Finite Group Databasebeta

← #30609198

Subgroups — #30609198

Order 15616.

2142 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519856C211yesmaximal, characteristic
2C2×(C2×#6923)⋊Q8C212yesmaximal, complemented
3C2×#16840704C211yesmaximal, characteristic
4#21461004C214yesmaximal, complemented
5#21461003C214yesmaximal
6#21460996C212yesmaximal, complemented
7C2×C22⋊Q8×D61C211yesmaximal, characteristic
8C2×#14519856—28no—
9C2×#14519856—24no—
10C22×C4⋊#6923C2212yes—
11C22×C4⋊#6923C2212yes—
12C22×C22⋊#6923C2212yes—
13C22×C22⋊#6923C2212yes—
14C22×C4×#6923C2212yes—
15(C2×#6923)⋊Q8—28no—
16#16840704—24no—
17C2×(C2×#6923)⋊C4C2218yes—
18C2×(C2×#6923)⋊C4C2212yes—
19C2×(C2×#6923)⋊C4C2218yes—
20C22×Q8×D61C2211yescharacteristic
21C23×C4×D61C2211yescharacteristic
22C2×#14519757C411yescharacteristic, complemented
23C2×#14519756C411yescharacteristic, complemented
24C2×#14519755C412yescomplemented
25C2×#14519754C412yescomplemented
26C2×#14519753C411yescharacteristic, complemented
27C22⋊Q8×D61—24no—
28C22⋊Q8×D61—24no—
29C2×C4⋊C4×D61C2211yescharacteristic
30C2×C4⋊C4×D61C2212yes—
31C2×C22⋊C4×D61C2212yes—
32C2×C22⋊Q8×C61C411yescharacteristic, complemented
33C23×#6923C2314yes—
34C23×#6923C2314yes—
35#14519856—232no—
36C2×C4⋊#6923—28no—
37C2×C4⋊#6923—24no—
38C2×C4⋊#6923—24no—
39C2×C4⋊#6923—24no—
40C2×C4⋊#6923—24no—
41C2×C22⋊#6923—24no—
42C2×C22⋊#6923—28no—
43C2×C22⋊#6923—24no—
44C2×C22⋊#6923—44no—
45C2×C4×#6923—24no—
46C2×C4×#6923—24no—
47C2×C4×#6923D418yes—
48(C2×#6923)⋊C4—216no—
49(C2×#6923)⋊C4—24no—
50(C2×#6923)⋊C4—48no—
51C24×D61C2311yescharacteristic
52C22×Dic122C2×C412yes—
53C22×Dic122C2×C411yescharacteristic
54C2×Q8×D61—24no—
55C2×Q8×D61—22no—
56C2×Q8×D61—22no—
57C2×Q8×D61—24no—
58C23×Dic61C2×C411yescharacteristic
59C22×C4×D61—24no—
60C22×C4×D61—24no—
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61C2×C411yescharacteristic
63C22×C4×D61D414yes—
64C22×C4×D61D414yes—
65C22×C4×D61C2312yes—
66C22×C4×D61C2312yes—
67C22×C4×D61C2311yescharacteristic
68C22×C4×D61C2311yescharacteristic
69#14519757—24no—
70#14519756—24no—
71#14519755—28no—
72#14519754—28no—
73#14519753—24no—
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×C4⋊Dic61C2×C412yes—
76C2×#11758196C2×C412yes—
77C2×#11758196C2×C412yes—
78C2×#11758196C2×C412yes—
79C4⋊C4×D61—24no—
80C4⋊C4×D61—24no—
81C4⋊C4×D61—22no—
82C4⋊C4×D61—22no—
83C2×C22⋊Dic61C2×C412yes—
84C2×#11758198C2×C412yes—
85C2×#11758198C2×C412yes—
86C22⋊C4×D61—24no—
87C22⋊C4×D61—24no—
88C22×Q8×C61C2×C411yescharacteristic
89C23×C4×C61C2×C411yescharacteristic
90C22⋊Q8×C61—24no—
91C2×C4⋊C4×C61C2×C412yes—
92C2×C4⋊C4×C61C2×C411yescharacteristic
93C2×C22⋊C4×C61C2×C412yes—
94C24×C61C22×C411yescharacteristic
95C23×D61—22no—
96C23×D61—22no—
97C23×D61—22no—
98C23×D61—24no—
99C23×D61—22no—
100C23×D61C22×C411yescharacteristic
101C23×D61C2×D412yes—
102C23×D61C2×D412yes—
103C23×D61C2411yescharacteristic
104C22×#6923—24no—
105C22×#6923—24no—
106C22×#6923—24no—
107C22×#6923—24no—
108C22×#6923—24no—
109C22×#6923—24no—
110C22×#6923—24no—
111C22×#6923—24no—
112C22×#6923—24no—
113C22×#6923—28no—
114C22×#6923—28no—
115C22×#6923C2×D418yes—
116C2×Q8×C61—22no—
117C2×Q8×C61—24no—
118C22×C4×C61—24no—
