Finite Group Databasebeta

← #29722588

Subgroups — #29722588

Order 15616.

844 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840034C211yesmaximal, characteristic, complemented
2#21455445C212yesmaximal
3#21454522C211yesmaximal, characteristic
4C2×#16839811C211yesmaximal, characteristic, complemented
5#21447184C212yesmaximal, complemented
6#21447119C212yesmaximal, complemented
7#21446735C211yesmaximal, characteristic, complemented
8#21446597C211yesmaximal, characteristic, complemented
9#21446288C212yesmaximal, complemented
10#21445879C211yesmaximal, characteristic, complemented
11C2×C61×#470C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519783C2211yescharacteristic
14C22×#11758194C2212yes—
15#16840273C2211yescharacteristic
16#16840034—24no—
17#16840018—22no—
18#16840010—22no—
19#16840001—22no—
20#16839993—22no—
21#16839838—22no—
22#16839831—22no—
23#16839811—24no—
24C2×#14519713C2212yes—
25C2×#14519713C2211yescharacteristic
26C2×#14519713C2211yescharacteristic
27#16839802C2212yes—
28#16839788C2211yescharacteristic
29#16839771C2212yes—
30#16839761C2212yes—
31C2×#14519696C2212yes—
32C2×#14519696C2212yes—
33#16839728C2211yescharacteristic
34#16839726C2211yescharacteristic
35#16839717C2211yescharacteristic
36#16839531C2212yes—
37#16839531C2212yes—
38C22×Q16×C61C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C2×C22⋊Q8×C61C2212yes—
41C61×#470—22no—
42C61×#470—22no—
43C2×#106×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C2×#11758222C2311yescharacteristic
46#14519808—24no—
47#14519808—42no—
48C2×#11758202—22no—
49C2×#11758201—22no—
50#14519783—22no—
51#14519783—22no—
52C2×#11758194—22no—
53C2×#11758194—24no—
54C2×#11758194—42no—
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2312yes—
58C2×#11758194C2312yes—
59C2×#11758189—21no—
60C22×#92418—22no—
61C2×C8×D61—21no—
62#14519760—21no—
63#14519756D412yes—
64#14519751D412yes—
65#14519746—21no—
66C4×C22⋊D61—22no—
67#14519736C2311yescharacteristic
68#14519713—22no—
69#14519713—24no—
70#14519713D414yescomplemented
71#14519710—22no—
72#14519708—22no—
73#14519706—22no—
74#14519701—22no—
75#14519701—22no—
76#14519700—22no—
77#14519697—22no—
78#14519696—24no—
79#14519696—42no—
80C22⋊#92418—22no—
81#14519689—22no—
82#14519687—21no—
83C8×Dic61—21no—
84C22×Q8×C61C2312yes—
85C23×C4×C61C2311yescharacteristic
86C2×Q16×C61—22no—
87C2×Q16×C61—21no—
88C2×Q16×C61—21no—
89C2×Q16×C61—22no—
90C2×M16×C61—22no—
91C2×M16×C61—22no—
92C2×M16×C61D412yes—
93C2×M16×C61D412yes—
94C22×C8×C61C2311yescharacteristic
95C22×C8×C61C2311yescharacteristic
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61C2312yes—
102C2×C22⋊C4×C61—22no—
103#106×C61—21no—
104#106×C61—21no—
105#103×C61—24no—
106C24×C61#4111yescharacteristic
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—21no—
124C2×C22⋊D61—22no—
125#11758222—21no—
126#11758222—21no—
127#11758222—21no—
128#11758222—21no—
129C2×D244C2×D411yescharacteristic
130C2×C4×D61—21no—
131C2×Dic122C2×D411yescharacteristic
132Q16×C61—42no—
133Q16×C61—42no—
134M16×C61—24no—
135M16×C61—24no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61C2×D412yes—
139C2×C8×C61C2×D412yes—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61#4112yes—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—42no—
146C4⋊C4×C61—42no—
147C22⋊C4×C61—42no—
148C22⋊C4×C61—42no—
149C22⋊Dic61—22no—
150C22⋊Dic61—21no—
151#11758202—44no—
