Finite Group Databasebeta

← #29572733

Subgroups — #29572733

Order 15616.

742 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519751C211yesmaximal, characteristic
2#21445799C211yesmaximal, characteristic, complemented
3C2×#16839495C212yesmaximal
4#21443698C212yesmaximal, complemented
5C61×C4⋊#95C211yesmaximal, characteristic, complemented
6C2×#14519751C2211yescharacteristic
7C2×C4×D244C411yescharacteristic, complemented
8C2×C4×C22⋊D61—21no—
9C2×C4⋊#92418—21no—
10C2×C22⋊#92418C2212yes—
11#16839664C411yescharacteristic, complemented
12#16839660C411yescharacteristic, complemented
13#16839653—21no—
14#16839641—21no—
15#16839622C2211yescharacteristic
16C4×C4⋊Dic61C411yescharacteristic, complemented
17C4×#11758198—21no—
18#16839517—21no—
19#16839495—24no—
20C22×C4⋊C4×C61—21no—
21C22×C42×C61C2211yescharacteristic
22C2×#95×C61—21no—
23C2×#95×C61C2212yes—
24C22×D244C2×C411yescharacteristic
25C22×C22⋊D61—21no—
26C22×C4×D61—21no—
27C22×#92418—22no—
28#14519751—21no—
29#14519751—21no—
30#14519751D412yes—
31#14519751Q812yes—
32C4×D244—42no—
33C4×C22⋊D61—42no—
34C4×C22⋊D61—42no—
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×#11758196—21no—
39C2×C22⋊Dic61—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×C4×Dic61—21no—
45C4⋊#92418—42no—
46C22⋊#92418—22no—
47C22⋊#92418—22no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51C23×C4×C61—22no—
52C23×C4×C61D411yescharacteristic
53C23×C4×C61D411yescharacteristic
54C23×C4×C61C2311yescharacteristic
55C2×C4⋊C4×C61—24no—
56C2×C4⋊C4×C61—42no—
57C2×C4⋊C4×C61—42no—
58C2×C42×C61—42no—
59C2×C42×C61C2×C411yescharacteristic
60C2×C42×C61C2×C411yescharacteristic
61C2×C42×C61D411yescharacteristic
62C2×C42×C61D411yescharacteristic
63#95×C61—22no—
64#95×C61—22no—
65#95×C61—42no—
66C24×C61C2×D411yescharacteristic
67C23×D61—21no—
68C22×C4×C61—22no—
69C22×C4×C61—22no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—24no—
73C22×C4×C61—41no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61—41no—
77C22×C4×C61—42no—
78C22×C4×C61—42no—
79C22×C4×C61—42no—
80C22×C4×C61—42no—
81C22×C4×C61—42no—
82C22×C4×C61—42no—
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61D812yes—
88C22×C4×C61SD1612yes—
89C22×C4×C61C22×C411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×Q811yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C2×C22⋊D61—22no—
96C2×C22⋊D61—22no—
97C2×C22⋊D61—41no—
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C2×C22⋊D61—41no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C2×D244—21no—
105C2×D244—41no—
106C2×D244—41no—
107C2×D244C4⋊C412yescomplemented
108C2×C4×D61—42no—
109C2×C4×D61—42no—
110C2×C4×D61—42no—
111C4⋊C4×C61—48no—
112C42×C61—22no—
113C42×C61—22no—
114C42×C61—41no—
115C42×C61—41no—
116C22⋊Dic61—41no—
117C22⋊Dic61—41no—
118#11758198—22no—
119#11758198—22no—
120#11758198—42no—
121#11758198—42no—
122#11758198—42no—
123C4⋊Dic61—21no—
124C4⋊Dic61—41no—
125C4⋊Dic61—41no—
126C4⋊Dic61C4⋊C412yescomplemented
127#11758196—42no—
128C4×Dic61—42no—
129C2×#92418—42no—
130C2×#92418—42no—
131C2×#92418—42no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61C2×C22⋊C411yescharacteristic
137C23×C61#12111yescharacteristic
138C23×C61C22⋊Q811yescharacteristic
139C23×C61C2×D811yescharacteristic
140C23×C61C2×SD1611yescharacteristic
141C23×C61AGL(1,ℤ/8)11yescharacteristic
142C23×C61#13711yescharacteristic
