Finite Group Databasebeta

← #29572319

Subgroups — #29572319

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446669C211yesmaximal, characteristic, complemented
2#21444549C211yesmaximal, characteristic, complemented
3#21444426C211yesmaximal, characteristic, complemented
4#21444284C211yesmaximal, characteristic, complemented
5#21444240C211yesmaximal, characteristic, complemented
6#21444171C211yesmaximal, characteristic, complemented
7C61×#3561C211yesmaximal, characteristic
8#16839819C2211yescharacteristic
9#16839815C2211yescharacteristic
10#16839786C2211yescharacteristic
11C2×#14519696—21no—
12C2×#14519694—21no—
13C4×#11758194C2211yescharacteristic
14#16839511—21no—
15#16839508—21no—
16#16839504—21no—
17#16839502—21no—
18Q8⋊#92418—22no—
19Q8⋊#92418—21no—
20D4⋊#92418—21no—
21#16839486—22no—
22C61×#563—21no—
23C61×#562—21no—
24C61×#556—21no—
25C61×#551—22no—
26D4×Q8×C61—21no—
27C61×#548—21no—
28C61×#546—21no—
29C61×#543C412yes—
30C61×#542—21no—
31C61×#542—22no—
32C61×#538C412yes—
33C61×#536—21no—
34C61×#533C2211yescharacteristic
35C61×#529C2211yescharacteristic
36C2×#124×C61C2211yescharacteristic
37C2×#11758194—21no—
38C2×#11758194—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43#14519696—42no—
44#14519694—42no—
45C22⋊#92418—21no—
46C22⋊#92418—21no—
47#14519688—21no—
48#14519688—21no—
49C4×#92418—21no—
50C4×#92418—21no—
51C2×#41×C61—21no—
52C22×Q8×C61—21no—
53C22×Q8×C61D411yescharacteristic
54C22×D4×C61D411yescharacteristic
55C4⋊Q8×C61—21no—
56C4⋊Q8×C61—22no—
57C4⋊Q8×C61D411yescharacteristic
58C4⋊Q8×C61D411yescharacteristic
59#126×C61—22no—
60#126×C61—22no—
61#126×C61—22no—
62#126×C61—21no—
63#125×C61—21no—
64#125×C61D411yescharacteristic
65#124×C61—21no—
66#124×C61—22no—
67#124×C61—22no—
68#124×C61—21no—
69#124×C61—21no—
70#124×C61—21no—
71#124×C61—21no—
72#124×C61D411yescharacteristic
73#123×C61—22no—
74#123×C61—22no—
75#123×C61—22no—
76#123×C61—21no—
77#123×C61—22no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—21no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—21no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61C2×C412yes—
88C22⋊Q8×C61C2×C411yescharacteristic
89#121×C61—21no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61C2×C411yescharacteristic
93#120×C61—22no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61D412yes—
97C4×Q8×C61D411yescharacteristic
98C4×D4×C61—21no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—21no—
102C4×D4×C61D411yescharacteristic
103#117×C61—22no—
104#117×C61—21no—
105#117×C61—21no—
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61C2×C412yes—
108C2×C22⋊C4×C61D412yes—
109C2×C42×C61C2311yescharacteristic
110C24×C61C22⋊C412yes—
111#41×C61—81no—
112#41×C61—81no—
113C2×Q8×C61—22no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—41no—
118C2×Q8×C61—41no—
119C2×Q8×C61—81no—
120C2×Q8×C61—81no—
121C2×Q8×C61C22⋊C411yescharacteristic
122C2×Q8×C61C22⋊C412yes—
123C2×D4×C61—21no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—41no—
129C2×D4×C61—41no—
130C2×D4×C61C22⋊C411yescharacteristic
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—21no—
135C22×C4×C61C22⋊C412yes—
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61C22⋊C411yescharacteristic
153C4⋊C4×C61C22⋊C411yescharacteristic
154C4⋊C4×C61C22⋊C412yes—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—21no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—21no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—21no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—21no—
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177#11758194—22no—
178#11758194—41no—
179#11758194—41no—
180#11758194—41no—
181C2×#92418—41no—
182C2×#92418—41no—
183C2×#92418—41no—
184C2×#92418—41no—
185C23×C61—22no—
186C23×C61—21no—
187C23×C61—42no—
188C23×C61—42no—
189C23×C61—42no—
190C23×C61C2×C22⋊C411yescharacteristic
191Q8×C61—41no—
192Q8×C61—42no—
193Q8×C61—41no—
194Q8×C61—41no—
195Q8×C61—41no—
196Q8×C61—41no—
197Q8×C61—81no—
198Q8×C61—82no—
199D4×C61—41no—
200D4×C61—42no—
201D4×C61—41no—
202D4×C61—41no—
203D4×C61—81no—
204D4×C61—82no—
205D4×C61—82no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—82no—
220C2×C4×C61—82no—
221C2×C4×C61—82no—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228#92418—42no—
229#92418—81no—
230#92418—81no—
231#92418—81no—
232#8484—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—41no—
237C22×C61—42no—
238C22×C61—41no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—41no—
242C22×C61—162no—
243C22×C61#37911yescharacteristic
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—41no—
250C4×C61—42no—
251C4×C61—42no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—42no—
255C4×C61#45311yescharacteristic
256C4×C61#45511yescharacteristic
257#3561D6111yescharacteristic
258#1967—611no—
259#1559—611no—
260#1529—611no—
261#1495—611no—
262#1485—611no—
263#1469—611no—