152#11758201—44no—
153#11758198—22no—
154C4⋊Dic61C2×D412yes—
155#11758196—22no—
156C4×Dic61—21no—
157#11758194—22no—
158#11758194—21no—
159#11758194—22no—
160#11758194—21no—
161#11758194—21no—
162#11758194—22no—
163#11758194—21no—
164#11758194—44no—
165#11758194C2×D414yes—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—44no—
171#11758189—42no—
172C8×D61—42no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—42no—
176C23×C61#12112yes—
177C23×C61C2×#4111yescharacteristic
178C22×D61—21no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—42no—
182Q8×C61—42no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—82no—
194C2×C4×C61#12112yes—
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61H(5,𝔽2)11yescharacteristic
200C2×C4×C612−1+411yescharacteristic
201C22⋊D61—41no—
202C22⋊D61—41no—
203C22⋊D61—82no—
204C2×Dic61—22no—
205C2×Dic61—21no—
206D244—21no—
207D244—21no—
208C4×D61—41no—
209C4×D61—41no—
210Dic122—21no—
211Dic122—21no—
212C8×C61—22no—
213C8×C61#12112yes—
214C8×C61#12112yes—
215#92418—24no—
216#92418—41no—
217#92418—41no—
218#92418—42no—
219#24807—611nomaximal, Sylow
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61C2×#12111yescharacteristic
227C22×C61#54811yescharacteristic
228C22×C61#55311yescharacteristic
229C2×D61—41no—
230C2×D61—42no—
231C4×C61—22no—
232C4×C61—82no—
233C4×C61—82no—
234C4×C61C2×#12111yescharacteristic
235C4×C61#53711yescharacteristic
236C4×C61D4211yescharacteristic
237C4×C61#54711yescharacteristic
238Dic61—41no—
239Dic61—42no—
240C2×#473—611no—
241#3056—612no—
242C2×#470D6111yescharacteristic, complemented
243#2930—611no—
244C2×#430—611no—
245#2064—612no—
246#2052—612no—
247#1979—611no—
248#1951—611no—
249#1901—612no—
250#1815—611no—
251C2×C61—21no—
252C2×C61—21no—
253C2×C61—42no—
254C2×C61#241611yescharacteristic
255C2×C61#306912yes—
256D61—81no—
257C2×#137—612no—
258C22×Q16C2×D6111yescharacteristic
259C2×#131—611no—
260C22×M16—612no—
261C22×M16C2×D6111yescharacteristic
262#525—611no—
263C2×C22⋊Q8C2×D6112yes—
264#473—1224no—
265#470—22no—
266#470—22no—
267#470—1222no—
268#468—1222no—
269#467—1222no—
270#465—1222no—
271#436—1222no—
272#435—1222no—
273#430—1224no—
274C2×#108—611no—
275C2×#108—612no—
276C2×#108—611no—
277#428—612no—
278C2×#106C2×D6111yescharacteristic
279#424—611no—
280#419—612no—
281#417—612no—
282C2×#103C2×D6112yes—
283C2×#101—612no—
284C2×#101—612no—
285#408—611no—
286#407—611no—
287#405—611no—
288#343—612no—
289#343—612no—
290C61#2480711yescharacteristic, complemented, Sylow
291C2×#41—611no—
292C22×Q8C22×D6112yes—
293C23×C4C22×D6111yescharacteristic
294#137—1224no—
295#137—2442no—
296C2×Q16—22no—
297C2×Q16—22no—
298C2×Q16—21no—
299C2×Q16—21no—
300C2×Q16—1222no—
301C2×SD16—1222no—
302#131—1222no—
303#131—1222no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16—612no—
307C2×M16—612no—
308C2×M16—611no—
309C2×M16—611no—
310C2×M16—1222no—
311C2×M16—1221no—
312C2×M16—1224no—
313C2×M16—2442no—
314C2×M16C22⋊D6112yes—
315C2×M16C22⋊D6112yes—
316C22×C8—1222no—
317C22×C8—1221no—
318C22×C8C22×D6111yescharacteristic
319C22×C8C22×D6111yescharacteristic
320#123—1221no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—612no—
326#121—612no—
327#120—1221no—
328C4×D4—1222no—
329#117—611no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4C22×D6112yes—
332C2×C22⋊C4—22no—
333#108—614no—
334#108—1224no—
335#108—1222no—
