Finite Group Databasebeta

← #29736145

Subgroups — #29736145

Order 15616.

1076 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840172C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic
3#21450804C211yesmaximal, characteristic, complemented
4#21447154C212yesmaximal, complemented
5#21447149C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446760C211yesmaximal, characteristic, complemented
8#21446080C212yesmaximal
9#21446066C212yesmaximal
10#21445644C211yesmaximal, characteristic
11C2×C61×#554C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13C2×#14519775—21no—
14#16840172—24no—
15#16840052—24no—
16C2×#14519710C2212yes—
17C2×#14519708C2212yes—
18C2×#14519708C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2212yes—
21C2×#14519698C2212yes—
22C2×#14519688C2211yescharacteristic
23#16839692—21no—
24#16839692—21no—
25#16839667—21no—
26#16839660C2212yes—
27#16839660C2212yes—
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—
52#14519775—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2312yes—
55C2×C4⋊Dic61C2312yes—
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×C4×Dic61—21no—
59#14519710—24no—
60#14519708—24no—
61#14519708—24no—
62C4⋊#92418—22no—
63#14519698—24no—
64#14519698Q818yescomplemented
65#14519688—22no—
66#14519684—21no—
67#14519684—22no—
68#14519684—22no—
69C22×Q8×C61—21no—
70C22×Q8×C61D411yescharacteristic
71C22×D4×C61C2311yescharacteristic
72C23×C4×C61—21no—
73C23×C4×C61—21no—
74C23×C4×C61—21no—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61—22no—
77#125×C61—22no—
78#125×C61—22no—
79#125×C61—22no—
80#125×C61D414yes—
81C22⋊Q8×C61—28no—
82C22⋊Q8×C61—28no—
83C22⋊Q8×C61—28no—
84C4×Q8×C61—22no—
85C4×Q8×C61—22no—
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61D414yes—
92C2×C4⋊C4×C61—28no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C24×C61—21no—
111C2×Q8×C61—24no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—41no—
115C2×Q8×C61—82no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61C2×D412yes—
119C2×D4×C61C2×Q814yes—
120C22×C4×C61—24no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—24no—
125C22×C4×C61—24no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61—82no—
129C22×C4×C61—82no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—22no—
138C22×Dic61—22no—
139C2×Dic122—42no—
140C2×Dic122—42no—
141C2×Dic122—42no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×D412yes—
154C4⋊C4×C61C2×D412yes—
155C4⋊C4×C61C2×D412yes—
156C22⋊C4×C61—24no—
157C22⋊C4×C61—24no—
158C22⋊C4×C61—24no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61C2×D412yes—
164C42×C61C2×D412yes—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—42no—
169C4⋊Dic61C2×Q814yes—
170#11758196—44no—
171#11758196—44no—
172C4×Dic61—42no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—24no—
177C2×#92418C2×Q814yes—
178C23×C61—22no—
179C23×C61—24no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—42no—
183C23×C61C22×D411yescharacteristic
184Q8×C61—44no—
185Q8×C61—84no—
186D4×C61—22no—
187D4×C61—24no—
188D4×C61C4⋊Q814yescomplemented
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—22no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—48no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—84no—
210C2×C4×C61—84no—
211C2×C4×C61#12012yes—
212C2×C4×C61#12012yes—
213C2×C4×C61AGL(1,ℤ/8)12yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×Q812yes—
217C2×C4×C612−1+412yes—
218C2×Dic61—24no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224Dic122—84no—
225#92418—24no—
226#92418—42no—
227#25851—611nomaximal, Sylow
228C22×C61—24no—
229C22×C61—44no—
230C22×C61—44no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61C2×#12011yescharacteristic
234C22×C61D4×Q812yes—
235C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
236C22×C61#57511yescharacteristic
237C22×C61#57712yes—
238C4×C61—22no—
239C4×C61—22no—
240C4×C61—22no—
241C4×C61—41no—
242C4×C61—42no—
243C4×C61—41no—
244C4×C61—41no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61C2×C4⋊Q811yescharacteristic
248C4×C61#53611yescharacteristic
249C4×C61D4×Q812yes—
250Dic61—44no—
251Dic61—82no—
252C2×#554D6111yescharacteristic
253C2×#501—611no—
254C2×#476—611no—
255#2546—611no—
256#2059—612no—
257#2058—612no—
258#1989—611no—
259#1985—611no—
260#1864—612no—
261#1862—612no—
262#1762—611no—
263C2×C61—44no—
264C2×C61#243411yescharacteristic
265C2×C61#300611yescharacteristic
266C2×C61#300911yescharacteristic