143C22×D61—22no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—24no—
157C2×C4×C61—24no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—44no—
171C2×C4×C61—44no—
172C2×C4×C61—44no—
173C2×C4×C61—82no—
174C2×C4×C61—82no—
175C2×C4×C61—82no—
176C2×C4×C61—82no—
177C2×C4×C61#9912yes—
178C2×C4×C61#10212yes—
179C2×C4×C61#10212yes—
180C2×C4×C61#10412yes—
181C2×C4×C61C2×C4⋊C411yescharacteristic
182C2×C4×C61#11711yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61C22⋊Q811yescharacteristic
185C22⋊D61—42no—
186C22⋊D61—42no—
187C22⋊D61—42no—
188C22⋊D61—42no—
189C2×Dic61—22no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198D244—42no—
199D244—42no—
200C4×D61—84no—
201#92418—82no—
202#8571—611nomaximal, Sylow
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—21no—
215C22×C61#38011yescharacteristic
216C22×C61C2×#9911yescharacteristic
217C22×C61#41011yescharacteristic
218C22×C61C2×#10211yescharacteristic
219C22×C61#41711yescharacteristic
220C22×C61C2×#10411yescharacteristic
221C22×C61#42111yescharacteristic
222C22×C61#45911yescharacteristic
223C22×C61#46111yescharacteristic
224C22×C61#47411yescharacteristic
225C22×C61#47611yescharacteristic
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—42no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—81no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—22no—
238C4×C61—41no—
239C4×C61—41no—
240C4×C61—84no—
241C4×C61#32714yescomplemented
242Dic61—42no—
243Dic61—81no—
244Dic61—81no—
245C4×#121—611no—
246#1797—611no—
247C2×#327—612no—
248C4⋊#95D6111yescharacteristic, complemented
249#1294—612no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61—21no—
253C2×C61—21no—
254C2×C61C2×#32711yescharacteristic
255C2×C61#150911yescharacteristic
256C2×C61#151311yescharacteristic
257C2×C61#151711yescharacteristic
258C2×C61#179911yescharacteristic
259C2×C61#180211yescharacteristic
260C2×C61#180611yescharacteristic
261D61—82no—
262C2×#121—611no—
263C2×C4×D4—611no—
264C2×C4×D4—1221no—
265C22×C4⋊C4—21no—
266C22×C42C2×D6111yescharacteristic
267C2×C4⋊C8—1221no—
268C2×C22⋊C8—612no—
269#390—611no—
270#389—611no—
271#388—1221no—
272#386—1221no—
273#380—611no—
274C4×C4⋊C4—611no—
275C4×C22⋊C4—1221no—
276C2×#95—21no—
277C2×#95C2×D6112yes—
278#336—1221no—
279#327—1224no—
280C61#857111yescharacteristic, complemented, Sylow
281C22×D4—611no—
282C22×D4—1221no—
283C23×C4—22no—
284C23×C4—1221no—
285C23×C4C22⋊D6111yescharacteristic
286C23×C4C22⋊D6111yescharacteristic
287C23×C4C22×D6111yescharacteristic
288C22×C8—1222no—
289#121—612no—
290#121—612no—
291#121—1221no—
292#121—1221no—
293C4×D4—2442no—
294C4×D4—2442no—
295C4×D4—2442no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—611no—
301C2×C4⋊C4—611no—
302C2×C4⋊C4—1221no—
303C2×C22⋊C4—1221no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1221no—
307C2×C22⋊C4—1221no—
308C2×C42—42no—
309C2×C42—1221no—
310C2×C42C4×D6111yescharacteristic
311C2×C42C4×D6111yescharacteristic
312C2×C42D24411yescharacteristic
313C2×C42D24411yescharacteristic
314C4⋊C8—2442no—
315C22⋊C8—1222no—
316C22⋊C8—1222no—
317#95—22no—
318#95—22no—
319#95—42no—
320#95—1221no—
321#95—1221no—
322#95—1221no—
323C24—1221no—
324C24C2×C22⋊D6111yescharacteristic
325C2×D4—612no—
326C2×D4—1222no—
327C2×D4—1222no—
328C2×D4—1221no—
329C2×D4—2441no—
330C2×D4—2441no—
331C2×D4—2441no—
332C2×D4—2441no—
333C2×D4—2441no—
334C2×D4—2441no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—2442no—
