Finite Group Databasebeta

← #30600719

Subgroups — #30600719

Order 15616.

522 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 86 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#21462351C211yesmaximal, characteristic
2#21462008C211yesmaximal, characteristic
3#21461257C211yesmaximal, characteristic
4C2×#16840616C211yesmaximal, characteristic
5#21460694C211yesmaximal, characteristic
6#21460660C211yesmaximal, characteristic
7#21456592C211yesmaximal, characteristic
8C2×#14519834C2211yescharacteristic
9(C2×#6923)⋊Q8—21no—
10#16840711—21no—
11#16840711—21no—
12#16840709—21no—
13#16840704C2211yescharacteristic
14C4⋊C4×#6923—21no—
15C22⋊C4×#6923—21no—
16C4×C22⋊#6923C2211yescharacteristic
17C2×(C2×#6923)⋊C4C2211yescharacteristic
18#16840640—21no—
19#16840640—21no—
20#16840616—21no—
21#16840616—21no—
22#16840616—21no—
23#16840615—21no—
24C2×#14519753C2211yescharacteristic
25C22⋊Q8×D61C2211yescharacteristic
26#16840098C411yescharacteristic, complemented
27#16840096—21no—
28#16840095C411yescharacteristic, complemented
29#16840002—21no—
30#16840000—21no—
31#16839918C411yescharacteristic, complemented
32#16839916—21no—
33C22⋊C8×D61C2211yescharacteristic
34C61×#451C411yescharacteristic, complemented
35C23×#6923—21no—
36C22×#92420—21no—
37C2×C4⋊#6923—21no—
38C2×C4⋊#6923—21no—
39C2×C4⋊#6923—21no—
40C2×C22⋊#6923—21no—
41C2×C22⋊#6923D411yescharacteristic
42C2×C22⋊#6923D411yescharacteristic
43C2×C4×#6923—21no—
44C2×C4×#6923—21no—
45C2×C4×#6923—21no—
46#14519834—21no—
47#14519834—21no—
48(C2×#6923)⋊C4—21no—
49(C2×#6923)⋊C4—21no—
50(C2×#6923)⋊C4—21no—
51(C2×#6923)⋊C4—21no—
52(C2×#6923)⋊C4—21no—
53(C2×#6923)⋊C4—21no—
54(C2×#6923)⋊C4—21no—
55(C2×#6923)⋊C4D411yescharacteristic
56(C2×#6923)⋊C4D411yescharacteristic
57C2×Q8×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C2×#11758202—21no—
60C2×C8×D61—21no—
61#14519757D411yescharacteristic
62#14519756C2×C411yescharacteristic
63#14519755—21no—
64#14519754—21no—
65#14519753—21no—
66#14519753—21no—
67#14519753—21no—
68#14519753—21no—
69#14519753D411yescharacteristic
70C2×C4⋊Dic61C2×C411yescharacteristic
71C2×#11758196—21no—
72C4⋊C4×D61—21no—
73C4⋊C4×D61D411yescharacteristic
74C2×#11758198—21no—
75C22⋊C4×D61—21no—
76#14519711—21no—
77#14519710—21no—
78#14519702—21no—
79#14519701—21no—
80#14519701—21no—
81C22⋊#92418C2×C411yescharacteristic
82#14519689—21no—
83C22×Q8×C61C2×C411yescharacteristic
84C2×Q16×C61—21no—
85C22⋊Q8×C61C2×C411yescharacteristic
86#103×C61—21no—
87C22⋊C8×C61C2×C411yescharacteristic
88C23×D61C2×D411yescharacteristic
89C22×#6923—21no—
90C22×#6923—21no—
91C22×#6923—21no—
92C22×#6923—21no—
93C22×#6923—42no—
94C22×#6923—41no—
95C22×#6923—41no—
96C22×#6923—41no—
97C22×#6923—81no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—81no—
101C2×Q8×C61C22⋊C411yescharacteristic
102C22×C4×C61—21no—
103C22×C4×C61C22×C411yescharacteristic
104C22×Dic61—21no—
105C22×Dic61#4111yescharacteristic
106Q8×D61—81no—
107C2×C4×D61—21no—
108C2×C4×D61—21no—
109C2×C4×D61—81no—
110C2×C4×D61C2×D411yescharacteristic
111C2×C4×D61C2×D411yescharacteristic
112C2×C4×D61C2×D411yescharacteristic
113C2×C4×D61#4111yescharacteristic
114C2×Dic122—21no—
115C2×Dic122C22⋊C411yescharacteristic
116C22⋊#6923—21no—
117C22⋊#6923—21no—
118C22⋊#6923—21no—
119C22⋊#6923—41no—
120C22⋊#6923—41no—
121C22⋊#6923Q1612yescomplemented
122C2×#92420—41no—
123C2×#92420—81no—
124C4⋊#6923—42no—
125C4⋊#6923—81no—
126C4⋊#6923—81no—
127C4×#6923—42no—
128C4×#6923—81no—
129C4×#6923—81no—
