Finite Group Databasebeta

← #29722581

Subgroups — #29722581

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×#16840038C211yesmaximal, characteristic, complemented
2#21455449C212yesmaximal
3C2×#16840020C211yesmaximal, characteristic, complemented
4#21454522C211yesmaximal, characteristic
5#21447181C212yesmaximal, complemented
6#21447120C212yesmaximal, complemented
7#21446732C211yesmaximal, characteristic, complemented
8#21446600C211yesmaximal, characteristic, complemented
9#21446287C212yesmaximal, complemented
10#21445875C211yesmaximal, characteristic, complemented
11C2×C61×#430C211yesmaximal, characteristic, complemented
12C2×#14519810C2212yes—
13C22×Dic244C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840273C2211yescharacteristic
16C2×#14519756C2212yes—
17#16840038—24no—
18#16840020—22no—
19#16840020—22no—
20#16840020—22no—
21#16840008—22no—
22#16840003—22no—
23#16839991—22no—
24#16839838—22no—
25#16839831—22no—
26#16839804C2212yes—
27C2×#14519709C2211yescharacteristic
28#16839788C2211yescharacteristic
29#16839772C2212yes—
30#16839764C2212yes—
31C2×#14519702C2212yes—
32C2×#14519695C2212yes—
33C2×#14519695C2212yes—
34#16839728C2211yescharacteristic
35#16839726C2211yescharacteristic
36#16839717C2211yescharacteristic
37#16839530C2212yes—
38#16839530C2212yes—
39C22×M16×C61C2212yes—
40C22×M16×C61C2211yescharacteristic
41C61×#430—24no—
42C2×#108×C61C2212yes—
43C2×#108×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C2×#11758222C2311yescharacteristic
46C22×Dic122C2312yes—
47#14519810—24no—
48#14519810—42no—
49C2×Dic244—22no—
50C2×Dic244—21no—
51C2×Dic244—21no—
52C2×Dic244—22no—
53C2×Dic244—22no—
54C2×#11758190—22no—
55#14519783—22no—
56#14519783—22no—
57C2×#11758194C2311yescharacteristic
58C2×#11758194C2311yescharacteristic
59C2×#11758189—21no—
60C2×C8×D61—21no—
61#14519760—21no—
62#14519756—22no—
63#14519756—22no—
64#14519756—22no—
65#14519756—22no—
66#14519756D412yes—
67#14519751D412yes—
68#14519746—21no—
69C4×C22⋊D61—22no—
70#14519736C2311yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—22no—
73C2×C22⋊Dic61—22no—
74#14519711—22no—
75#14519709—21no—
76#14519709—21no—
77#14519709—22no—
78#14519706—22no—
79#14519702—22no—
80#14519702—24no—
81#14519702—22no—
82#14519702—22no—
83#14519699—22no—
84#14519695—24no—
85#14519695—42no—
86C22⋊#92418—22no—
87#14519689—22no—
88#14519687—21no—
89C8×Dic61—21no—
90C23×C4×C61C2311yescharacteristic
91C2×M16×C61—22no—
92C2×M16×C61—22no—
93C2×M16×C61—22no—
94C2×M16×C61—24no—
95C2×M16×C61—42no—
96C2×M16×C61D412yes—
97C2×M16×C61D412yes—
98C2×M16×C61C2312yes—
99C2×M16×C61C2312yes—
100C22×C8×C61—22no—
101C22×C8×C61C2311yescharacteristic
102C22×C8×C61C2311yescharacteristic
103#108×C61—24no—
104#108×C61—22no—
105#108×C61D414yescomplemented
106C24×C61#4111yescharacteristic
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D412yes—
112C22×C4×C61C2×D412yes—
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61C2411yescharacteristic
115C2×C22⋊D61—21no—
116C2×C22⋊D61—22no—
117C22×Dic61—22no—
118C22×Dic61—22no—
119#11758222—21no—
120#11758222—21no—
121#11758222—21no—
122#11758222—21no—
123C2×D244C2×D411yescharacteristic
124C2×C4×D61—21no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122C2×D411yescharacteristic
132M16×C61—22no—
133M16×C61—24no—
134M16×C61—24no—