119C22×C4×C61C22⋊C414yes—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×Dic61—24no—
125C22×Dic61C22⋊C414yes—
126C22×Dic61C22×C412yes—
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C411yescharacteristic
129C22×Dic61C22×C411yescharacteristic
130Q8×D61—44no—
131Q8×D61—48no—
132Q8×D61—44no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—24no—
136C2×C4×D61—24no—
137C2×C4×D61—28no—
138C2×C4×D61—216no—
139C2×C4×D61—28no—
140C2×C4×D61—24no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—24no—
146C2×C4×D61—24no—
147C2×C4×D61—24no—
148C2×C4×D61C22⋊C414yes—
149C2×C4×D61C22⋊C414yes—
150C2×C4×D61C2×D412yes—
151C2×C4×D61C2×D412yes—
152C2×C4×D61C2×D414yes—
153C2×C4×D61C2×D412yes—
154C2×C4×D61C2×D412yes—
155C2×C4×D61#4114yes—
156C2×Dic122—22no—
157C2×Dic122—24no—
158C2×Dic122—24no—
159C2×Dic122—28no—
160C22⋊#6923—216no—
161C22⋊#6923—48no—
162C4⋊#6923—216no—
163C4⋊#6923—48no—
164C4×#6923—216no—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—22no—
167C22⋊C4×C61—24no—
168C22⋊Dic61—24no—
169#11758198—24no—
170#11758198—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—24no—
173#11758196—24no—
174#11758196—24no—
175#11758196—24no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61C2×C22⋊C412yes—
180C23×C61C23×C411yescharacteristic
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—24no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—24no—
189C22×D61—24no—
190C22×D61—22no—
191C22×D61—24no—
192C22×D61—44no—
193C22×D61—44no—
194C22×D61—44no—
195C22×D61—44no—
196C22×D61C2×C22⋊C412yes—
197C22×D61C22×D412yes—
198C22×D61C22×D411yescharacteristic
199C22×D61C22×D411yescharacteristic
200C22×D61C2×#4112yes—
201C22×D61H(5,𝔽2)12yes—
202C22×D612−1+412yes—
203C22×D612−1+414yes—
204C2×#6923—28no—
205C2×#6923—28no—
206C2×#6923—28no—
207C2×#6923—44no—
208C2×#6923—44no—
209C2×#6923—44no—
210C2×#6923—44no—
211C2×#6923—44no—
212C2×#6923—44no—
213C2×#6923—44no—
214C2×#6923—48no—
215C2×#6923—48no—
216Q8×C61—44no—
217C2×C4×C61—22no—
218C2×C4×C61—28no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61C2×C22⋊C412yes—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C4×D414yes—
226C2×Dic61—22no—
227C2×Dic61—28no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—24no—
231C2×Dic61—24no—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C412yes—
234C2×Dic61C4×D414yes—
235C4×D61—22no—
236C4×D61—24no—
237C4×D61—22no—
238C4×D61—28no—
239C4×D61—28no—
240C4×D61—42no—
241C4×D61—44no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61—42no—
245C4×D61—44no—
246C4×D61#12014yes—
247C4×D61#12118yes—
248C4×D61#12314yes—
249Dic122—44no—
250Dic122—48no—
251#14258—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—44no—
256C22×C61C22×C22⋊C411yescharacteristic
257C22×C61C2×C4×D412yes—
258C22×C61#51912yes—
259C22×C61#52012yes—
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—44no—
264C2×D61—44no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61—44no—
268C2×D61C2×#12011yescharacteristic
269C2×D61C2×#12112yes—
270C2×D61C2×#12311yescharacteristic
271C2×D61#53511yescharacteristic
272C2×D61#53614yes—
273C2×D61#53612yes—
274C2×D61#53612yes—
275C2×D61#53611yescharacteristic
276C2×D61#53714yes—
277C2×D61#53712yes—