267C2×C61#338412yes—
268C2×C61#338812yes—
269#554—24no—
270#554—24no—
271C2×C4⋊Q8—611no—
272C2×C4⋊Q8—1221no—
273C2×C4⋊Q8C2×D6111yescharacteristic
274C2×#125C2×D6111yescharacteristic
275C2×#125C2×D6111yescharacteristic
276C2×C22⋊Q8—21no—
277C2×C22⋊Q8—21no—
278C2×C22⋊Q8—21no—
279C2×C4×Q8C2×D6111yescharacteristic
280C2×C4×D4C2×D6111yescharacteristic
281C2×C4×D4C2×D6111yescharacteristic
282C2×C4×D4C2×D6111yescharacteristic
283C22×C4⋊C4—21no—
284#501—1224no—
285#476—1224no—
286C2×#107—612no—
287C2×#106—612no—
288C2×#106—612no—
289C2×C4⋊C8—611no—
290C2×#102—612no—
291C2×#102—612no—
292C2×#97—611no—
293#398—1221no—
294#398—1221no—
295#391—1221no—
296#389—612no—
297#389—612no—
298C4×C4⋊C4—611no—
299#340—612no—
300#340—612no—
301#340—612no—
302#336—612no—
303#336—612no—
304C61#2585111yescharacteristic, complemented, Sylow
305C22×Q8—21no—
306C22×Q8—1221no—
307C22×Q8C22⋊D6111yescharacteristic
308C22×D4C22×D6111yescharacteristic
309C23×C4—21no—
310C23×C4—21no—
311C23×C4—21no—
312C22×C8—612no—
313C22×C8—612no—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C4⋊Q8—1222no—
317C4⋊Q8—1222no—
318C4⋊Q8—2444no—
319#125—22no—
320#125—22no—
321#125—22no—
322#125C22⋊D6114yes—
323C22⋊Q8—28no—
324C22⋊Q8—28no—
325C22⋊Q8—28no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4C22⋊D6114yes—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4C22⋊D6111yescharacteristic
346C2×C4⋊C4C22⋊D6111yescharacteristic
347C2×C4⋊C4C22⋊D6111yescharacteristic
348C2×C4⋊C4C22×D6111yescharacteristic
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C42—1221no—
355C2×C42C22×D6111yescharacteristic
356C2×C42C22×D6111yescharacteristic
357C2×C42C22×D6111yescharacteristic
358#107—1224no—
359#106—1224no—
360#106—1224no—
361C4⋊C8—1222no—
362#102—618no—
363#102—1224no—
364#97—1222no—
365#95—1221no—
366#95—1222no—
367#95—1222no—
368C24—21no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—41no—
373C2×Q8—82no—
374C2×Q8—2442no—
375C2×Q8—2442no—
376C2×Q8—2442no—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4Q8×D6114yes—
380C2×D4C2×C22⋊D6112yes—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—1222no—
392C22×C4—1222no—
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398C22×C4C2×C22⋊D6111yescharacteristic
399C22×C4C23×D6111yescharacteristic
400C2×C8—614no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—1224no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—614no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4—2444no—
421C4⋊C4—2442no—
422C4⋊C4—2444no—
423C4⋊C4C2×C22⋊D6112yes—
424C4⋊C4C2×C22⋊D6112yes—
425C4⋊C4C2×C22⋊D6112yes—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—2442no—
434C42C2×C22⋊D6112yes—
435C42C2×C22⋊D6112yes—
436C23—22no—
437C23—24no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23C22×C22⋊D6111yescharacteristic
442Q8—44no—
443Q8—84no—
444Q8—4884no—
445D4—22no—
446D4—24no—
447D4#1451977514yescomplemented
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—48no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—1224no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4#1451974612yes—
477C2×C4#1451974612yes—
478C2×C4#1451980212yes—
479C2×C4C22×C22⋊D6111yescharacteristic
480C2×C4C22×C22⋊D6111yescharacteristic
481C2×C4C2×Q8×D6112yes—
482C2×C4#1451982512yes—
483C8—1224no—
484C8—2442no—
485C22—24no—
486C22—44no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22C2×#1451974611yescharacteristic
491C22Q8×C22⋊D6112yes—
492C22C2×#1451980211yescharacteristic
493C22#1684056011yescharacteristic
494C22#1684057312yes—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—4882no—
505C4—2444no—
506C4C2×#1451977511yescharacteristic
507C4#1684033611yescharacteristic
508C4Q8×C22⋊D6112yes—
509C2—44no—
510C2#2144945911yescharacteristic
511C2#2145507111yescharacteristic
512C2#2145509611yescharacteristic
513C2#2145871812yes—
514C2#2145876412yes—
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#25851
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. #21450804C2semidirect
4. #21447154C2semidirect
5. #21447149C2semidirect
6. #21446777C2semidirect
7. #21446760C2semidirect
64. #14519698Q8semidirect
188. D4×C61C4⋊Q8semidirect
304. C61#25851semidirect
447. D4#14519775semidirect

Maximal subgroups

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

subgroupindex
C2×#168401722
C2×#168400522
#214508042
#214471542
#214471492
#214467772
#214467602
#214460802
#214460662
#214456442
C2×C61×#5542
#2585161