Finite Group Databasebeta

← #29734534

Subgroups — #29734534

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456366C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic
3#21451031C211yesmaximal, characteristic, complemented
4#21447033C214yesmaximal, complemented
5#21446764C211yesmaximal, characteristic, complemented
6#21446739C211yesmaximal, characteristic
7#21446052C214yesmaximal, complemented
8C4×#14519696C211yesmaximal, characteristic, complemented
9C61×#3507C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—22no—
12#16840165—21no—
13#16840158—21no—
14#16840121—22no—
15#16840112—21no—
16#16840108—21no—
17#16839805C2214yes—
18C2×#14519704C2214yes—
19#16839763C2214yes—
20C2×#14519696C2212yes—
21C4×#11758194C2211yescharacteristic
22C4×#11758194C2211yescharacteristic
23#16839701—22no—
24#16839669C2212yes—
25#16839668—22no—
26#16839627C2212yes—
27C4×C4⋊Dic61C2211yescharacteristic
28#16839531C2212yes—
29#16839521C2214yes—
30C61×#559—21no—
31C61×#557—21no—
32C61×#556—21no—
33C61×#556—22no—
34C61×#554—21no—
35C61×#552—22no—
36C61×#539—22no—
37C61×#536C2212yes—
38C61×#533C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C61×#529—22no—
41C61×#520C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C22×Dic122—21no—
44C2×#11758194C2312yes—
45C2×#11758194C2312yes—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61C2312yes—
53C2×C4×Dic61C2312yes—
54#14519710—24no—
55#14519708—24no—
56#14519704—28no—
57#14519701—24no—
58#14519698—24no—
59#14519696—24no—
60#14519688—21no—
61#14519688—21no—
62C4×#92418—21no—
63C4×#92418—21no—
64#14519684—21no—
65#14519684—24no—
66#14519684—24no—
67C2×#41×C61C2312yes—
68C22×Q8×C61C2311yescharacteristic
69C4⋊Q8×C61—22no—
70C4⋊Q8×C61D411yescharacteristic
71C4⋊Q8×C61D411yescharacteristic
72C4⋊Q8×C61D411yescharacteristic
73#127×C61D411yescharacteristic
74#126×C61—24no—
75#126×C61—24no—
76#125×C61—24no—
77#125×C61—22no—
78#125×C61D412yes—
79#124×C61—22no—
80#124×C61—22no—
81#124×C61D412yes—
82#123×C61—24no—
83#123×C61—24no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61—21no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—22no—
89#121×C61—21no—
90#121×C61—21no—
91C4×Q8×C61—22no—
92C4×Q8×C61—22no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—24no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104#117×C61—24no—
105#117×C61—22no—
106#117×C61C2312yes—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61D412yes—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C42×C61D412yes—
111C2×C42×C61C2311yescharacteristic
112#41×C61—24no—
113#41×C61—24no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61C2×D412yes—
121C2×D4×C61—22no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61C2×D412yes—
125C22×C4×C61—24no—
126C22×C4×C61—22no—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—22no—
133C22×Dic61—22no—
134C2×Dic122—44no—
135C2×Dic122—41no—
136C2×Dic122—41no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C42×C61—24no—
159C42×C61—22no—
160C42×C61—22no—
161C42×C61—22no—
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—42no—
169C4⋊Dic61—41no—
170C4⋊Dic61—41no—
171C4×Dic61—24no—
172#11758194—22no—
173#11758194—22no—
174#11758194—22no—
175#11758194#4114yes—
176C2×#92418—22no—
177C2×#92418—22no—
178C23×C61—22no—
179C23×C61C22×D411yescharacteristic
180Q8×C61—24no—
181Q8×C61—42no—
182Q8×C61—42no—
183Q8×C61—42no—
184Q8×C61—81no—
185D4×C61—24no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—81no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—44no—
200C2×C4×C61—42no—
201C2×C4×C61—84no—
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#12011yescharacteristic
205C2×C4×C61#12011yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4112yes—
209C2×C4×C612−1+412yes—
210C2×Dic61—42no—
211C2×Dic61—44no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215Dic122—82no—
216Dic122—82no—
217#92418—24no—
218#92418—42no—
219#25696—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61C2×#12011yescharacteristic
226C22×C61#53211yescharacteristic
227C22×C61#53611yescharacteristic
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—42no—
234C4×C61—42no—
235C4×C61—42no—
236C4×C61#45412yes—
237C4×C61#54712yes—
238C4×C61#54812yes—
239Dic61—82no—
240Dic61—82no—
241#3507D6111yescharacteristic
242#3165—611no—
243#3141—611no—
244#2568—611no—
245#2038—614no—
246#1986—611no—
247#1980—611no—
248#1860—614no—
249C4×#101—611no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61—82no—