278C2×D61#54114yes—
279C2×D61#54214yes—
280#6923—48no—
281#6923—84no—
282C4×C61—22no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61#38614yes—
286Dic61—22no—
287Dic61—42no—
288Dic61—42no—
289Dic61#38614yes—
290C22×C22⋊Q8—611no—
291C22×#117—611no—
292C2×#391—612no—
293C2×#380—611no—
294#1441—614no—
295#1440—614no—
296C4⋊#95—612no—
297C2×C61—42no—
298C2×C61C2×#38611yescharacteristic
299C2×C61#231011yescharacteristic
300C2×C61#231111yescharacteristic
301C2×C61#232314yes—
302D61—42no—
303D61#261612yes—
304D61#262012yes—
305D61#262112yes—
306D61#262512yes—
307C23×Q8—611no—
308C24×C4—611no—
309C2×C22⋊Q8—611no—
310C2×C22⋊Q8—611no—
311C2×C22⋊Q8—612no—
312C2×C22⋊Q8—611no—
313C2×C22⋊Q8—612no—
314C2×C22⋊Q8—1224no—
315C2×C22⋊Q8—1224no—
316C2×C22⋊Q8#692311yescharacteristic, complemented
317C2×#117—1228no—
318C2×#117—1224no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—612no—
321C22×C4⋊C4—612no—
322C22×C4⋊C4—612no—
323C22×C22⋊C4—612no—
324C22×C22⋊C4—612no—
325C22×C22⋊C4—612no—
326C22×C42—612no—
327#391—1228no—
328#380—1224no—
329C2×#95—618no—
330C2×#95—612no—
331C2×#95—618no—
332C61#1425811yescharacteristic, complemented, Sylow
333C25—611no—
334C22×Q8—612no—
335C22×Q8—611no—
336C22×Q8—1222no—
337C22×Q8—1222no—
338C22×Q8—1224no—
339C22×Q8—1224no—
340C22×Q8C2×#692311yescharacteristic
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—614no—
344C23×C4—614no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—614no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4—614no—
352C23×C4—1224no—
353C23×C4—1224no—
354C23×C4C2×#692311yescharacteristic
355C22⋊Q8—24no—
356C22⋊Q8—1228no—
357C22⋊Q8—1228no—
358C22⋊Q8—1224no—
359C22⋊Q8—1224no—
360C22⋊Q8—1224no—
361#117—12232no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1224no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1228no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4—1224no—
375C2×C4⋊C4—1224no—
376C2×C4⋊C4C2×#692311yescharacteristic
377C2×C4⋊C4C2×#692312yes—
378C2×C22⋊C4—612no—
379C2×C22⋊C4—612no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—1224no—
382C2×C22⋊C4—1224no—
383C2×C22⋊C4—1228no—
384C2×C22⋊C4—1224no—
385C2×C22⋊C4—1224no—
386C2×C22⋊C4—2444no—
387C2×C22⋊C4C2×#692312yes—
388C2×C42—618no—
389C2×C42—1224no—
390C2×C42—1224no—
391#95—1224no—
392#95—12216no—
393#95—2448no—
394C24—611no—
395C24—612no—
396C24—612no—
397C24—611no—
398C24—1222no—
399C24—1222no—
400C24—1222no—
401C24—1224no—
402C24—1222no—
403C24C22×#692311yescharacteristic
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—1224no—
407C2×Q8—1228no—
408C2×Q8—1222no—
409C2×Q8—1224no—
410C2×Q8—2444no—
411C2×Q8—2448no—
412C2×Q8—2444no—
413C22×C4—24no—
414C22×C4—611no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—614no—
418C22×C4—614no—
419C22×C4—614no—
420C22×C4—611no—
421C22×C4—612no—
422C22×C4—612no—
423C22×C4—614no—
424C22×C4—614no—
425C22×C4—618no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—1228no—
429C22×C4—1224no—
430C22×C4—12216no—
431C22×C4—1228no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1224no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4—1224no—
450C22×C4—1224no—
451C22×C4—1224no—
452C22×C4—1228no—
453C22×C4—1224no—
