8#6923semidirect
291. C61#9146semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊#69232
#214616422
C2×#168406162
#214606922
#214541032
#914661