264C2×C61—41no—
265C2×C61—81no—
266C2×C61—82no—
267C2×C61#179111yescharacteristic
268C2×C61#179211yescharacteristic
269C2×C61#179611yescharacteristic
270#563—21no—
271#562—21no—
272#556—21no—
273#551—22no—
274D4×Q8—21no—
275#548—21no—
276#546—21no—
277#543Dic6112yes—
278#542—21no—
279#542—22no—
280#538Dic6112yes—
281#536—21no—
282#533C2×D6111yescharacteristic
283#529C2×D6111yescharacteristic
284C2×#124C2×D6111yescharacteristic
285#432—611no—
286#431—611no—
287#423—611no—
288C2×#101—1221no—
289C2×#100—1221no—
290C4×M16—611no—
291#333—1221no—
292#332—1221no—
293#330—1221no—
294#329—1221no—
295Q8⋊C8—1222no—
296Q8⋊C8—1221no—
297D4⋊C8—1221no—
298#323—1222no—
299C61#848411yescharacteristic, complemented, Sylow
300C2×#41—21no—
301C22×Q8—21no—
302C22×Q8C22⋊D6111yescharacteristic
303C22×D4C22⋊D6111yescharacteristic
304C2×M16—1221no—
305C2×M16—1221no—
306C4⋊Q8—21no—
307C4⋊Q8—22no—
308C4⋊Q8C22⋊D6111yescharacteristic
309C4⋊Q8C22⋊D6111yescharacteristic
310#126—22no—
311#126—22no—
312#126—22no—
313#126—21no—
314#125—21no—
315#125C22⋊D6111yescharacteristic
316#124—21no—
317#124—22no—
318#124—22no—
319#124—21no—
320#124—21no—
321#124—21no—
322#124—21no—
323#124C22⋊D6111yescharacteristic
324#123—22no—
325#123—22no—
326#123—22no—
327#123—21no—
328#123—22no—
329C22⋊Q8—22no—
330C22⋊Q8—21no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—21no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8C2×Dic6111yescharacteristic
339C22⋊Q8C2×Dic6112yes—
340#121—21no—
341#121—22no—
342#121—22no—
343#121C2×Dic6111yescharacteristic
344#120—22no—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8C22⋊D6111yescharacteristic
348C4×Q8C22⋊D6112yes—
349C4×D4—21no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—21no—
353C4×D4C22⋊D6111yescharacteristic
354#117—22no—
355#117—21no—
356#117—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4C2×Dic6112yes—
359C2×C22⋊C4C22⋊D6112yes—
360C2×C42C22×D6111yescharacteristic
361C4⋊C8—1221no—
362C4⋊C8—1221no—
363C4⋊C8—1221no—
364C4⋊C8—1221no—
365#101—2442no—
366#100—2442no—
367C22⋊C8—1221no—
368C22⋊C8—1221no—
369#97—1221no—
370#97—1221no—
371C4×C8—1221no—
372C4×C8—1221no—
373C24C22⋊Dic6112yes—
374#41—81no—
375#41—81no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×Q8C22⋊Dic6111yescharacteristic
385C2×Q8C22⋊Dic6112yes—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4C22⋊Dic6111yescharacteristic
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4C22⋊Dic6112yes—
399C22×C4C22×Dic6111yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402M16—1222no—
403M16—2441no—
404M16—2441no—
405M16—2441no—
406C2×C8—2441no—
407C2×C8—2441no—
408C2×C8—2441no—
409C2×C8—2441no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4C22⋊Dic6111yescharacteristic
424C4⋊C4C22⋊Dic6111yescharacteristic
425C4⋊C4C22⋊Dic6112yes—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—21no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—21no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C42—22no—
442C42—22no—
443C42—21no—
444C42C2×C22⋊D6111yescharacteristic
445C42C2×C22⋊D6111yescharacteristic
446C42C2×C22⋊D6111yescharacteristic
447C42C2×C22⋊D6111yescharacteristic
448C23—22no—
449C23—21no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23C2×C22⋊Dic6111yescharacteristic
454Q8—41no—
455Q8—42no—
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—81no—
461Q8—82no—
462D4—41no—
463D4—42no—
464D4—41no—
465D4—41no—
466D4—81no—
467D4—82no—
468D4—82no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4C2×C22⋊Dic6111yescharacteristic
486C2×C4C2×C22⋊Dic6111yescharacteristic
487C2×C4#1451974611yescharacteristic
488C2×C4#1451974611yescharacteristic
489C2×C4#1451974611yescharacteristic
490C2×C4#1451974611yescharacteristic
491C8—4881no—
492C8—4881no—
493C8—4881no—
494C8—2442no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—41no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—162no—
505C22#1683961811yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—41no—
512C4—42no—
513C4—41no—
514C4—42no—
515C4—42no—
516C4—41no—
517C4#1683993011yescharacteristic
518C4#1683994011yescharacteristic
519C2—41no—
520C2—82no—
521C2—81no—
522C2#2144577011yescharacteristic
523C2#2144577311yescharacteristic
524C2#2144579411yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3561C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3561D61
hypocenterC61#8484
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3561

Complements

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

NH ≅ G/Ntype
1. #21446669C2semidirect
2. #21444549C2semidirect
3. #21444426C2semidirect
4. #21444284C2semidirect
5. #21444240C2semidirect
6. #21444171C2semidirect
299. C61#8484semidirect

Maximal subgroups

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

subgroupindex
#214466692
#214445492
#214444262
#214442842
#214442402
#214441712
C61×#35612
#848461