135M16×C61—22no—
136M16×C61—22no—
137M16×C61—44no—
138M16×C61C2×D414yes—
139C2×C8×C61—21no—
140C2×C8×C61—22no—
141C2×C8×C61—21no—
142C2×C8×C61—22no—
143C2×C8×C61—44no—
144C2×C8×C61C2×D412yes—
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61C2×D412yes—
148C2×C8×C61#4112yes—
149C22⋊Dic61—22no—
150C22⋊Dic61—21no—
151C22⋊Dic61—42no—
152C22⋊Dic61—42no—
153#11758198—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61C2×D412yes—
157#11758196—22no—
158#11758196—42no—
159#11758196—42no—
160C4×Dic61—21no—
161#11758194—21no—
162#11758194—21no—
163#11758194—21no—
164#11758194—21no—
165C2×#92418—21no—
166C2×#92418—21no—
167Dic244—44no—
168Dic244—42no—
169Dic244—42no—
170#11758190—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—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61#12112yes—
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61H(5,𝔽2)11yescharacteristic
191C2×C4×C612−1+411yescharacteristic
192C22⋊D61—41no—
193C22⋊D61—41no—
194C22⋊D61—82no—
195C2×Dic61—22no—
196C2×Dic61—21no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—82no—
202D244—21no—
203D244—21no—
204C4×D61—41no—
205C4×D61—41no—
206Dic122—21no—
207Dic122—21no—
208Dic122—42no—
209Dic122—42no—
210Dic122—42no—
211Dic122—42no—
212C8×C61—22no—
213C8×C61—24no—
214C8×C61—42no—
215C8×C61#12112yes—
216C8×C61#12112yes—
217#92418—41no—
218#92418—41no—
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×C61C2×#12111yescharacteristic
233C4×C61#53711yescharacteristic
234C4×C61D4211yescharacteristic
235C4×C61#54711yescharacteristic
236Dic61—41no—
237Dic61—42no—
238Dic61—82no—
239Dic61—82no—
240C2×#473—611no—
241#3056—612no—
242C2×#470—611no—
243#2930—611no—
244C2×#430D6111yescharacteristic, complemented
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×Q16—611no—
259C2×#131—611no—
260C22×M16C2×D6112yes—
261C22×M16C2×D6111yescharacteristic
262#525—611no—
263C2×C22⋊Q8—612no—
264#473—1224no—
265#470—1222no—
266#470—1222no—
267#470—1222no—
268#468—1222no—
269#467—1222no—
270#465—1222no—
271#436—1222no—
272#435—1222no—
273#430—24no—
274C2×#108C2×D6112yes—
275C2×#108C2×D6111yescharacteristic
276C2×#108C2×D6111yescharacteristic
277#428—612no—
278C2×#106—611no—
279#424—611no—
280#419—612no—
281#417—612no—
282C2×#103—612no—
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×Q8—612no—
293C23×C4C22×D6111yescharacteristic
294#137—1224no—
295#137—2442no—
296C2×Q16—1222no—
297C2×Q16—1221no—
298C2×Q16—1221no—
299C2×Q16—1222no—
300C2×Q16—1222no—
301C2×SD16—1222no—
302#131—1222no—
303#131—1222no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16—22no—
307C2×M16—24no—
308C2×M16—42no—
309C2×M16—611no—
310C2×M16—611no—
311C2×M16—1221no—
312C2×M16D24412yes—
313C2×M16D24412yes—
314C2×M16C22×D6112yes—
315C2×M16C22×D6112yes—
316C22×C8—22no—
317C22×C8—1221no—
318C22×C8C22×D6111yescharacteristic
319C22×C8C22×D6111yescharacteristic
320#123—1221no—
321C22⋊Q8—612no—
322C22⋊Q8—1222no—
323C22⋊Q8—1222no—
324C22⋊Q8—1222no—
325C22⋊Q8—1222no—
326#121—612no—
327#120—1221no—
328C4×D4—1222no—
329#117—611no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—1222no—
332C2×C22⋊C4—1222no—
333#108—24no—
334#108—22no—
335#108C22⋊D6114yescomplemented
336#107—1222no—
337#106—1222no—
338#106—1221no—
339#106—1221no—
340C4⋊C8—1222no—
341#103—1222no—
342#103—1222no—
343#103—1222no—
