Finite Group Databasebeta

← #29658704

Subgroups — #29658704

Order 15616.

976 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456744C211yesmaximal, characteristic
2#21455850C211yesmaximal, characteristic
3#21455291C211yesmaximal, characteristic
4#21445040C214yesmaximal, complemented
5#21444897C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C2×Q82×C61C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840136C2212yes—
13#16840112C2212yes—
14#16840080—21no—
15#16840047C2212yes—
16#16839984—22no—
17#16839972C2212yes—
18#16839964—22no—
19#16839962C2212yes—
20#16839962C2212yes—
21#16839901—22no—
22#16839901—21no—
23C4×#11758201—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839761—21no—
28#16839511C2214yes—
29Q8⋊#92418C2214yes—
30Q8⋊#92418C2214yes—
31Q82×C61—22no—
32Q82×C61—22no—
33Q82×C61—22no—
34Q82×C61C2214yes—
35C2×C4⋊Q8×C61—22no—
36C2×C4⋊Q8×C61—21no—
37C2×C4⋊Q8×C61—21no—
38C2×C4⋊Q8×C61C2211yescharacteristic
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×#11758201—24no—
49C2×#11758194—21no—
50#14519777C2311yescharacteristic
51#14519774C2311yescharacteristic
52#14519765—21no—
53#14519751—21no—
54#14519751—21no—
55C4×D244—21no—
56C4×D244—21no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519701—24no—
63#14519701—22no—
64#14519700—22no—
65#14519697—22no—
66#14519697—22no—
67C22⋊#92418—21no—
68C22⋊#92418—21no—
69C4×#92418—21no—
70C4×#92418C2312yes—
71C22×Q8×C61—22no—
72C22×Q8×C61D411yescharacteristic
73C22×Q8×C61D411yescharacteristic
74C22×Q8×C61D412yes—
75C4⋊Q8×C61—28no—
76C4⋊Q8×C61—24no—
77C4⋊Q8×C61—24no—
78C4⋊Q8×C61—28no—
79C4⋊Q8×C61—24no—
80C4⋊Q8×C61—24no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61D412yes—
84C4⋊Q8×C61C2312yes—
85C4×Q8×C61—24no—
86C4×Q8×C61—24no—
87C4×Q8×C61—24no—
88C4×Q8×C61—24no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61D412yes—
94C4×Q8×C61D412yes—
95C4×Q8×C61C2312yes—
96C4×Q8×C61C2312yes—
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×C61—21no—
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—28no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—82no—
116C2×Q8×C61—84no—
117C2×Q8×C61SD1614yescomplemented
118C2×Q8×C61C2×D412yes—
119C2×Q8×C61C2×D412yes—
120C2×Q8×C61C2×D414yes—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C2×C22⋊D61—41no—
130#11758222—42no—
131#11758222—42no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×C4×D61—41no—
136C2×Dic122—21no—
137C4⋊C4×C61—28no—
138C4⋊C4×C61—28no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D412yes—
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#11758201—84no—
164#11758198—41no—
165#11758198—41no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168#11758194—42no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61#12011yescharacteristic
173C22×D61—41no—
174Q8×C61—24no—
175Q8×C61—22no—
176Q8×C61—22no—
177Q8×C61—24no—
178Q8×C61—44no—
179Q8×C61—82no—
180Q8×C61—88no—
181Q8×C61C2×SD1614yes—
182Q8×C61#13714yescomplemented
183C2×C4×C61—24no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—24no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—44no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61C2×SD1612yes—
199C2×C4×C61#13712yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C22⋊D61—82no—
204C2×Dic61—41no—
205D244—42no—
206D244—41no—
207D244—42no—
208D244—41no—
209C4×D61—82no—
210Dic122—41no—
211Dic122—41no—
212#92418—42no—
213#92418—42no—
214#92418—42no—
215#18768—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61#44812yes—
219C22×C61C2×#12011yescharacteristic
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—24no—
224C4×C61—24no—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—44no—
229C4×C61—42no—
230C4×C61—42no—
231C4×C61#44812yes—
232C4×C61C22×SD1611yescharacteristic
233C4×C61C2×#13711yescharacteristic
234C4×C61C2×#13712yes—
235Dic61—81no—
236C2×Q82D6111yescharacteristic, complemented
237#3206—611no—
238#3103—611no—
239#3037—611no—
240#1645—614no—
241#1623—614no—
242#1548—611no—
243#1487—611no—
244#1479—611no—
245C2×C61—22no—
246C2×C61C2×#44811yescharacteristic
247C2×C61#300411yescharacteristic
248C2×C61#301911yescharacteristic
249D61—161no—
250C2×#137—1221no—
251Q82—22no—
252Q82—22no—
253Q82—22no—
254Q82C2×D6114yes—
255#532—611no—
256C2×C4⋊Q8—22no—
257C2×C4⋊Q8—21no—
258C2×C4⋊Q8—21no—
259C2×C4⋊Q8C2×D6111yescharacteristic
