Finite Group Databasebeta

← #29658620

Subgroups — #29658620

Order 15616.

932 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456940C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic
3#21455291C211yesmaximal, characteristic
4#21445061C214yesmaximal, complemented
5#21444915C214yesmaximal, complemented
6#21444491C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#557C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840165C2212yes—
13#16840110C2212yes—
14#16840080—21no—
15#16840075—22no—
16#16840047C2212yes—
17#16840039—22no—
18#16839984—22no—
19#16839972C2212yes—
20#16839962C2212yes—
21#16839962C2212yes—
22#16839901—21no—
23C4×#11758201—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
28#16839508C2214yes—
29Q8⋊#92418C2214yes—
30Q8⋊#92418C2214yes—
31C61×#557—22no—
32C61×#557—22no—
33C61×#557—22no—
34C61×#557C2214yes—
35C2×C4⋊Q8×C61—21no—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#125×C61—22no—
38C2×#125×C61—21no—
39C2×C4×Q8×C61—21no—
40C2×C4×Q8×C61—21no—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C2×#11758222—21no—
44#14519808—44no—
45C2×#11758202—22no—
46C2×#11758201—22no—
47C2×#11758201—22no—
48C2×#11758194—21no—
49#14519777C2311yescharacteristic
50#14519774C2311yescharacteristic
51#14519765—21no—
52#14519751—21no—
53#14519751—21no—
54C4×D244—21no—
55C4×D244—21no—
56#14519710—24no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—22no—
63#14519700—22no—
64#14519697—22no—
65#14519697—22no—
66#14519697—24no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C22×Q8×C61—21no—
72C22×Q8×C61D411yescharacteristic
73C22×Q8×C61D411yescharacteristic
74C4⋊Q8×C61—24no—
75C4⋊Q8×C61—24no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61D412yes—
79C4⋊Q8×C61C2312yes—
80#125×C61—28no—
81#125×C61—24no—
82#125×C61—28no—
83#125×C61—24no—
84C4×Q8×C61—24no—
85C4×Q8×C61—24no—
86C4×Q8×C61—24no—
87C4×Q8×C61—24no—
88C4×Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61D412yes—
93C4×Q8×C61D412yes—
94C4×Q8×C61C2312yes—
95C4×Q8×C61C2312yes—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D412yes—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C42×C61—22no—
106C2×C42×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C2×Q8×C61—24no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—42no—
117C2×Q8×C61—42no—
118C2×Q8×C61C2×D412yes—
119C2×Q8×C61C2×D412yes—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C2×C22⋊D61—41no—
129#11758222—42no—
130#11758222—42no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×C4×D61—41no—
135C2×Dic122—21no—
136C4⋊C4×C61—28no—
137C4⋊C4×C61—28no—
138C4⋊C4×C61—28no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D414yes—
151C42×C61—28no—
152C42×C61—22no—
153C42×C61—22no—
154C42×C61—22no—
155C42×C61—22no—
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2411yescharacteristic
160#11758202—44no—
161#11758201—44no—
162#11758201—44no—
163#11758198—41no—
164#11758198—41no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167#11758194—42no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61#12011yescharacteristic
172C22×D61—41no—
173Q8×C61—24no—
174Q8×C61—22no—
175Q8×C61—22no—
176Q8×C61—22no—
177Q8×C61—42no—
178Q8×C61—42no—
179Q8×C61—42no—
180Q8×C61—44no—
181Q8×C61#13714yescomplemented
182C2×C4×C61—24no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—24no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—44no—
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#13512yes—
198C2×C4×C61AGL(1,ℤ/8)12yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C22⋊D61—82no—
203C2×Dic61—41no—
204D244—42no—
205D244—41no—
206D244—41no—
207D244—42no—
208C4×D61—82no—
209Dic122—41no—
210Dic122—41no—
211#92418—42no—
212#92418—42no—
213#92418—42no—
214#18760—611nomaximal, Sylow
215C22×C61—21no—
216C22×C61—21no—
217C22×C61#44912yes—
218C22×C61C2×#12011yescharacteristic
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C4×C61—24no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—42no—
227C4×C61—44no—
228C4×C61—42no—
229C4×C61—42no—
230C4×C61#44912yes—
231C4×C61C2×#13511yescharacteristic
232C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
233C4×C61C2×#13712yes—
234Dic61—81no—
235C2×#557D6111yescharacteristic, complemented
236#3226—611no—
237#3103—611no—
238#3037—611no—
239#1648—614no—
240#1626—614no—
241#1546—611no—
242#1487—611no—
243#1479—611no—
244C2×C61—22no—
245C2×C61C2×#44911yescharacteristic
246C2×C61#300411yescharacteristic
