Finite Group Databasebeta

← #29658806

Subgroups — #29658806

Order 15616.

932 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456947C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic
3#21455291C211yesmaximal, characteristic
4#21445096C214yesmaximal, complemented
5#21444915C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#557C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840158C2212yes—
13#16840110C2212yes—
14#16840080—22no—
15#16840080—21no—
16#16840049—22no—
17#16840047C2212yes—
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#14519708—22no—
57#14519708—24no—
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—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—42no—
115C2×Q8×C61—44no—
116C2×Q8×C61SD1614yescomplemented
117C2×Q8×C61C2×D412yes—
118C2×Q8×C61C2×D412yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C2×C22⋊D61—41no—
128#11758222—42no—
129#11758222—42no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×C4×D61—41no—
134C2×Dic122—21no—
135C4⋊C4×C61—28no—
136C4⋊C4×C61—28no—
137C4⋊C4×C61—28no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61C2×D412yes—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×D414yes—
150C42×C61—28no—
151C42×C61—22no—
152C42×C61—22no—
153C42×C61—22no—
154C42×C61—22no—
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2411yescharacteristic
159#11758202—44no—
160#11758201—44no—
161#11758201—44no—
162#11758198—41no—
163#11758198—41no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166#11758194—42no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C23×C61#12011yescharacteristic
171C22×D61—41no—
172Q8×C61—22no—
173Q8×C61—24no—
174Q8×C61—22no—
175Q8×C61—22no—
176Q8×C61—42no—
177Q8×C61—44no—
178Q8×C61—44no—
179Q8×C61C2×SD1614yes—
180C2×C4×C61—22no—
181C2×C4×C61—24no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—44no—
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61C2×SD1612yes—
196C2×C4×C61#13712yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C22⋊D61—82no—
201C2×Dic61—41no—
202D244—42no—
203D244—41no—
204D244—41no—
205D244—42no—
206C4×D61—82no—
207Dic122—41no—
208Dic122—41no—
209#92418—42no—
210#92418—42no—
211#92418—42no—
212#18778—611nomaximal, Sylow
213C22×C61—21no—
214C22×C61—21no—
215C22×C61#44812yes—
216C22×C61C2×#12011yescharacteristic
217C2×D61—81no—
218C2×D61—81no—
219C2×D61—81no—
220C4×C61—24no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—42no—
225C4×C61—44no—
226C4×C61—42no—
227C4×C61—42no—
228C4×C61#44812yes—
229C4×C61C22×SD1611yescharacteristic
230C4×C61C2×#13711yescharacteristic
231C4×C61#57512yes—
232Dic61—81no—
233C2×#557D6111yescharacteristic, complemented
234#3227—611no—
235#3103—611no—
236#3037—611no—
237#1653—614no—
238#1626—614no—
239#1549—611no—
240#1487—611no—
241#1479—611no—
242C2×C61—22no—
243C2×C61C2×#44811yescharacteristic
244C2×C61#300511yescharacteristic
245C2×C61#302011yescharacteristic
246D61—161no—
247C2×#137—1221no—
248#557—22no—
249#557—22no—
250#557—22no—
251#557C2×D6114yes—
252#532—611no—
253C2×C4⋊Q8—21no—
254C2×C4⋊Q8C2×D6111yescharacteristic
255C2×#125—22no—
256C2×#125—21no—
257C2×C4×Q8—21no—
258C2×C4×Q8—21no—
259C2×C4×Q8C2×D6111yescharacteristic
260C2×C4×Q8C2×D6111yescharacteristic
261#498—612no—
262#486—612no—
263#481—1221no—
264#481—1222no—
265#476—1222no—
266#475—612no—
267#464—1222no—
268#462—612no—
269#460—612no—
270#460—612no—
271#448—1221no—
272C4×SD16—1222no—
273#431—611no—
274#431—611no—
275#423—611no—
276#417—1221no—
277#332—614no—
278Q8⋊C8—614no—
279Q8⋊C8—614no—
280C61#1877811yescharacteristic, complemented, Sylow
281C2×#41—1221no—
282C22×Q8—21no—
283C22×Q8C22⋊D6111yescharacteristic