354C22×C4—2442no—
355C22×C4—2442no—
356C22×C4#1175819811yescharacteristic
357C22×C4#1175819811yescharacteristic
358C22×C4#1175819811yescharacteristic
359C22×C4#1175819811yescharacteristic
360C22×C4#1175819912yes—
361C22×C4#1175820112yes—
362C22×C4C2×C4×D6111yescharacteristic
363C22×C4C2×D24411yescharacteristic
364C22×C4D4×D6111yescharacteristic
365C22×C4#1175822411yescharacteristic
366C22×C4Q8×D6111yescharacteristic
367C22×C4#1175822611yescharacteristic
368C2×C8—2442no—
369C2×C8—2442no—
370C2×C8—2442no—
371C4⋊C4—48no—
372C4⋊C4—612no—
373C4⋊C4—1221no—
374C4⋊C4—2441no—
375C4⋊C4—2441no—
376C4⋊C4—2442no—
377C22⋊C4—1222no—
378C22⋊C4—1222no—
379C22⋊C4—2442no—
380C22⋊C4—2442no—
381C22⋊C4—2441no—
382C22⋊C4—2441no—
383C22⋊C4—2442no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—2442no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—1222no—
394C23—2441no—
395C23—2441no—
396C23—2441no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23C2×#1175819811yescharacteristic
401C23#1451974811yescharacteristic
402C23#1451975311yescharacteristic
403C23C2×#1175819911yescharacteristic
404C23C2×#1175820111yescharacteristic
405C23#1451980311yescharacteristic
406C23#1451980811yescharacteristic
407D4—2442no—
408D4—2442no—
409D4—2442no—
410D4—2442no—
411D4—2442no—
412D4—2442no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—1222no—
441C2×C4—4884no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4#1451969112yes—
451C2×C4#1451969712yes—
452C2×C4#1451969712yes—
453C2×C4#1451970512yes—
454C2×C4C4⋊C4×D6111yescharacteristic
455C2×C4#1451973511yescharacteristic
456C2×C4#1451975211yescharacteristic
457C2×C4#1451975711yescharacteristic
458C8—4882no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—4881no—
472C22—4881no—
473C22—4881no—
474C22—4881no—
475C22—2442no—
476C22—2442no—
477C22—2442no—
478C22#1683961911yescharacteristic
479C22C2×#1451969111yescharacteristic
480C22#1683973511yescharacteristic
481C22C2×#1451969711yescharacteristic
482C22#1683976111yescharacteristic
483C22C2×#1451970511yescharacteristic
484C22#1683978011yescharacteristic
485C22#1683995411yescharacteristic
486C22#1683996411yescharacteristic
487C22#1684003911yescharacteristic
488C22#1684004911yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—41no—
495C4—41no—
496C4—84no—
497C4—4881no—
498C4—4881no—
499C4—2442no—
500C4#1683949514yescomplemented
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—4882no—
506C2C2×#1683949511yescharacteristic
507C2#2144434211yescharacteristic
508C2#2144436011yescharacteristic
509C2#2144437811yescharacteristic
510C2#2144580411yescharacteristic
511C2#2144581611yescharacteristic
512C2#2144584011yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×C4⋊#95C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenterC4⋊#95D61
hypocenterC61#8571
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. #21445799C2semidirect
4. #21443698C2semidirect
5. C61×C4⋊#95C2semidirect
7. C2×C4×D244C4semidirect
11. #16839664C4semidirect
12. #16839660C4semidirect
16. C4×C4⋊Dic61C4semidirect
107. C2×D244C4⋊C4semidirect
126. C4⋊Dic61C4⋊C4semidirect
241. C4×C61#327semidirect
248. C4⋊#95D61semidirect
280. C61#8571semidirect
500. C4#16839495semidirect

Maximal subgroups

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

subgroupindex
C4×#145197512
#214457992
C2×#168394952
#214436982
C61×C4⋊#952
#857161