247C2×C61#302011yescharacteristic
248D61—161no—
249C2×#137—1221no—
250#557—22no—
251#557—22no—
252#557—22no—
253#557C2×D6114yes—
254#532—611no—
255C2×C4⋊Q8—21no—
256C2×C4⋊Q8C2×D6111yescharacteristic
257C2×#125—22no—
258C2×#125—21no—
259C2×C4×Q8—21no—
260C2×C4×Q8—21no—
261C2×C4×Q8C2×D6111yescharacteristic
262C2×C4×Q8C2×D6111yescharacteristic
263#500—612no—
264#486—612no—
265#481—1221no—
266#480—1222no—
267#475—612no—
268#474—1222no—
269#464—1222no—
270#462—612no—
271#460—612no—
272#460—612no—
273#448—1221no—
274C4×SD16—1222no—
275#431—611no—
276#431—611no—
277#423—611no—
278#417—1221no—
279#332—614no—
280Q8⋊C8—614no—
281Q8⋊C8—614no—
282C61#1876011yescharacteristic, complemented, Sylow
283C2×#41—1221no—
284C22×Q8—21no—
285C22×Q8C22⋊D6111yescharacteristic
286C22×Q8C22⋊D6111yescharacteristic
287#137—2444no—
288C2×Q16—1222no—
289C2×SD16—1222no—
290C2×SD16—1222no—
291C2×M16—1221no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—611no—
297C4⋊Q8C22⋊D6112yes—
298C4⋊Q8C22×D6112yes—
299#127—611no—
300#125—28no—
301#125—24no—
302#125—28no—
303#125—24no—
304#124—1221no—
305#121—1221no—
306#121—1221no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8C22⋊D6112yes—
316C4×Q8C22⋊D6112yes—
317C4×Q8C22×D6112yes—
318C4×Q8C22×D6112yes—
319C4×D4—1221no—
320C4×D4—1221no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C22⋊D6111yescharacteristic
328C2×C4⋊C4C22⋊D6111yescharacteristic
329C2×C4⋊C4C22⋊D6112yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42C22×D6111yescharacteristic
334#107—1224no—
335#106—1222no—
336C4⋊C8—612no—
337C4⋊C8—612no—
338C4⋊C8—1221no—
339C4⋊C8—1221no—
340#103—1222no—
341#103—1222no—
342#102—1222no—
343#102—1222no—
344#102—1224no—
345C22⋊C8—1221no—
346C22⋊C8—1221no—
347C4×C8—612no—
348C4×C8—1221no—
349#41—2442no—
350#41—2442no—
351C2×Q8—24no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—1221no—
361C2×Q8C2×C22⋊D6112yes—
362C2×Q8C2×C22⋊D6112yes—
363C2×D4—1221no—
364C2×D4—1221no—
365C2×D4—1221no—
366C2×D4—2441no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—2441no—
373C22×C4C2×C22⋊D6111yescharacteristic
374C22×C4C2×C22⋊D6111yescharacteristic
375C22×C4C2×C22⋊D6111yescharacteristic
376Q16—2444no—
377SD16—2444no—
378SD16—2444no—
379M16—2442no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1222no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4C2×C22⋊D6112yes—
398C4⋊C4C2×C22⋊D6112yes—
399C4⋊C4C2×C22⋊D6114yes—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C42—28no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42C2×C22⋊D6111yescharacteristic
408C42C2×C22⋊D6111yescharacteristic
409C42C2×C22⋊D6111yescharacteristic
410C42C23×D6111yescharacteristic
411C23—2441no—
412C23#1451974611yescharacteristic
413Q8—24no—
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—42no—
418Q8—42no—
419Q8—42no—
420Q8—44no—
421Q8—2441no—
422Q8—2441no—
423Q8#1451980814yescomplemented
424D4—4882no—
425D4—2442no—
426D4—2441no—
427D4—2441no—
428D4—2442no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—4882no—
442C2×C4—2441no—
443C2×C4#1451974611yescharacteristic
444C2×C4#1451974611yescharacteristic
445C2×C4#1451974611yescharacteristic
446C2×C4#1451979712yes—
447C2×C4#1451980212yes—
448C2×C4C22×C22⋊D6111yescharacteristic
449C2×C4C22×C22⋊D6111yescharacteristic
450C2×C4C22×C22⋊D6111yescharacteristic
451C8—2442no—
452C8—2442no—
453C8—2442no—
454C22—21no—
455C22—21no—
456C22—4881no—
457C22—4881no—
458C22—4881no—
459C22#1683990712yes—
460C22C2×#1451974611yescharacteristic
461C4—21no—
462C4—21no—
463C4—24no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—44no—
469C4—4881no—
470C4#1683990712yes—
471C4C2×#1451979711yescharacteristic
472C4C2×#1451980211yescharacteristic
473C4C2×#1451980812yes—
474C2—22no—
475C2—9761no—
476C2C2×#1683990711yescharacteristic
477C2#2145505311yescharacteristic
478C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#557C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenterC2×#557D61
hypocenterC61#18760
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #21445061C2semidirect
5. #21444915C2semidirect
6. #21444491C2semidirect
7. #21444248C2semidirect
8. #21444214C2semidirect
9. C2×C61×#557C2semidirect
181. Q8×C61#137semidirect
235. C2×#557D61semidirect
282. C61#18760semidirect
423. Q8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214569402
#214558502
#214552912
#214450612
#214449152
#214444912
#214442482
#214442142
C2×C61×#5572
#1876061