Finite Group Databasebeta

← #29729817

Subgroups — #29729817

Order 15616.

732 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456195C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21447176C212yesmaximal, complemented
5#21447079C212yesmaximal, complemented
6#21446721C212yesmaximal, complemented
7#21446658C211yesmaximal, characteristic
8#21446592C211yesmaximal, characteristic, complemented
9#21446480C212yesmaximal, complemented
10C4×#14519696C211yesmaximal, characteristic, complemented
11C61×#3078C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519774C2211yescharacteristic
14#16840128—22no—
15#16840110—21no—
16#16840110—21no—
17#16839962—21no—
18#16839962—21no—
19#16839962—22no—
20#16839893—21no—
21#16839893—22no—
22Q8×#92418—21no—
23C2×#14519712C2212yes—
24#16839789C2212yes—
25C2×C4⋊#92418C2211yescharacteristic
26#16839766C2212yes—
27#16839766C2212yes—
28C2×#14519697C2212yes—
29C2×#14519696C2211yescharacteristic
30C2×#14519696C2211yescharacteristic
31C2×#14519693C2212yes—
32C2×#14519693C2212yes—
33#16839718C2212yes—
34C4×#11758194C2211yescharacteristic
35C4×#11758194C2211yescharacteristic
36#16839531C2211yescharacteristic
37#16839531C2211yescharacteristic
38#16839526C2212yes—
39C61×#533C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C61×#479—22no—
42C61×#477—21no—
43C61×#475—21no—
44C61×#428C2212yes—
45C61×#423C2211yescharacteristic
46C61×#419C2212yes—
47C22×D244—21no—
48C22×D244—21no—
49C22×D244C2312yes—
50C2×#11758201—24no—
51C2×#11758201—22no—
52C2×#11758201—22no—
53C2×#11758201—42no—
54C2×#11758194C2311yescharacteristic
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2311yescharacteristic
58C22×#92418C2312yes—
59#14519774—21no—
60#14519774—21no—
61#14519774—21no—
62#14519774—21no—
63#14519712—42no—
64C4⋊#92418—22no—
65C4⋊#92418—22no—
66C4⋊#92418D411yescharacteristic
67C4⋊#92418D412yes—
68C4⋊#92418D411yescharacteristic
69#14519697—22no—
70#14519697—22no—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75#14519696—22no—
76#14519696—22no—
77#14519693—24no—
78#14519693—42no—
79#14519688—21no—
80#14519688—21no—
81#14519688—22no—
82C4×#92418—21no—
83C4×#92418—21no—
84C4×#92418—22no—
85C22×Q8×C61C2311yescharacteristic
86C2×M16×C61C2312yes—
87C4⋊Q8×C61D411yescharacteristic
88C4⋊Q8×C61D411yescharacteristic
89#125×C61D412yes—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98#117×C61C2312yes—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104#106×C61—22no—
105C4⋊C8×C61—21no—
106C4⋊C8×C61—21no—
107#103×C61—22no—
108#103×C61—22no—
109C23×D61—22no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61#4112yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—41no—
130C2×D244—41no—
131C2×D244—41no—
132C2×D244—41no—
133C2×D244—82no—
134C2×D244—82no—
135M16×C61—22no—
136M16×C61—22no—
137C2×C8×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61C2×D412yes—
147C4⋊C4×C61C2×D412yes—
148C22⋊C4×C61—22no—
149C42×C61—22no—
150C42×C61—41no—
151C42×C61—41no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156#11758201—44no—
157#11758201—44no—
158#11758194—21no—
159#11758194—21no—
160#11758194—21no—
161#11758194—21no—
162#11758194—21no—
163#11758194—21no—
164#11758194—21no—
165#11758194#4112yes—
166C2×#92418—21no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418C2×D412yes—
173C2×#92418C2×D412yes—
174C23×C61C2×#4111yescharacteristic
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—82no—
179C22×D61—82no—
180Q8×C61—22no—
181Q8×C61—22no—
182Q8×C61—41no—
183Q8×C61—41no—
184Q8×C61—42no—
185Q8×C61—81no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—81no—
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C2×#4111yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61H(5,𝔽2)11yescharacteristic
204C2×C4×C612−1+411yescharacteristic
205D244—42no—
206D244—42no—
207D244—42no—
208D244—42no—
209D244—82no—
210D244—82no—
211D244—82no—
212D244—82no—
213C8×C61—42no—
214#92418—22no—
215#92418—22no—
216#92418—22no—
217#92418—41no—
218#92418—41no—
219#92418—41no—
220#25315—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61C2×#12111yescharacteristic
224C22×C61#52911yescharacteristic
225C22×C61#54211yescharacteristic
226C2×D61—82no—
227C2×D61—82no—
228C2×D61—82no—
229C2×D61—162no—
230C2×D61—162no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—42no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61—81no—
238C4×C61#47112yes—
239C4×C61#54611yescharacteristic
240C4×C61#54811yescharacteristic
241C4×C61#54811yescharacteristic
242C4×C61#55011yescharacteristic
243#3138—611no—
244#3078D6111yescharacteristic, complemented
245C2×#460—611no—
246#2962—611no—
247#2063—612no—
248#2046—612no—
249#1977—612no—
250#1965—611no—
251#1950—611no—
252#1931—612no—
253C4×#101—611no—
254C2×C61—21no—
255C2×C61—21no—
256C2×C61#242611yescharacteristic
257C2×C61C2×#47111yescharacteristic
258C2×C61#306811yescharacteristic
259D61—162no—
260C22×SD16—612no—
