Finite Group Databasebeta

← #30600683

Subgroups — #30600683

Order 15616.

708 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 96 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
1Q8×C22⋊#6923C211yesmaximal, characteristic
2#21462008C211yesmaximal, characteristic
3#21461658C211yesmaximal, characteristic
4#21461259C211yesmaximal, characteristic
5#21460682C211yesmaximal, characteristic
6#21460661C211yesmaximal, characteristic
7C2×#16839916C211yesmaximal, characteristic
8C2×Q8×#6923—21no—
9C2×#14519840—21no—
10C2×#14519840—21no—
11D61⋊(C22⋊C8)C2211yescharacteristic
12(C2×#6923)⋊Q8—22no—
13(C2×#6923)⋊Q8—21no—
14#16840704—21no—
15#16840704C2211yescharacteristic
16#16840704C2211yescharacteristic
17C4×C22⋊#6923—21no—
18C4×C22⋊#6923C2211yescharacteristic
19#16840640—21no—
20#16840640—21no—
21#16840616—22no—
22C22×Q8×D61C2211yescharacteristic
23C22×#11758202C411yescharacteristic, complemented
24C2×#14519753C2211yescharacteristic
25#16839916—21no—
26#16839916—21no—
27#16839916—21no—
28#16839916—21no—
29C2×#14519702C411yescharacteristic, complemented
30C2×#14519701C411yescharacteristic, complemented
31C2×#14519689C2211yescharacteristic
32C22×Q16×C61C411yescharacteristic, complemented
33C2×D61⋊C8—21no—
34Q8×#6923—42no—
35Q8×#6923—42no—
36C2×C4⋊#6923—22no—
37C2×C4⋊#6923—21no—
38C2×C4⋊#6923—21no—
39C2×C22⋊#6923—21no—
40C2×C22⋊#6923—21no—
41C2×C22⋊#6923D411yescharacteristic
42C2×C22⋊#6923D411yescharacteristic
43C2×C4×#6923—22no—
44C2×C4×#6923—21no—
45#14519840—42no—
46#14519840—42no—
47#14519834—21no—
48C22⋊#92420—21no—
49(C2×#6923)⋊C4—22no—
50(C2×#6923)⋊C4—22no—
51(C2×#6923)⋊C4—21no—
52(C2×#6923)⋊C4—21no—
53(C2×#6923)⋊C4D412yes—
54C22×Dic122—21no—
55C22×Dic122C2×C411yescharacteristic
56C2×Q8×D61—21no—
57C2×Q8×D61—21no—
58C2×Q8×D61—42no—
59C2×Q8×D61—42no—
60C2×Q8×D61D411yescharacteristic
61C2×Q8×D61D411yescharacteristic
62C2×Q8×D61D411yescharacteristic
63C2×Q8×D61D411yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61C2311yescharacteristic
66C2×#11758202—21no—
67C2×#11758202—21no—
68C2×#11758202—42no—
69C22×#92418C2×C411yescharacteristic
70#14519753—21no—
71#14519753—21no—
72#14519753—21no—
73#14519753—21no—
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×#11758196—21no—
76C2×#11758198—21no—
77#14519702—21no—
78#14519702—21no—
79#14519701—21no—
80#14519701—21no—
81#14519689—21no—
82#14519689—21no—
83C22×Q8×C61C2×C411yescharacteristic
84C22×Q8×C61C2×C411yescharacteristic
85C2×Q16×C61—21no—
86C2×Q16×C61—21no—
87C2×Q16×C61—42no—
88C22×C8×C61C2×C411yescharacteristic
89C23×D61C2×D411yescharacteristic
90C22×#6923—21no—
91C22×#6923—21no—
92C22×#6923—22no—
93C22×#6923—41no—
94C2×Q8×C61—21no—
95C2×Q8×C61—22no—
96C2×Q8×C61—21no—
97C2×Q8×C61—21no—
98C2×Q8×C61—41no—
99C2×Q8×C61—41no—
100C2×Q8×C61—41no—
101C2×Q8×C61C22⋊C411yescharacteristic
102C2×Q8×C61C22⋊C411yescharacteristic