344#103—1224no—
345#102—1222no—
346#101—1224no—
347#101—2442no—
348C22⋊C8—1222no—
349C22⋊C8—1222no—
350#97—1221no—
351C4×C8—1221no—
352C24#1175822611yescharacteristic
353#41—1221no—
354#41—1221no—
355#41—1221no—
356#41—1221no—
357C2×Q8—611no—
358C2×Q8—1222no—
359C2×Q8—1222no—
360C2×Q8—1222no—
361C2×Q8—1222no—
362C2×Q8—1222no—
363C2×Q8—1222no—
364C2×D4—611no—
365C2×D4—1222no—
366C2×D4—1221no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—1221no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4C2×D24412yes—
375C22×C4D4×D6112yes—
376C22×C4#1175822611yescharacteristic
377C22×C4C23×D6111yescharacteristic
378Q16—2444no—
379Q16—2442no—
380Q16—2442no—
381SD16—2444no—
382M16—22no—
383M16—24no—
384M16—24no—
385M16—22no—
386M16—22no—
387M16—44no—
388M16—1221no—
389M16—1221no—
390M16—1221no—
391M16—1221no—
392M16—2442no—
393M16C2×C22⋊D6114yes—
394C2×C8—22no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—44no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—2442no—
402C2×C8C2×D24412yes—
403C2×C8C2×D24411yescharacteristic
404C2×C8C2×D24411yescharacteristic
405C2×C8#1175822212yes—
406C2×C8C2×C22⋊D6112yes—
407C4⋊C4—612no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—2442no—
412C4⋊C4—2442no—
413C22⋊C4—1221no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C22⋊C4—2442no—
417C22⋊C4—2442no—
418C42—1221no—
419C23—21no—
420C23—21no—
421C23—42no—
422C23—1221no—
423C23#1451975212yes—
424C23C2×#1175822611yescharacteristic
425Q8—1221no—
426Q8—1221no—
427Q8—2442no—
428Q8—2442no—
429Q8—2442no—
430Q8—2442no—
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—1222no—
443C2×C4—1221no—
444C2×C4—4882no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4#1451975212yes—
452C2×C4C2×D4×D6111yescharacteristic
453C2×C4C22×C22⋊D6111yescharacteristic
454C2×C4C22×D24411yescharacteristic
455C2×C4C2×#1175822211yescharacteristic
456C2×C4#1451982211yescharacteristic
457C2×C4#1451982511yescharacteristic
458C8—22no—
459C8—24no—
460C8—42no—
461C8—2441no—
462C8—2441no—
463C8#1451975112yes—
464C8#1451975112yes—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—81no—
470C22—81no—
471C22—2441no—
472C22—2442no—
473C22C2×#1451975211yescharacteristic
474C22#1684042311yescharacteristic
475C22#1684046011yescharacteristic
476C4—22no—
477C4—4882no—
478C4—4882no—
479C4—2441no—
480C4—2442no—
481C4C2×#1451975111yescharacteristic
482C4#1684034311yescharacteristic
483C4D4×C22⋊D6111yescharacteristic
484C4#1684041311yescharacteristic
485C2—21no—
486C2—21no—
487C2—42no—
488C2—4881no—
489C2#2144925311yescharacteristic
490C2#2145555712yes—
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#430C2
socleC22×C61C2×#121
prime radical#16840273C22
hypercenterC2×#430D61
hypocenterC61#24807
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
1. C2×#16840038C2semidirect
3. C2×#16840020C2semidirect
5. #21447181C2semidirect
6. #21447120C2semidirect
7. #21446732C2semidirect
8. #21446600C2semidirect
9. #21446287C2semidirect
10. #21445875C2semidirect
11. C2×C61×#430C2semidirect
105. #108×C61D4semidirect
244. C2×#430D61semidirect
290. C61#24807semidirect
335. #108C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400382
#214554492
C2×#168400202
#214545222
#214471812
#214471202
#214467322
#214466002
#214462872
#214458752
C2×C61×#4302
#2480761