454C22×C4—1228no—
455C22×C4C22⋊#692314yes—
456C22×C4C22×#692312yes—
457C22×C4C22×#692312yes—
458C22×C4C22×#692311yescharacteristic
459C22×C4C22×#692311yescharacteristic
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—1224no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—12216no—
466C4⋊C4—1224no—
467C4⋊C4—1222no—
468C4⋊C4—2448no—
469C22⋊C4—24no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—12216no—
473C22⋊C4—1224no—
474C22⋊C4—2448no—
475C42—12216no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—611no—
480C23—611no—
481C23—612no—
482C23—612no—
483C23—614no—
484C23—612no—
485C23—612no—
486C23—612no—
487C23—1222no—
488C23—1222no—
489C23—1224no—
490C23—1222no—
491C23—1224no—
492C23—1224no—
493C23—1224no—
494C23—1222no—
495C23—1222no—
496C23—1222no—
497C23—1222no—
498C23—2444no—
499C23—2444no—
500C23—2444no—
501C23—2444no—
502C23C2×C22⋊#692312yes—
503C23C23×#692311yescharacteristic
504Q8—44no—
505Q8—2448no—
506Q8—2444no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—612no—
514C2×C4—614no—
515C2×C4—614no—
516C2×C4—618no—
517C2×C4—614no—
518C2×C4—612no—
519C2×C4—1228no—
520C2×C4—1222no—
521C2×C4—1224no—
522C2×C4—1222no—
523C2×C4—1228no—
524C2×C4—1222no—
525C2×C4—1228no—
526C2×C4—1228no—
527C2×C4—1228no—
528C2×C4—1228no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1224no—
532C2×C4—1224no—
533C2×C4—2442no—
534C2×C4—2444no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2444no—
538C2×C4—2442no—
539C2×C4—2444no—
540C2×C4—2444no—
541C2×C4—2444no—
542C2×C4—2448no—
543C2×C4—2448no—
544C2×C4—2444no—
545C2×C4—2444no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4C2×C22⋊#692312yes—
549C2×C4C2×C22⋊#692312yes—
550C2×C4D4×#692314yes—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—44no—
555C22—612no—
556C22—612no—
557C22—611no—
558C22—612no—
559C22—611no—
560C22—614no—
561C22—614no—
562C22—614no—
563C22—614no—
564C22—611no—
565C22—612no—
566C22—611no—
567C22—1222no—
568C22—1222no—
569C22—1222no—
570C22—2444no—
571C22—2444no—
572C22—2444no—
573C22—2444no—
574C22—2444no—
575C22C22×C22⋊#692311yescharacteristic
576C22C2×D4×#692312yes—
577C22#1684083012yes—
578C22#1684083112yes—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—614no—
583C4—1222no—
584C4—4884no—
585C4—2448no—
586C4—2442no—
587C4—2442no—
588C4#1684071014yes—
589C2—42no—
590C2—612no—
591C2—612no—
592C2—612no—
593C2—612no—
594C2—2442no—
595C2C2×#1684071011yescharacteristic
596C2#2146229311yescharacteristic
597C2#2146229411yescharacteristic
598C2#2146230614yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×C22⋊Q8×C61C4
prime radicalC24×D61C23
cosocleC23×D61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#14258
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×(C2×#6923)⋊Q8C2semidirect
4. #21461004C2semidirect
6. #21460996C2semidirect
22. C2×#14519757C4semidirect
23. C2×#14519756C4semidirect
24. C2×#14519755C4semidirect
25. C2×#14519754C4semidirect
26. C2×#14519753C4semidirect
32. C2×C22⋊Q8×C61C4semidirect
316. C2×C22⋊Q8#6923semidirect
332. C61#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#145198562
C2×(C2×#6923)⋊Q82
C2×#168407042
#214610042
#214610032
#214609962
C2×C22⋊Q8×D612
#1425861