130Q16×C61—81no—
131C2×C8×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61C22⋊C411yescharacteristic
134C22⋊C4×C61—21no—
135C22⋊Dic61—21no—
136#11758202—81no—
137#11758198—21no—
138#11758198—21no—
139#11758198—41no—
140#11758198—41no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61C22⋊C411yescharacteristic
145#11758196—21no—
146#11758196—21no—
147#11758196—21no—
148#11758196—41no—
149#11758196—41no—
150C2×#92418—21no—
151C8×D61—81no—
152C23×C61C4×D411yescharacteristic
153C22×D61—41no—
154C22×D61—41no—
155C22×D61#12011yescharacteristic
156C22×D61#12111yescharacteristic
157C22×D61C2×Q1611yescharacteristic
158C22×D61#13511yescharacteristic
159C22×D61#13711yescharacteristic
160C22×D61#13711yescharacteristic
161C2×#6923—22no—
162C2×#6923—22no—
163C2×#6923—41no—
164C2×#6923—41no—
165C2×#6923—41no—
166C2×#6923—41no—
167C2×#6923—41no—
168C2×#6923—41no—
169C2×#6923—41no—
170C2×#6923—41no—
171C2×#6923—41no—
172C2×#6923—41no—
173C2×#6923—82no—
174C2×#6923—82no—
175C2×#6923—81no—
176C2×#6923—81no—
177C2×#6923—81no—
178Q8×C61—41no—
179Q8×C61—41no—
180Q8×C61—41no—
181Q8×C61—81no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—81no—
187C2×C4×C61C2×C22⋊C411yescharacteristic
188C2×C4×C61C4×D411yescharacteristic
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—81no—
194C2×Dic61#12111yescharacteristic
195C2×Dic61#12311yescharacteristic
196C4×D61—41no—
197C4×D61—41no—
198C4×D61—41no—
199C4×D61—41no—
200C4×D61—41no—
201C4×D61—41no—
202C4×D61—41no—
203C4×D61—41no—
204Dic122—41no—
205Dic122—81no—
206#92420—81no—
207C8×C61—41no—
208#92418—41no—
209#9276—611nomaximal, Sylow
210C22×C61—21no—
211C22×C61—21no—
212C22×C61#38611yescharacteristic
213C22×C61C4×Q1611yescharacteristic
214C22×C61#44111yescharacteristic
215C2×D61—21no—
216C2×D61—21no—
217C2×D61—21no—
218C2×D61—21no—
219C2×D61—41no—
220C2×D61—41no—
221C2×D61—41no—
222C2×D61—41no—
223C2×D61#45111yescharacteristic
224C2×D61#45211yescharacteristic
225C2×D61#46211yescharacteristic
226C2×D61#46411yescharacteristic
227#6923—42no—
228#6923—42no—
229#6923—81no—
230#6923—81no—
231#6923—161no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—41no—
235C4×C61—41no—
236C4×C61—81no—
237Dic61—21no—
238Dic61—21no—
239Dic61—41no—
240Dic61—41no—
241Dic61—81no—
242#3189—611no—
243#2368—611no—
244#1996—611no—
245#1520—611no—
246C2×#328—611no—
247#1294—611no—
248#1282—611no—
249C2×C61—21no—
250C2×C61—21no—
251C2×C61#187811yescharacteristic
252C2×C61#190611yescharacteristic
253C2×C61#191311yescharacteristic
254D61—21no—
255D61—21no—
256D61—21no—
257D61—21no—
258C2×C22⋊Q8—611no—
259C2×C22⋊Q8—611no—
260#484—611no—
261#484—611no—
262#484—1221no—
263#467—1221no—
264#467—1221no—
265#451—611no—
266#451—1221no—
267#451#692311yescharacteristic, complemented
268C2×C22⋊C8—611no—
269C2×C22⋊C8—611no—
270#391—1221no—
271#387—1221no—
272#387—1221no—
273#385—1221no—
274#380—611no—
275C4×C4⋊C4—1221no—
276C4×C22⋊C4—611no—
277C4×C22⋊C4—1221no—
278C2×#95—611no—
279#342—1221no—
280#342—1221no—
281#328—1221no—
282#328—1221no—
283#328—1221no—
284#328—1221no—
285C61#927611yescharacteristic, complemented, Sylow
286C22×Q8—611no—
287C22×Q8C2×#692311yescharacteristic
288C23×C4—611no—
289C23×C4—1221no—
290C2×Q16—21no—
291C2×Q16—1221no—
292C22×C8—1221no—
293C22×C8—1221no—
294C22⋊Q8—611no—
295C22⋊Q8—611no—
296C22⋊Q8—611no—
297C22⋊Q8—1221no—
298C22⋊Q8—1221no—
299C22⋊Q8—1221no—
300C22⋊Q8—1221no—
301C22⋊Q8—1221no—
302C22⋊Q8—1221no—
303C22⋊Q8C2×#692311yescharacteristic
304C2×C4⋊C4—611no—
