Finite Group Databasebeta

← #29736134

Subgroups — #29736134

Order 15616.

1056 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840172C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic
3#21451024C211yesmaximal, characteristic, complemented
4#21447154C212yesmaximal, complemented
5#21447144C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446760C211yesmaximal, characteristic, complemented
8#21446080C212yesmaximal
9#21446066C212yesmaximal
10#21445644C211yesmaximal, characteristic
11C2×C61×#554C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840172—24no—
14#16840042—24no—
15C2×#14519710C2212yes—
16C2×#14519710C2212yes—
17C2×#14519708C2212yes—
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519698C2212yes—
20C2×#14519698C2212yes—
21C2×#14519688C2211yescharacteristic
22#16839682—21no—
23#16839682—21no—
24#16839669—21no—
25#16839660C2212yes—
26#16839660C2212yes—
27#16839631—21no—
28C4×C4⋊Dic61C2211yescharacteristic
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839517C2212yes—
33#16839517C2212yes—
34C61×#554—24no—
35C61×#554—24no—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#125×C61C2211yescharacteristic
38C2×#125×C61C2211yescharacteristic
39C2×C22⋊Q8×C61—21no—
40C2×C22⋊Q8×C61—21no—
41C2×C22⋊Q8×C61—21no—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×C4×D4×C61C2211yescharacteristic
45C2×C4×D4×C61C2211yescharacteristic
46C22×C4⋊C4×C61—21no—
47C22×Dic122—21no—
48C22×#92418C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×C4×Dic61—21no—
58#14519710—24no—
59#14519710—24no—
60#14519708—24no—
61C4⋊#92418—22no—
62#14519698—24no—
63#14519698Q818yescomplemented
64#14519688—22no—
65#14519684—21no—
66#14519684—22no—
67#14519684—22no—
68C22×Q8×C61—21no—
69C22×Q8×C61D411yescharacteristic
70C22×D4×C61C2311yescharacteristic
71C23×C4×C61—21no—
72C23×C4×C61—21no—
73C23×C4×C61—21no—
74C4⋊Q8×C61—22no—
75C4⋊Q8×C61—22no—
76#125×C61—22no—
77#125×C61—22no—
78#125×C61—22no—
79#125×C61D414yes—
80C22⋊Q8×C61—28no—
81C22⋊Q8×C61—28no—
82C22⋊Q8×C61—28no—
83C4×Q8×C61—22no—
84C4×Q8×C61—22no—
85C4×D4×C61—22no—
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C4×D4×C61—22no—
90C4×D4×C61D414yes—
91C2×C4⋊C4×C61—28no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C2×Q8×C61—24no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—41no—
114C2×Q8×C61—82no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61C2×D412yes—
118C2×D4×C61C2×Q814yes—
119C22×C4×C61—24no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—24no—
124C22×C4×C61—24no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61—82no—
128C22×C4×C61—82no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—22no—
137C22×Dic61—22no—
138C2×Dic122—42no—
139C2×Dic122—82no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61C2×D412yes—
152C4⋊C4×C61C2×D412yes—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—24no—
156C22⋊C4×C61—24no—
157C42×C61—21no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61C2×D412yes—
162C42×C61C2×D412yes—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—42no—
167C4⋊Dic61C2×Q814yes—
168#11758196—82no—
169#11758196—82no—
170C4×Dic61—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—24no—
175C2×#92418C2×Q814yes—
176C23×C61—22no—
177C23×C61—24no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—42no—
181C23×C61C22×D411yescharacteristic
182Q8×C61—44no—
183Q8×C61—84no—
184D4×C61—22no—
185D4×C61—24no—
186D4×C61C4⋊Q814yes—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—22no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—48no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—84no—
208C2×C4×C61—84no—
209C2×C4×C61#12012yes—
210C2×C4×C61#12012yes—
211C2×C4×C61#13712yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×Q812yes—
215C2×C4×C612−1+412yes—
216C2×Dic61—24no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222Dic122—84no—
223#92418—24no—
224#92418—42no—
225#25850—611nomaximal, Sylow
226C22×C61—24no—
227C22×C61—44no—
228C22×C61—44no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61C2×#12011yescharacteristic
232C22×C61D4×Q812yes—
233C22×C61C2×#13711yescharacteristic
234C22×C61#57511yescharacteristic
235C22×C61#57612yes—
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—22no—
239C4×C61—41no—
240C4×C61—42no—
241C4×C61—41no—
242C4×C61—41no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61C2×C4⋊Q811yescharacteristic
246C4×C61#53611yescharacteristic
247C4×C61D4×Q812yes—
248Dic61—44no—
249Dic61—82no—
250C2×#554D6111yescharacteristic
251C2×#501—611no—
252C2×#474—611no—
253#2567—611no—
254#2059—612no—
255#2057—612no—
256#1989—611no—
257#1985—611no—