336#107—1222no—
337#106—21no—
338#106—21no—
339#106—1222no—
340C4⋊C8—1222no—
341#103—24no—
342#103—1222no—
343#103—1222no—
344#103—1222no—
345#102—1222no—
346#101—1224no—
347#101—2442no—
348C22⋊C8—1222no—
349C22⋊C8—1222no—
350#97—1221no—
351C4×C8—1221no—
352C24#1175822411yescharacteristic
353#41—1221no—
354#41—1221no—
355#41—1221no—
356#41—1221no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—611no—
364C2×D4—611no—
365C2×D4—1222no—
366C2×D4—1221no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—1221no—
374C22×C4D4×D6112yes—
375C22×C4#1175822411yescharacteristic
376C22×C4C2×C22⋊D6112yes—
377C22×C4C23×D6111yescharacteristic
378Q16—42no—
379Q16—42no—
380Q16—2444no—
381SD16—2444no—
382M16—24no—
383M16—24no—
384M16—614no—
385M16—1222no—
386M16—1221no—
387M16—1222no—
388M16—1221no—
389M16—1221no—
390M16—1222no—
391M16—1221no—
392M16—2444no—
393M16—2442no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—2444no—
401C2×C8—2442no—
402C2×C8D4×D6112yes—
403C2×C8#1175822412yes—
404C2×C8C2×C22⋊D6112yes—
405C2×C8C2×C22⋊D6111yescharacteristic
406C2×C8C2×C22⋊D6111yescharacteristic
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—612no—
412C4⋊C4—1222no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—1221no—
416C22⋊C4—1222no—
417C22⋊C4—1222no—
418C42—1221no—
419C23—21no—
420C23—21no—
421C23—42no—
422C23—1221no—
423C23#1451974812yes—
424C23C2×#1175822411yescharacteristic
425Q8—42no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429Q8—1221no—
430Q8—1221no—
431D4—1221no—
432D4—1221no—
433D4—4882no—
434D4—2441no—
435D4—2441no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—82no—
447C2×C4—1222no—
448C2×C4—1221no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4#1451974812yes—
452C2×C4C2×D4×D6111yescharacteristic
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C22×C22⋊D6111yescharacteristic
455C2×C4C2×#1175822411yescharacteristic
456C2×C4#1451982311yescharacteristic
457C2×C4#1451982411yescharacteristic
458C8—22no—
459C8—1224no—
460C8—2441no—
461C8—2441no—
462C8—2442no—
463C8#1451974912yes—
464C8#1451974912yes—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—81no—
470C22—81no—
471C22—2441no—
472C22—2442no—
473C22C2×#1451974811yescharacteristic
474C22#1684041811yescharacteristic
475C22#1684046411yescharacteristic
476C4—22no—
477C4—82no—
478C4—82no—
479C4—2441no—
480C4—2442no—
481C4C2×#1451974911yescharacteristic
482C4#1684034011yescharacteristic
483C4#1684039611yescharacteristic
484C4#1684041211yescharacteristic
485C2—21no—
486C2—21no—
487C2—42no—
488C2—4881no—
489C2#2144926011yescharacteristic
490C2#2145555212yes—
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#470C2
socleC22×C61C2×#121
prime radical#16840273C22
hypercenterC2×#470D61
hypocenterC61#24807
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#470

Complements

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

NH ≅ G/Ntype
1. C2×#16840034C2semidirect
4. C2×#16839811C2semidirect
5. #21447184C2semidirect
6. #21447119C2semidirect
7. #21446735C2semidirect
8. #21446597C2semidirect
9. #21446288C2semidirect
10. #21445879C2semidirect
11. C2×C61×#470C2semidirect
70. #14519713D4semidirect
242. C2×#470D61semidirect
290. C61#24807semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
#214554452
#214545222
C2×#168398112
#214471842
#214471192
#214467352
#214465972
#214462882
#214458792
C2×C61×#4702
#2480761