261#533C2×D6111yescharacteristic
262C2×#127—611no—
263C2×C4×Q8C2×D6111yescharacteristic
264#490—1222no—
265#486—1221no—
266#486—1221no—
267#479—22no—
268#477—21no—
269#475—21no—
270#460—1221no—
271#460—1222no—
272#460—1221no—
273#446—1222no—
274#446—1221no—
275C8×Q8—1221no—
276C2×#108—612no—
277#428C2×D6112yes—
278#424—612no—
279#423C2×D6111yescharacteristic
280C2×C4⋊C8—611no—
281#419C2×D6112yes—
282#418—612no—
283#418—612no—
284C2×#102—612no—
285C2×#101—611no—
286C2×#101—611no—
287C2×#100—612no—
288C2×#100—612no—
289#405—612no—
290C4×M16—611no—
291C4×M16—611no—
292#343—611no—
293#343—611no—
294#341—612no—
295C61#2531511yescharacteristic, complemented, Sylow
296C22×Q8C22×D6111yescharacteristic
297C22×D4—612no—
298C22×D4—1221no—
299C22×D4—1221no—
300C2×SD16—1222no—
301C2×SD16—1222no—
302C2×SD16—1224no—
303C2×SD16—2442no—
304C2×M16—611no—
305C2×M16—611no—
306C2×M16—611no—
307C2×M16—611no—
308C2×M16C22×D6112yes—
309C22×C8—612no—
310C4⋊Q8C22⋊D6111yescharacteristic
311C4⋊Q8C22⋊D6111yescharacteristic
312#127—1221no—
313#127—1221no—
314#127—1221no—
315#127—1221no—
316#125C22⋊D6112yes—
317C22⋊Q8—21no—
318C22⋊Q8—21no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—21no—
325#117C22×D6112yes—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C22×D6111yescharacteristic
328C2×C42—21no—
329C2×C42C22×D6111yescharacteristic
330#108—2442no—
331#107—22no—
332#106—22no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—611no—
336C4⋊C8—611no—
337C4⋊C8—612no—
338C4⋊C8—1222no—
339C4⋊C8—1222no—
340#103—22no—
341#103—22no—
342#102—1222no—
343#102—1222no—
344#102—1222no—
345#102—1222no—
346#102—1222no—
347#102—1222no—
348#101—1222no—
349#101—1222no—
350#100—1224no—
351#100—2442no—
352#97—1222no—
353#97—1221no—
354#97—1221no—
355C4×C8—1222no—
356C4×C8—1221no—
357C4×C8—1221no—
358C24—1222no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8#1175822612yes—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—1222no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—4882no—
374C2×D4—4882no—
375C2×D4—2441no—
376C2×D4—2441no—
377C2×D4—2441no—
378C2×D4—2441no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4#1175822411yescharacteristic
382C22×C4#1175822411yescharacteristic
383C22×C4C23×D6111yescharacteristic
384SD16—2444no—
385SD16—2444no—
386M16—22no—
387M16—22no—
388M16—612no—
389M16—1221no—
390M16—1221no—
391M16—1221no—
392M16—1221no—
393M16—1221no—
394M16—1221no—
395M16—1221no—
396C2×C8—22no—
397C2×C8—612no—
398C2×C8—612no—
399C2×C8—1221no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—1221no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4C2×C22⋊D6112yes—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—22no—
416C42—22no—
417C42—41no—
418C42—41no—
419C42D4×D6111yescharacteristic
420C42D4×D6111yescharacteristic
421C42C2×C22⋊D6111yescharacteristic
422C42C2×C22⋊D6111yescharacteristic
423C23—4882no—
424C23—4882no—
425C23—2442no—
426C23—2442no—
427C23—2442no—
428C23C2×#1175822411yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—42no—
432Q8—41no—
433Q8—41no—
434Q8—81no—
435D4—4882no—
436D4—4882no—
437D4—4882no—
438D4—4882no—
439D4—2442no—
440D4—2442no—
441D4—2442no—
442D4—2442no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—81no—
452C2×C4#1451974911yescharacteristic
453C2×C4#1451974911yescharacteristic
454C2×C4#1451974911yescharacteristic
455C2×C4#1451974911yescharacteristic
456C2×C4C2×D4×D6111yescharacteristic
457C2×C4C22×C22⋊D6111yescharacteristic
458C2×C4#41×D6111yescharacteristic
459C2×C4C2×#1175822611yescharacteristic
460C2×C4#1451982311yescharacteristic
461C2×C4#1451982511yescharacteristic
462C8—42no—
463C8—1222no—
464C8—1222no—
465C8—1222no—
466C8—2441no—
467C8—2441no—
468C8—2441no—
469C22—21no—
470C22—21no—
471C22—9762no—
472C22—9762no—
473C22—4882no—
474C22—4882no—
475C22—4882no—
476C22C2×#1451974911yescharacteristic
477C22#1684029811yescharacteristic
478C22#1684038511yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4#1684002712yes—
487C4#1684040311yescharacteristic
488C4#1684042211yescharacteristic
489C4#1684042311yescharacteristic
490C4#1684043111yescharacteristic
491C2—21no—
492C2—21no—
493C2—9762no—
494C2#2144938711yescharacteristic
495C2C2×#1684002711yescharacteristic
496C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3078C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#3078D61
hypocenterC61#25315
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
3. #21454768C2semidirect
4. #21447176C2semidirect
5. #21447079C2semidirect
6. #21446721C2semidirect
8. #21446592C2semidirect
9. #21446480C2semidirect
10. C4×#14519696C2semidirect
11. C61×#3078C2semidirect
244. #3078D61semidirect
295. C61#25315semidirect

Maximal subgroups

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

subgroupindex
#214561952
C2×#168399622
#214547682
#214471762
#214470792
#214467212
#214466582
#214465922
#214464802
C4×#145196962
C61×#30782
#2531561