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61C22×C411yescharacteristic
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61#4111yescharacteristic
109Q8×D61—22no—
110Q8×D61—22no—
111Q8×D61—22no—
112Q8×D61—22no—
113Q8×D61—41no—
114Q8×D61—41no—
115Q8×D61—41no—
116Q8×D61—41no—
117Q8×D61—44no—
118Q8×D61—44no—
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—41no—
124C2×C4×D61—41no—
125C2×C4×D61—44no—
126C2×C4×D61C2×D411yescharacteristic
127C2×C4×D61C2×D411yescharacteristic
128C2×C4×D61C2×D411yescharacteristic
129C2×C4×D61#4111yescharacteristic
130C2×Dic122—21no—
131C2×Dic122—22no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122—21no—
135C2×Dic122—21no—
136C2×Dic122—41no—
137C2×Dic122—44no—
138C2×Dic122C22⋊C411yescharacteristic
139C2×Dic122C22⋊C411yescharacteristic
140C22⋊#6923—21no—
141C22⋊#6923—21no—
142C22⋊#6923—21no—
143C22⋊#6923—41no—
144C22⋊#6923—41no—
145C22⋊#6923—41no—
146C22⋊#6923—41no—
147C22⋊#6923Q1612yescomplemented
148C2×#92420—41no—
149C4⋊#6923—44no—
150C4⋊#6923—41no—
151C4⋊#6923—41no—
152C4⋊#6923—41no—
153C4⋊#6923—41no—
154C4×#6923—44no—
155C4×#6923—41no—
156C4×#6923—41no—
157D61⋊C8—41no—
158D61⋊C8—41no—
159Q16×C61—42no—
160Q16×C61—42no—
161C2×C8×C61—21no—
162C2×C8×C61—41no—
163#11758202—42no—
164#11758202—42no—
165#11758198—41no—
166#11758198—41no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169#11758196—41no—
170#11758196—41no—
171C2×#92418—21no—
172C2×#92418—41no—
173C23×C61C4×D411yescharacteristic
174C22×D61—41no—
175C22×D61—41no—
176C22×D61#12011yescharacteristic
177C22×D61#12111yescharacteristic
178C22×D61C2×Q1611yescharacteristic
179C22×D61#13511yescharacteristic
180C22×D61#13711yescharacteristic
181C22×D61#13711yescharacteristic
182C2×#6923—22no—
183C2×#6923—41no—
184C2×#6923—41no—
185C2×#6923—41no—
186C2×#6923—42no—
187C2×#6923—41no—
188C2×#6923—42no—
189C2×#6923—41no—
190C2×#6923—42no—
191C2×#6923—81no—
192C2×#6923—81no—
193C2×#6923—81no—
194Q8×C61—21no—
195Q8×C61—21no—
196Q8×C61—21no—
197Q8×C61—42no—
198Q8×C61—41no—
199Q8×C61—41no—
200Q8×C61—81no—
201Q8×C61#9912yes—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—81no—
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61C4×D411yescharacteristic
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215C2×Dic61—41no—
216C2×Dic61—81no—
217C2×Dic61#12111yescharacteristic
218C2×Dic61#12311yescharacteristic
219C4×D61—24no—
220C4×D61—21no—
221C4×D61—21no—
222C4×D61—21no—
223C4×D61—21no—
224C4×D61—41no—
225C4×D61—41no—
226C4×D61—42no—
227C4×D61—42no—
228C4×D61—42no—
229Dic122—24no—
230Dic122—21no—
231Dic122—21no—
232Dic122—21no—
233Dic122—42no—
234Dic122—42no—
235Dic122—42no—
236Dic122—42no—
237Dic122#9912yes—
238#92420—81no—
239C8×C61—41no—
240#92418—41no—
241#9258—611nomaximal, Sylow
242C22×C61—21no—
243C22×C61—21no—
244C22×C61#38611yescharacteristic
245C22×C61C4×Q1611yescharacteristic
246C22×C61#44111yescharacteristic
247C2×D61—21no—
248C2×D61—21no—
