Finite Group Databasebeta

← #29729796

Subgroups — #29729796

Order 15616.

712 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
1#21455612C211yesmaximal, characteristic
2C2×#16839964C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21447186C212yesmaximal, complemented
5#21447079C212yesmaximal
6#21446721C212yesmaximal
7#21446664C211yesmaximal, characteristic, complemented
8#21446594C211yesmaximal, characteristic, complemented
9#21446477C212yesmaximal, complemented
10C4×#14519693C211yesmaximal, characteristic
11C61×#3139C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×C4×D244C2211yescharacteristic
14#16840059—22no—
15#16840049—21no—
16#16840039—21no—
17#16839964—21no—
18#16839964—21no—
19#16839964—22no—
20#16839893—21no—
21#16839893—22no—
22Q8×#92418—21no—
23C2×#14519712C2212yes—
24#16839805C2212yes—
25#16839789C2212yes—
26#16839786C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28#16839766C2212yes—
29C2×#14519701C2212yes—
30C2×#14519696C2212yes—
31C2×#14519696C2212yes—
32C2×#14519693C2211yescharacteristic
33C2×#14519693C2211yescharacteristic
34#16839718C2212yes—
35C4×#11758194C2211yescharacteristic
36#16839530C2211yescharacteristic
37#16839530C2211yescharacteristic
38#16839526C2212yes—
39C61×#533C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C61×#491—22no—
42C61×#487—21no—
43C61×#487—21no—
44C61×#419C2212yes—
45C61×#419C2212yes—
46C4×M16×C61C2211yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C2×#11758201—24no—
50C2×#11758201—22no—
51C2×#11758201—22no—
52C2×#11758201—42no—
53C2×#11758194C2311yescharacteristic
54C2×#11758194C2312yes—
55C2×#11758194C2311yescharacteristic
56C22×#92418C2312yes—
57C4×D244—21no—
58C4×D244—21no—
59C4×D244—21no—
60C4×D244—21no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758198—21no—
63#14519712—42no—
64#14519710—22no—
65#14519708—22no—
66C4⋊#92418—21no—
67C4⋊#92418—21no—
68C4⋊#92418—22no—
69C4⋊#92418—22no—
70C4⋊#92418D411yescharacteristic
71C4⋊#92418D412yes—
72C4⋊#92418D411yescharacteristic
73#14519701—22no—
74#14519701—22no—
75#14519697—22no—
76#14519697—22no—
77#14519696—24no—
78#14519696—42no—
79#14519693—22no—
80#14519693—22no—
81#14519688—21no—
82#14519688—22no—
83C4×#92418—21no—
84C4×#92418—22no—
85C22×Q8×C61C2312yes—
86C2×M16×C61C2311yescharacteristic
87C2×M16×C61C2311yescharacteristic
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61D411yescharacteristic
93C4⋊Q8×C61D411yescharacteristic
94#125×C61D412yes—
95C22⋊Q8×C61—21no—
96C22⋊Q8×C61—21no—
97C4×Q8×C61—21no—
98C4×Q8×C61—21no—
99#117×C61C2312yes—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#103×C61—22no—
104#103×C61—22no—
105#103×C61—22no—
106#103×C61—22no—
107#97×C61—21no—
108C4×C8×C61—21no—
109C23×D61—21no—
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—22no—
117C2×Q8×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61#4111yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244#4112yes—
129C2×C4×D61—41no—
130C2×C4×D61—41no—
131C2×C4×D61—82no—
132M16×C61—21no—
133M16×C61—21no—
134M16×C61—21no—
135M16×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61C2×D412yes—
147C4⋊C4×C61C2×D412yes—
148C22⋊C4×C61—22no—
149C42×C61—22no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154#11758201—44no—
155#11758201—44no—
156#11758198—41no—
157#11758198—41no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160#11758194—21no—
161#11758194—21no—
162#11758194—22no—
163#11758194—21no—
164#11758194—22no—
165#11758194#4112yes—
166C2×#92418—21no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—22no—
171C2×#92418C2×D412yes—
172C2×#92418C2×D412yes—
173C23×C61C2×#4111yescharacteristic
174C22×D61—22no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—81no—
178C22×D61—81no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—42no—
182Q8×C61—42no—
183Q8×C61—42no—
184Q8×C61—81no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61H(5,𝔽2)11yescharacteristic
201C2×C4×C612−1+411yescharacteristic
202C2×Dic61—41no—
203C2×Dic61—81no—
204D244—22no—
205D244—22no—
206D244—41no—
207D244—41no—
208C4×D61—82no—
209C4×D61—82no—
210C8×C61—41no—
211C8×C61—41no—
212#92418—22no—
213#92418—22no—
214#92418—22no—
215#92418—41no—
216#92418—42no—
217#25313—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61C2×#12111yescharacteristic
221C22×C61#52911yescharacteristic
222C22×C61#54211yescharacteristic
223C2×D61—42no—
224C2×D61—82no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—82no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—82no—
234C4×C61#47112yes—
235C4×C61#54611yescharacteristic
236C4×C61#54811yescharacteristic
237C4×C61#54811yescharacteristic
238C4×C61#55011yescharacteristic
239Dic61—81no—
240#3139D6111yescharacteristic, complemented
241#3077—611no—
242C2×#461—611no—
243#2962—611no—
244#2064—612no—
245#2046—612no—
246#1977—612no—
247#1966—611no—
248#1950—611no—
249#1930—612no—
250C4×#100—611no—
251C2×C61—21no—
252C2×C61—21no—
253C2×C61#242611yescharacteristic