305C2×C4⋊C4—611no—
306C2×C4⋊C4—1221no—
307C2×C4⋊C4—1221no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C22⋊C4—611no—
312C2×C22⋊C4—611no—
313C2×C22⋊C4—1221no—
314C2×C22⋊C4—1221no—
315C2×C22⋊C4—1221no—
316C2×C42—1221no—
317C2×C42—1221no—
318C2×C42—1221no—
319#107—1221no—
320#107—1221no—
321#103—21no—
322#103—1221no—
323#103—1221no—
324#103—1221no—
325C22⋊C8—611no—
326C22⋊C8—1221no—
327C22⋊C8—1221no—
328C22⋊C8—1221no—
329C22⋊C8C2×#692311yescharacteristic
330#95—611no—
331#95—611no—
332#95—1221no—
333#95—1221no—
334#95—1221no—
335#95—1221no—
336#95—1221no—
337#95—1221no—
338#95—1221no—
339C24—611no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—81no—
343C2×Q8—611no—
344C2×Q8—1221no—
345C2×Q8—4881no—
346C2×Q8C22⋊#692311yescharacteristic
347C22×C4—21no—
348C22×C4—611no—
349C22×C4—611no—
350C22×C4—611no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—4881no—
361C22×C4—4881no—
362C22×C4—2442no—
363C22×C4—2441no—
364C22×C4—2441no—
365C22×C4—2441no—
366C22×C4C22×#692311yescharacteristic
367Q16—81no—
368Q16—4881no—
369C2×C8—21no—
370C2×C8—1221no—
371C2×C8—4881no—
372C2×C8—4881no—
373C2×C8—2441no—
374C4⋊C4—21no—
375C4⋊C4—611no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—4881no—
383C4⋊C4—4881no—
384C4⋊C4—2441no—
385C4⋊C4—2441no—
386C4⋊C4—2442no—
387C4⋊C4C22⋊#692311yescharacteristic
388C22⋊C4—21no—
389C22⋊C4—612no—
390C22⋊C4—1221no—
391C22⋊C4—1221no—
392C22⋊C4—1221no—
393C22⋊C4—1221no—
394C22⋊C4—1221no—
395C22⋊C4—1221no—
396C22⋊C4—2441no—
397C22⋊C4—2441no—
398C22⋊C4—2441no—
399C22⋊C4—2441no—
400C42—4881no—
401C42—4881no—
402C42—2442no—
403C23—611no—
404C23—611no—
405C23—611no—
406C23—611no—
407C23—611no—
408C23—611no—
409C23—2441no—
410C23—2441no—
411C23D4×#692311yescharacteristic
412Q8—41no—
413Q8—41no—
414Q8—41no—
415Q8—81no—
416Q8—4881no—
417Q8—2441no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—81no—
423C2×C4—611no—
424C2×C4—611no—
425C2×C4—1222no—
426C2×C4—1222no—
427C2×C4—1221no—
428C2×C4—1221no—
429C2×C4—4881no—
430C2×C4—4881no—
431C2×C4—4881no—
432C2×C4—4882no—
433C2×C4—4882no—
434C2×C4—4881no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4C2×C22⋊#692311yescharacteristic
456C2×C4D4×#692311yescharacteristic
457C8—41no—
458C8—4881no—
459C8—2441no—
460C22—21no—
461C22—21no—
462C22—611no—
463C22—611no—
464C22—611no—
465C22—611no—
466C22—1221no—
467C22—1221no—
468C22—1221no—
469C22—1221no—
470C22—2441no—
471C22—2441no—
472C22—2441no—
473C22—2441no—
474C22#1684071011yescharacteristic
475C22Q16×#692311yescharacteristic
476C22#1684080211yescharacteristic
477C4—21no—
478C4—21no—
479C4—41no—
480C4—41no—
481C4—81no—
482C4—1221no—
483C4—1221no—
484C4—9761no—
485C4—4881no—
486C4—4881no—
487C4—4881no—
488C4—2442no—
489C4—2442no—
490C4—2441no—
491C4—2441no—
492C2—21no—
493C2—21no—
494C2—1221no—
495C2—1221no—
496C2—1221no—
497C2—1221no—
498C2#2146181311yescharacteristic
499C2#2146185711yescharacteristic
500C2#2146186811yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#451C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#451#6923
hypocenterC61#9276
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
26. #16840098C4semidirect
28. #16840095C4semidirect
31. #16839918C4semidirect
34. C61×#451C4semidirect
121. C22⋊#6923Q16semidirect
267. #451#6923semidirect
285. C61#9276semidirect

Maximal subgroups

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

subgroupindex
#214623512
#214620082
#214612572
C2×#168406162
#214606942
#214606602
#214565922
#927661