253C2×C61#243211yescharacteristic
254C2×C61C2×#45411yescharacteristic
255C2×C61#302211yescharacteristic
256#559—21no—
257#557—21no—
258#556—21no—
259#556—22no—
260#554—21no—
261#552—22no—
262#539—22no—
263#536C2×D6112yes—
264#533C2×D6111yescharacteristic
265#532C2×D6111yescharacteristic
266C2×C4⋊Q8—611no—
267#529—22no—
268#520C2×D6112yes—
269C2×C4×Q8C2×D6111yescharacteristic
270#501—1222no—
271#500—1221no—
272#498—1221no—
273#489—1222no—
274#487—1221no—
275#486—1221no—
276#428—614no—
277C2×#104—614no—
278#417—614no—
279C2×#101—612no—
280C4×M16—611no—
281C4×M16—611no—
282#400—1222no—
283#391—612no—
284#391—1222no—
285#382—612no—
286C4×C4⋊C4—611no—
287#343—612no—
288#339—614no—
289C61#2569611yescharacteristic, complemented, Sylow
290C2×#41C22×D6112yes—
291C22×Q8—1221no—
292C22×Q8C22×D6111yescharacteristic
293C2×M16—612no—
294C2×M16—612no—
295C4⋊Q8—22no—
296C4⋊Q8—1221no—
297C4⋊Q8—1221no—
298C4⋊Q8—1221no—
299C4⋊Q8—1221no—
300C4⋊Q8C22⋊D6111yescharacteristic
301C4⋊Q8C22⋊D6111yescharacteristic
302C4⋊Q8C22⋊D6111yescharacteristic
303#127C22⋊D6111yescharacteristic
304#126—24no—
305#126—24no—
306#125—22no—
307#125—24no—
308#125C22⋊D6112yes—
309#124—22no—
310#124—22no—
311#124C22⋊D6112yes—
312#123—24no—
313#123—24no—
314C22⋊Q8—24no—
315C22⋊Q8—21no—
316C22⋊Q8—21no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319#121—21no—
320#121—21no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—24no—
332C4×D4—21no—
333C4×D4—21no—
334#117—24no—
335#117—22no—
336#117C22×D6112yes—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4C22⋊D6112yes—
342C2×C4⋊C4C22×D6111yescharacteristic
343C2×C42—612no—
344C2×C42C22⋊D6112yes—
345C2×C42C22×D6111yescharacteristic
346#107—1224no—
347#106—1224no—
348#104—1228no—
349#103—1224no—
350#102—1224no—
351#101—1224no—
352#97—1221no—
353#97—1221no—
354C4×C8—1221no—
355C4×C8—1221no—
356#95—1221no—
357#95—1224no—
358#95—1224no—
359#41—24no—
360#41—24no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—2444no—
368C2×Q8—2441no—
369C2×Q8—2441no—
370C2×Q8C2×C22⋊D6112yes—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4C2×C22⋊D6112yes—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4C2×C22⋊D6112yes—
380C22×C4C2×C22⋊D6111yescharacteristic
381C22×C4C2×C22⋊D6112yes—
382C22×C4C2×C22⋊D6111yescharacteristic
383C22×C4C23×D6111yescharacteristic
384M16—614no—
385M16—1222no—
386M16—1222no—
387M16—1222no—
388C2×C8—1222no—
389C2×C8—1222no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—1222no—
404C4⋊C4—1222no—
405C4⋊C4—2442no—
406C4⋊C4—2441no—
407C4⋊C4—2441no—
408C4⋊C4C2×C22⋊D6112yes—
409C4⋊C4C2×C22⋊D6112yes—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C42—24no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—1224no—
421C42C2×C22⋊D6111yescharacteristic
422C42C2×C22⋊D6111yescharacteristic
423C42C2×C22⋊D6111yescharacteristic
424C42C2×C22⋊D6111yescharacteristic
425C23—22no—
426C23C22×C22⋊D6111yescharacteristic
427Q8—24no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—81no—
432Q8—4882no—
433Q8—4882no—
434D4—24no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—81no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—84no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2444no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4#1451974611yescharacteristic
457C2×C4#1451974611yescharacteristic
458C2×C4#1451974611yescharacteristic
459C2×C4#1451974611yescharacteristic
460C2×C4C22×C22⋊D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C2×C4#41×D6112yes—
463C2×C4#1451982412yes—
464C8—1224no—
465C8—2442no—
466C22—21no—
467C22—21no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22C2×#1451974611yescharacteristic
472C22#1684030811yescharacteristic
473C22#1684033311yescharacteristic
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—4882no—
483C4—4882no—
484C4#1683993512yes—
485C4#1684041112yes—
486C4#1684041912yes—
487C2—21no—
488C2—21no—
489C2—82no—
490C2#2144944311yescharacteristic
491C2C2×#1683993511yescharacteristic
492C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3507D61
hypocenterC61#25696
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. #21456366C2semidirect
3. #21451031C2semidirect
4. #21447033C2semidirect
5. #21446764C2semidirect
7. #21446052C2semidirect
8. C4×#14519696C2semidirect
289. C61#25696semidirect

Maximal subgroups

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

subgroupindex
#214563662
#214562122
#214510312
#214470332
#214467642
#214467392
#214460522
C4×#145196962
C61×#35072
#2569661