254C2×C61C2×#47111yescharacteristic
255C2×C61#306911yescharacteristic
256D61—82no—
257C22×SD16—612no—
258#533C2×D6111yescharacteristic
259C2×C4⋊Q8C2×D6111yescharacteristic
260C2×C4×D4—611no—
261#491—22no—
262#487—21no—
263#487—21no—
264#478—1222no—
265#476—1221no—
266#474—1221no—
267#461—1221no—
268#461—1222no—
269#461—1221no—
270#446—1222no—
271#446—1221no—
272C8×Q8—1221no—
273C2×#108—612no—
274#428—612no—
275#424—612no—
276#423—611no—
277C2×C4⋊C8—611no—
278#419C2×D6112yes—
279#419C2×D6112yes—
280#418—612no—
281C2×#103—612no—
282C2×#101—612no—
283C2×#101—612no—
284C2×#100—611no—
285C2×#100—611no—
286#405—612no—
287C4×M16—611no—
288C4×M16C2×D6111yescharacteristic
289#343—611no—
290#343—611no—
291#341—612no—
292C61#2531311yescharacteristic, complemented, Sylow
293C22×Q8C22×D6112yes—
294C22×D4—611no—
295C23×C4—1221no—
296C2×SD16—1222no—
297C2×SD16—1222no—
298C2×SD16—1224no—
299C2×SD16—2442no—
300C2×M16—611no—
301C2×M16—612no—
302C2×M16—611no—
303C2×M16C22×D6111yescharacteristic
304C2×M16C22×D6111yescharacteristic
305C22×C8—612no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8C22⋊D6111yescharacteristic
311C4⋊Q8C22⋊D6111yescharacteristic
312#125C22⋊D6112yes—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×D4—1221no—
318C4×D4—1221no—
319C4×D4—1221no—
320C4×D4—1221no—
321#117C22×D6112yes—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—611no—
325C2×C22⋊C4—1221no—
326C2×C42C22×D6111yescharacteristic
327#108—2442no—
328#107—1222no—
329#106—1222no—
330C4⋊C8—611no—
331C4⋊C8—611no—
332C4⋊C8—612no—
333C4⋊C8—1222no—
334C4⋊C8—1222no—
335C4⋊C8—1221no—
336C4⋊C8—1221no—
337#103—22no—
338#103—22no—
339#103—22no—
340#103—22no—
341#103—1222no—
342#103—1222no—
343#102—1222no—
344#102—1222no—
345#101—1224no—
346#101—2442no—
347#100—1222no—
348#100—1222no—
349#97—21no—
350#97—1222no—
351#97—1221no—
352C4×C8—21no—
353C4×C8—1222no—
354C4×C8—1221no—
355C24—1221no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×D4—612no—
365C2×D4—1221no—
366C2×D4—1221no—
367C2×D4—1221no—
368C2×D4—1221no—
369C2×D4—1221no—
370C22×C4—22no—
371C22×C4—1221no—
372C22×C4—4882no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4#1175822411yescharacteristic
376C22×C4#1175822411yescharacteristic
377C22×C4C23×D6111yescharacteristic
378SD16—2444no—
379SD16—2444no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—612no—
385M16—1221no—
386M16—1221no—
387M16—1222no—
388M16—1221no—
389M16—1222no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—612no—
393C2×C8—612no—
394C2×C8—1221no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—1221no—
398C2×C8—1222no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4C2×C22⋊D6112yes—
410C4⋊C4C2×C22⋊D6112yes—
411C22⋊C4—22no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—22no—
415C42D4×D6111yescharacteristic
416C42D4×D6111yescharacteristic
417C42C2×C22⋊D6111yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C23—1222no—
420C23—4881no—
421C23—4881no—
422C23—2441no—
423C23—2441no—
424C23C2×#1175822411yescharacteristic
425Q8—42no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429Q8—42no—
430Q8—81no—
431D4—1222no—
432D4—1222no—
433D4—2441no—
434D4—2441no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—4882no—
443C2×C4—4882no—
444C2×C4—4881no—
445C2×C4—2441no—
446C2×C4#1451974911yescharacteristic
447C2×C4#1451974911yescharacteristic
448C2×C4#1451974911yescharacteristic
449C2×C4#1451974911yescharacteristic
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4C22×C22⋊D6111yescharacteristic
452C2×C4#41×D6111yescharacteristic
453C2×C4#41×D6111yescharacteristic
454C2×C4#1451982311yescharacteristic
455C2×C4#1451982411yescharacteristic
456C8—41no—
457C8—41no—
458C8—1222no—
459C8—1222no—
460C8—1222no—
461C8—2441no—
462C8—2442no—
463C22—21no—
464C22—21no—
465C22—4882no—
466C22—4882no—
467C22—4881no—
468C22—4881no—
469C22—2442no—
470C22C2×#1451974911yescharacteristic
471C22#1684029111yescharacteristic
472C22#1684038111yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—82no—
479C4—4881no—
480C4#1684002712yes—
481C4#1684040511yescharacteristic
482C4#1684041911yescharacteristic
483C4#1684042011yescharacteristic
484C4#1684043111yescharacteristic
485C2—21no—
486C2—21no—
487C2—4882no—
488C2#2144938811yescharacteristic
489C2C2×#1684002711yescharacteristic
490C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3139C2
socleC22×C61C2×#121
prime radicalC22×D244C23
hypercenter#3139D61
hypocenterC61#25313
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
3. #21454768C2semidirect
4. #21447186C2semidirect
7. #21446664C2semidirect
8. #21446594C2semidirect
9. #21446477C2semidirect
11. C61×#3139C2semidirect
240. #3139D61semidirect
292. C61#25313semidirect

Maximal subgroups

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

subgroupindex
#214556122
C2×#168399642
#214547682
#214471862
#214470792
#214467212
#214466642
#214465942
#214464772
C4×#145196932
C61×#31392
#2531361