260C2×C4×Q8—21no—
261C2×C4×Q8—21no—
262C2×C4×Q8C2×D6111yescharacteristic
263C2×C4×Q8C2×D6111yescharacteristic
264#492—612no—
265#487—612no—
266#481—1221no—
267#475—612no—
268#464—1222no—
269#462—612no—
270#461—1222no—
271#460—612no—
272#460—612no—
273#448—1221no—
274#448—1222no—
275C4×SD16—1222no—
276#431—611no—
277#431—611no—
278#423—611no—
279#417—1221no—
280#333—614no—
281Q8⋊C8—614no—
282Q8⋊C8—614no—
283C61#1876811yescharacteristic, complemented, Sylow
284C2×#41—1221no—
285C22×Q8—22no—
286C22×Q8C22⋊D6111yescharacteristic
287C22×Q8C22⋊D6111yescharacteristic
288C22×Q8C22⋊D6112yes—
289#137—2444no—
290C2×Q16—1222no—
291C2×SD16—1224no—
292C2×SD16—1222no—
293C2×SD16—1222no—
294C2×M16—1221no—
295C4⋊Q8—28no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—28no—
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—611no—
304C4⋊Q8C22⋊D6112yes—
305C4⋊Q8C22×D6112yes—
306#127—611no—
307#124—1221no—
308#121—1221no—
309#121—1221no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8C22⋊D6112yes—
319C4×Q8C22⋊D6112yes—
320C4×Q8C22×D6112yes—
321C4×Q8C22×D6112yes—
322C4×D4—1221no—
323C4×D4—1221no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C22⋊D6111yescharacteristic
331C2×C4⋊C4C22⋊D6111yescharacteristic
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C22×D6111yescharacteristic
336#106—1222no—
337C4⋊C8—612no—
338C4⋊C8—612no—
339C4⋊C8—1221no—
340C4⋊C8—1221no—
341#103—1222no—
342#103—1224no—
343#103—1222no—
344#102—1222no—
345#102—1222no—
346C22⋊C8—1221no—
347C22⋊C8—1221no—
348C4×C8—612no—
349C4×C8—1221no—
350#41—2442no—
351#41—2442no—
352C2×Q8—24no—
353C2×Q8—28no—
354C2×Q8—22no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—82no—
359C2×Q8—84no—
360C2×Q8—1221no—
361C2×Q8#1175820114yescomplemented
362C2×Q8C2×C22⋊D6112yes—
363C2×Q8C2×C22⋊D6112yes—
364C2×Q8C2×C22⋊D6114yes—
365C2×D4—1221no—
366C2×D4—1221no—
367C2×D4—1221no—
368C2×D4—2441no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—2441no—
375C22×C4C2×C22⋊D6111yescharacteristic
376C22×C4C2×C22⋊D6111yescharacteristic
377C22×C4C2×C22⋊D6111yescharacteristic
378Q16—2444no—
379SD16—4884no—
380SD16—2444no—
381SD16—2444no—
382M16—2442no—
383C2×C8—1222no—
384C2×C8—1222no—
385C2×C8—1222no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4C2×C22⋊D6112yes—
401C4⋊C4C2×C22⋊D6112yes—
402C22⋊C4—2441no—
403C22⋊C4—2441no—
404C42—28no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42C2×C22⋊D6111yescharacteristic
410C42C2×C22⋊D6111yescharacteristic
411C42C2×C22⋊D6111yescharacteristic
412C42C23×D6111yescharacteristic
413C23—2441no—
414C23#1451974611yescharacteristic
415Q8—24no—
416Q8—22no—
417Q8—22no—
418Q8—24no—
419Q8—44no—
420Q8—82no—
421Q8—88no—
422Q8—2441no—
423Q8—2441no—
424Q8C2×#1175820114yes—
425Q8#1451980814yescomplemented
426D4—4882no—
427D4—2442no—
428D4—2441no—
429D4—2442no—
430D4—2441no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—4882no—
444C2×C4—2441no—
445C2×C4#1451974611yescharacteristic
446C2×C4#1451974611yescharacteristic
447C2×C4#1451974611yescharacteristic
448C2×C4C2×#1175820112yes—
449C2×C4#1451980812yes—
450C2×C4C22×C22⋊D6111yescharacteristic
451C2×C4C22×C22⋊D6111yescharacteristic
452C2×C4C22×C22⋊D6111yescharacteristic
453C8—2442no—
454C8—2442no—
455C8—2442no—
456C22—21no—
457C22—21no—
458C22—4881no—
459C22—4881no—
460C22—4881no—
461C22#1683990112yes—
462C22C2×#1451974611yescharacteristic
463C4—21no—
464C4—21no—
465C4—24no—
466C4—24no—
467C4—21no—
468C4—44no—
469C4—42no—
470C4—42no—
471C4—4881no—
472C4#1683990112yes—
473C4C22×#1175820111yescharacteristic
474C4C2×#1451980811yescharacteristic
475C4C2×#1451980812yes—
476C2—22no—
477C2—9761no—
478C2C2×#1683990111yescharacteristic
479C2#2145505311yescharacteristic
480C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×Q82×C61C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenterC2×Q82D61
hypocenterC61#18768
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
4. #21445040C2semidirect
5. #21444897C2semidirect
6. #21444500C2semidirect
7. #21444248C2semidirect
8. #21444214C2semidirect
9. C2×Q82×C61C2semidirect
117. C2×Q8×C61SD16semidirect
182. Q8×C61#137semidirect
236. C2×Q82D61semidirect
283. C61#18768semidirect
361. C2×Q8#11758201semidirect
425. Q8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214558502
#214552912
#214450402
#214448972
#214445002
#214442482
#214442142
C2×Q82×C612
#1876861