258#1864—612no—
259#1862—612no—
260#1762—611no—
261C2×C61—44no—
262C2×C61#243411yescharacteristic
263C2×C61#300711yescharacteristic
264C2×C61#300811yescharacteristic
265C2×C61#338512yes—
266C2×C61#338912yes—
267#554—24no—
268#554—24no—
269C2×C4⋊Q8—611no—
270C2×C4⋊Q8C2×D6111yescharacteristic
271C2×#125C2×D6111yescharacteristic
272C2×#125C2×D6111yescharacteristic
273C2×C22⋊Q8—21no—
274C2×C22⋊Q8—21no—
275C2×C22⋊Q8—21no—
276C2×C4×Q8C2×D6111yescharacteristic
277C2×C4×D4C2×D6111yescharacteristic
278C2×C4×D4C2×D6111yescharacteristic
279C2×C4×D4C2×D6111yescharacteristic
280C22×C4⋊C4—21no—
281#501—1224no—
282#474—1224no—
283C2×#107—612no—
284C2×#107—612no—
285C2×#106—612no—
286C2×C4⋊C8—611no—
287C2×#102—612no—
288C2×#102—612no—
289C2×#97—611no—
290#395—1221no—
291#395—1221no—
292#391—1221no—
293#389—612no—
294#389—612no—
295#384—1221no—
296C4×C4⋊C4—611no—
297#340—612no—
298#340—612no—
299#340—612no—
300#336—612no—
301#336—612no—
302C61#2585011yescharacteristic, complemented, Sylow
303C22×Q8—21no—
304C22×Q8—1221no—
305C22×Q8C22⋊D6111yescharacteristic
306C22×D4C22×D6111yescharacteristic
307C23×C4—21no—
308C23×C4—21no—
309C23×C4—21no—
310C22×C8—612no—
311C22×C8—612no—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—1222no—
315C4⋊Q8—1222no—
316#125—22no—
317#125—22no—
318#125—22no—
319#125C22⋊D6114yes—
320C22⋊Q8—28no—
321C22⋊Q8—28no—
322C22⋊Q8—28no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4C22⋊D6114yes—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4C22⋊D6111yescharacteristic
343C2×C4⋊C4C22⋊D6111yescharacteristic
344C2×C4⋊C4C22⋊D6111yescharacteristic
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C4⋊C4C22×D6111yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C42—1221no—
352C2×C42C22×D6111yescharacteristic
353C2×C42C22×D6111yescharacteristic
354C2×C42C22×D6111yescharacteristic
355#107—1224no—
356#107—1224no—
357#106—1224no—
358C4⋊C8—1222no—
359#102—618no—
360#102—1224no—
361#97—1222no—
362#95—1221no—
363#95—1222no—
364#95—1222no—
365C24—21no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—41no—
370C2×Q8—82no—
371C2×Q8—4882no—
372C2×Q8—2442no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4Q8×D6114yes—
376C2×D4C2×C22⋊D6112yes—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—1222no—
388C22×C4—1222no—
389C22×C4C2×C22⋊D6111yescharacteristic
390C22×C4C2×C22⋊D6111yescharacteristic
391C22×C4C2×C22⋊D6111yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396C2×C8—614no—
397C2×C8—1222no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1224no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—614no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—4882no—
417C4⋊C4—4882no—
418C4⋊C4—2442no—
419C4⋊C4C2×C22⋊D6112yes—
420C4⋊C4C2×C22⋊D6112yes—
421C4⋊C4C2×C22⋊D6112yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—2442no—
430C42C2×C22⋊D6112yes—
431C42C2×C22⋊D6112yes—
432C23—22no—
433C23—24no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23C22×C22⋊D6111yescharacteristic
438Q8—44no—
439Q8—84no—
440Q8—4884no—
441D4—22no—
442D4—24no—
443D4#1451977514yes—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—48no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—1224no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4#1451974612yes—
473C2×C4#1451974612yes—
474C2×C4#1451980712yes—
475C2×C4C22×C22⋊D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C2×Q8×D6112yes—
478C2×C4#1451982512yes—
479C8—1224no—
480C8—2442no—
481C22—24no—
482C22—44no—
483C22—44no—
484C22—42no—
485C22—42no—
486C22C2×#1451974611yescharacteristic
487C22Q8×C22⋊D6112yes—
488C22C2×#1451980711yescharacteristic
489C22#1684056011yescharacteristic
490C22#1684056512yes—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—41no—
495C4—42no—
496C4—41no—
497C4—41no—
498C4—42no—
499C4—42no—
500C4—4882no—
501C4—2444no—
502C4C2×#1451977511yescharacteristic
503C4#1684033611yescharacteristic
504C4Q8×C22⋊D6112yes—
505C2—44no—
506C2#2144945911yescharacteristic
507C2#2145507911yescharacteristic
508C2#2145508711yescharacteristic
509C2#2145872912yes—
510C2#2145877312yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#554D61
hypocenterC61#25850
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16840172C2semidirect
3. #21451024C2semidirect
4. #21447154C2semidirect
5. #21447144C2semidirect
6. #21446777C2semidirect
7. #21446760C2semidirect
63. #14519698Q8semidirect
302. C61#25850semidirect

Maximal subgroups

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

subgroupindex
C2×#168401722
C2×#168400422
#214510242
#214471542
#214471442
#214467772
#214467602
#214460802
#214460662
#214456442
C2×C61×#5542
#2585061