249C2×D61—21no—
250C2×D61—21no—
251C2×D61—41no—
252C2×D61—41no—
253C2×D61—41no—
254C2×D61—41no—
255C2×D61#45111yescharacteristic
256C2×D61#45211yescharacteristic
257C2×D61#46211yescharacteristic
258C2×D61#46411yescharacteristic
259#6923—42no—
260#6923—81no—
261#6923—82no—
262#6923—161no—
263C4×C61—22no—
264C4×C61—21no—
265C4×C61—41no—
266C4×C61—81no—
267C4×C61C2×#9911yescharacteristic
268C4×C61C2×#9911yescharacteristic
269Dic61—21no—
270Dic61—21no—
271Dic61—22no—
272Dic61—41no—
273Dic61—81no—
274C2×#451—611no—
275Q8×C22⋊C4—611no—
276#1996—611no—
277#1787—611no—
278#1520—611no—
279#1289—611no—
280#1282—611no—
281C2×C61—21no—
282C2×C61—21no—
283C2×C61#187811yescharacteristic
284C2×C61#189811yescharacteristic
285C2×C61#190511yescharacteristic
286D61—21no—
287D61—21no—
288D61—21no—
289D61—21no—
290C23×Q8—611no—
291C22×Q16—611no—
292C22×Q16#692311yescharacteristic, complemented
293C2×C22⋊Q8—611no—
294C2×C4×Q8—1221no—
295#451—1221no—
296#451—1221no—
297#451—1221no—
298#451—1221no—
299C2×#103—611no—
300C2×#103—611no—
301C2×#103—1221no—
302C2×#103—1221no—
303C2×C22⋊C8—611no—
304C2×C22⋊C8—611no—
305#391—1222no—
306#391—1221no—
307#380—611no—
308#380—611no—
309#380—1221no—
310C4×C22⋊C4—611no—
311C4×C22⋊C4—1221no—
312#342—1221no—
313#342—1221no—
314#328—1222no—
315C61#925811yescharacteristic, complemented, Sylow
316C22×Q8—611no—
317C22×Q8—611no—
318C22×Q8—611no—
319C22×Q8—611no—
320C22×Q8—611no—
321C22×Q8—1221no—
322C22×Q8—1221no—
323C22×Q8—1221no—
324C22×Q8—2442no—
325C22×Q8—2442no—
326C22×Q8C2×#692311yescharacteristic
327C22×Q8C2×#692311yescharacteristic
328C23×C4—611no—
329C23×C4—1221no—
330C2×Q16—21no—
331C2×Q16—21no—
332C2×Q16—42no—
333C2×Q16—1221no—
334C2×Q16—1221no—
335C2×Q16—2442no—
336C22×C8—611no—
337C22×C8—1221no—
338C22×C8C2×#692311yescharacteristic
339C22⋊Q8—1221no—
340C22⋊Q8—1221no—
341C22⋊Q8—1221no—
342C22⋊Q8—1221no—
343C4×Q8—2442no—
344C4×Q8—2442no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1222no—
350C2×C22⋊C4—611no—
351C2×C22⋊C4—611no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C42—1221no—
356C2×C42—1222no—
357#103—1221no—
358#103—1221no—
359#103—1221no—
360#103—1221no—
361#103—2442no—
362#103—2442no—
363C22⋊C8—1221no—
364C22⋊C8—1221no—
365C22⋊C8—1221no—
366C22⋊C8—1221no—
367#95—612no—
368#95—1221no—
369#95—1221no—
370#95—1222no—
371#95—1222no—
372C24—611no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—611no—
381C2×Q8—611no—
382C2×Q8—1221no—
383C2×Q8—1221no—
384C2×Q8—1221no—
385C2×Q8—1221no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8—1222no—
389C2×Q8—1221no—
390C2×Q8—1222no—
391C2×Q8—1222no—
392C2×Q8—2441no—
393C2×Q8—2444no—
394C2×Q8—2441no—
395C2×Q8—2441no—
396C2×Q8—2441no—
397C2×Q8—2441no—
398C2×Q8—2444no—
399C2×Q8—2444no—
400C2×Q8C22⋊#692311yescharacteristic
401C2×Q8C22⋊#692311yescharacteristic
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—1221no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1222no—