284C22×Q8C22⋊D6111yescharacteristic
285#137—2444no—
286C2×Q16—1222no—
287C2×SD16—1222no—
288C2×SD16—1222no—
289C2×M16—1221no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—611no—
295C4⋊Q8C22⋊D6112yes—
296C4⋊Q8C22×D6112yes—
297#127—611no—
298#125—28no—
299#125—24no—
300#125—28no—
301#125—24no—
302#124—1221no—
303#121—1221no—
304#121—1221no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8C22⋊D6112yes—
314C4×Q8C22⋊D6112yes—
315C4×Q8C22×D6112yes—
316C4×Q8C22×D6112yes—
317C4×D4—1221no—
318C4×D4—1221no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C22⋊D6111yescharacteristic
326C2×C4⋊C4C22⋊D6111yescharacteristic
327C2×C4⋊C4C22⋊D6112yes—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C22×D6111yescharacteristic
332#106—1222no—
333#106—1224no—
334C4⋊C8—612no—
335C4⋊C8—612no—
336C4⋊C8—1221no—
337C4⋊C8—1221no—
338#103—1222no—
339#103—1222no—
340#102—1222no—
341#102—1222no—
342#102—1224no—
343C22⋊C8—1221no—
344C22⋊C8—1221no—
345C4×C8—612no—
346C4×C8—1221no—
347#41—2442no—
348#41—2442no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—42no—
355C2×Q8—44no—
356C2×Q8—1221no—
357C2×Q8#1175820114yescomplemented
358C2×Q8C2×C22⋊D6112yes—
359C2×Q8C2×C22⋊D6112yes—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—2441no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—2441no—
370C22×C4C2×C22⋊D6111yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372C22×C4C2×C22⋊D6111yescharacteristic
373Q16—2444no—
374SD16—2444no—
375SD16—2444no—
376M16—2442no—
377C2×C8—1222no—
378C2×C8—1222no—
379C2×C8—1222no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4C2×C22⋊D6112yes—
395C4⋊C4C2×C22⋊D6112yes—
396C4⋊C4C2×C22⋊D6114yes—
397C22⋊C4—2441no—
398C22⋊C4—2441no—
399C42—28no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42C2×C22⋊D6111yescharacteristic
405C42C2×C22⋊D6111yescharacteristic
406C42C2×C22⋊D6111yescharacteristic
407C42C23×D6111yescharacteristic
408C23—2441no—
409C23#1451974611yescharacteristic
410Q8—22no—
411Q8—24no—
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8—44no—
416Q8—44no—
417Q8—2441no—
418Q8—2441no—
419Q8C2×#1175820114yes—
420D4—4882no—
421D4—2442no—
422D4—2441no—
423D4—2441no—
424D4—2442no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—4882no—
438C2×C4—2441no—
439C2×C4#1451974611yescharacteristic
440C2×C4#1451974611yescharacteristic
441C2×C4#1451974611yescharacteristic
442C2×C4C2×#1175820112yes—
443C2×C4#1451980812yes—
444C2×C4C22×C22⋊D6111yescharacteristic
445C2×C4C22×C22⋊D6111yescharacteristic
446C2×C4C22×C22⋊D6111yescharacteristic
447C8—2442no—
448C8—2442no—
449C8—2442no—
450C22—21no—
451C22—21no—
452C22—4881no—
453C22—4881no—
454C22—4881no—
455C22#1683990112yes—
456C22C2×#1451974611yescharacteristic
457C4—21no—
458C4—21no—
459C4—24no—
460C4—21no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—44no—
465C4—4881no—
466C4#1683990112yes—
467C4C22×#1175820111yescharacteristic
468C4C2×#1451980811yescharacteristic
469C4#1684056012yes—
470C2—22no—
471C2—9761no—
472C2C2×#1683990111yescharacteristic
473C2#2145506111yescharacteristic
474C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#557C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenterC2×#557D61
hypocenterC61#18778
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. #21445096C2semidirect
5. #21444915C2semidirect
6. #21444504C2semidirect
7. #21444248C2semidirect
8. #21444214C2semidirect
9. C2×C61×#557C2semidirect
116. C2×Q8×C61SD16semidirect
233. C2×#557D61semidirect
280. C61#18778semidirect
357. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214569472
#214558502
#214552912
#214450962
#214449152
#214445042
#214442482
#214442142
C2×C61×#5572
#1877861