417C22×C4—1221no—
418C22×C4—2441no—
419C22×C4—2441no—
420C22×C4—2444no—
421C22×C4—2441no—
422C22×C4C22×#692311yescharacteristic
423Q16—42no—
424Q16—42no—
425Q16—2442no—
426Q16—2442no—
427C2×C8—21no—
428C2×C8—41no—
429C2×C8—1221no—
430C2×C8—2441no—
431C2×C8—2441no—
432C2×C8—2441no—
433C2×C8—2441no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—2441no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2444no—
441C4⋊C4—2441no—
442C4⋊C4—2441no—
443C22⋊C4—612no—
444C22⋊C4—1221no—
445C22⋊C4—1221no—
446C22⋊C4—1221no—
447C22⋊C4—2441no—
448C22⋊C4—2441no—
449C22⋊C4—2441no—
450C22⋊C4—2441no—
451C22⋊C4—2441no—
452C22⋊C4—2441no—
453C42—2444no—
454C42—2441no—
455C42—2441no—
456C23—611no—
457C23—611no—
458C23—611no—
459C23—611no—
460C23—611no—
461C23—611no—
462C23—2441no—
463C23—2441no—
464C23D4×#692311yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—21no—
468Q8—42no—
469Q8—41no—
470Q8—41no—
471Q8—81no—
472Q8—612no—
473Q8—1224no—
474Q8—1221no—
475Q8—1221no—
476Q8—1221no—
477Q8—2442no—
478Q8—2442no—
479Q8—2442no—
480Q8—2442no—
481Q8#1451983512yes—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—81no—
489C2×C4—611no—
490C2×C4—611no—
491C2×C4—1224no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1222no—
498C2×C4—1221no—
499C2×C4—4881no—
500C2×C4—4881no—
501C2×C4—4881no—
502C2×C4—4881no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2441no—
509C2×C4—2442no—
510C2×C4—2441no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4C2×C22⋊#692311yescharacteristic
520C2×C4D4×#692311yescharacteristic
521C8—41no—
522C8—4881no—
523C8—2441no—
524C22—21no—
525C22—21no—
526C22—611no—
527C22—611no—
528C22—611no—
529C22—611no—
530C22—1221no—
531C22—1221no—
532C22—1221no—
533C22—1221no—
534C22—2441no—
535C22—2441no—
536C22—2441no—
537C22—2441no—
538C22#1684071011yescharacteristic
539C22Q16×#692311yescharacteristic
540C22#1684080211yescharacteristic
541C4—21no—
542C4—22no—
543C4—41no—
544C4—81no—
545C4—1221no—
546C4—1221no—
547C4—1222no—
548C4—9761no—
549C4—4881no—
550C4—4882no—
551C4—4881no—
552C4—2442no—
553C4—2441no—
554C4C2×#1451983511yescharacteristic
555C4D61⋊#9911yescharacteristic
556C2—21no—
557C2—21no—
558C2—1221no—
559C2—1221no—
560C2—1221no—
561C2—1221no—
562C2#2146181311yescharacteristic
563C2#2146184011yescharacteristic
564C2#2146185411yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×Q16×C61C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC22×Q16#6923
hypocenterC61#9258
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
23. C22×#11758202C4semidirect
29. C2×#14519702C4semidirect
30. C2×#14519701C4semidirect
32. C22×Q16×C61C4semidirect
147. C22⋊#6923Q16semidirect
292. C22×Q16#6923semidirect
315. C61#9258semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊#69232
#214620082
#214616582
#214612592
#214606822
#214606612
C2×#168399162
#925861