Finite Group Databasebeta

← #29574781

Subgroups — #29574781

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839962C211yesmaximal, characteristic
2#21446746C211yesmaximal, characteristic
3#21446624C211yesmaximal, characteristic
4C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
5#21443702C211yesmaximal, characteristic, complemented
6#21443660C211yesmaximal, characteristic, complemented
7C61×#2355C211yesmaximal, characteristic, complemented
8C22×#11758201—21no—
9C2×#14519774C2211yescharacteristic
10#16839962—22no—
11#16839962C414yescomplemented
12C2×C4⋊#92418C2211yescharacteristic
13C2×C4⋊#92418C2211yescharacteristic
14C2×#14519697—21no—
15C2×#14519697—21no—
16C2×#14519697—21no—
17C2×#14519697—22no—
18C2×C4×#92418C2211yescharacteristic
19Q8⋊#92418—22no—
20C2×C4×Q8×C61C2211yescharacteristic
21C61×#391—21no—
22C61×#387—22no—
23C61×#384C2211yescharacteristic
24C61×#382—21no—
25C4×C4⋊C4×C61—21no—
26C4×C4⋊C4×C61—21no—
27C4×C4⋊C4×C61C2211yescharacteristic
28C22×D244—21no—
29C22×D244—21no—
30C22×D244—21no—
31C2×#11758201—24no—
32C2×#11758201—42no—
33C2×#11758201—42no—
34C22×#92418—21no—
35C22×#92418—21no—
36C22×#92418—21no—
37#14519774—21no—
38#14519774—21no—
39#14519774—21no—
40#14519774C2×C412yes—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43C4⋊#92418—21no—
44C4⋊#92418C2×C412yes—
45#14519697—24no—
46#14519697—44no—
47#14519697—42no—
48#14519697—42no—
49C4×#92418—21no—
50C4×#92418—21no—
51C22×Q8×C61D411yescharacteristic
52C4×Q8×C61—21no—
53C4×Q8×C61—21no—
54C4×Q8×C61C2×C412yes—
55C4×Q8×C61D412yes—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61D411yescharacteristic
60C2×C4⋊C4×C61D411yescharacteristic
61C2×C4⋊C4×C61D411yescharacteristic
62C2×C4⋊C4×C61D412yes—
63C2×C42×C61—21no—
64C2×C42×C61—21no—
65C2×C42×C61—22no—
66C2×C42×C61C2311yescharacteristic
67#95×C61—22no—
68#95×C61—22no—
69#95×C61—22no—
70#95×C61—21no—
71C23×D61—41no—
72C2×Q8×C61—21no—
73C2×Q8×C61—21no—
74C2×Q8×C61—21no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61C22⋊C412yes—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—22no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C2×D244—22no—
87C2×D244—22no—
88C2×D244—22no—
89C2×D244—42no—
90C2×D244—42no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C2×D244—41no—
98C2×D244—41no—
99C2×D244—41no—
100C2×D244—41no—
101C2×D244—41no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—42no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61C22⋊C412yes—
114C4⋊C4×C61SD1612yescomplemented
115C42×C61—22no—
116C42×C61—22no—
117C42×C61—41no—
118C42×C61—41no—
119C42×C61—82no—
120C42×C61C22×C411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C42×C61#4111yescharacteristic
124#11758201—48no—
125C2×#92418—22no—
126C2×#92418—22no—
127C2×#92418—41no—
128C2×#92418—41no—
129C2×#92418—41no—
130C2×#92418—41no—
131C2×#92418—41no—
132C2×#92418—41no—
133C2×#92418—41no—
134C23×C61#12011yescharacteristic
135C22×D61—42no—
136C22×D61—81no—
137C22×D61—81no—
138C22×D61—81no—
139C22×D61—81no—
140C22×D61—81no—
141C22×D61—81no—
142Q8×C61—22no—
143Q8×C61—22no—
144Q8×C61—41no—
145Q8×C61—41no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×C4×C61C4×D411yescharacteristic
165C2×C4×C61C4×D411yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12311yescharacteristic
169C2×C4×C61C2×SD1611yescharacteristic
170C2×C4×C61#13511yescharacteristic
171C2×C4×C61#13511yescharacteristic
172C2×C4×C61#13511yescharacteristic
173C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C61#13711yescharacteristic
177D244—44no—
178D244—42no—
179D244—42no—
180D244—42no—
181D244—42no—
182D244—82no—
183D244—81no—
184D244—81no—
185D244—81no—
186D244—81no—
187#92418—42no—
188#92418—42no—
189#92418—81no—
190#8978—611nomaximal, Sylow
191C22×C61#38611yescharacteristic
192C22×C61#44811yescharacteristic
193C22×C61#44911yescharacteristic
194C22×C61#44911yescharacteristic
195C22×C61#44911yescharacteristic
196C22×C61#45311yescharacteristic
197C22×C61#45411yescharacteristic
198C2×D61—82no—
199C2×D61—82no—
200C2×D61—82no—
201C2×D61—161no—
202C2×D61—161no—
203C2×D61—161no—
204C2×D61—161no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—42no—
208C4×C61—42no—
209C4×C61—42no—
210C4×C61—82no—
211C4×C61—81no—
212C4×C61#41611yescharacteristic
213C4×C61#41811yescharacteristic
214C4×C61C4×SD1611yescharacteristic
215C4×C61#44011yescharacteristic
216C4×C61#46011yescharacteristic
217C4×C61#46411yescharacteristic
218C4×C61#48211yescharacteristic
219C4×C61#48211yescharacteristic
220C2×#460—611no—
221#2355D6111yescharacteristic, complemented
222#1982—611no—
223#1957—611no—
224C2×Q8⋊C8—611no—
225#1296—611no—
226#1279—611no—
227C2×C61#162011yescharacteristic
228C2×C61#162211yescharacteristic
229C2×C61#162611yescharacteristic
230C2×C61#162811yescharacteristic
231C2×C61#187511yescharacteristic
232C2×C61#188011yescharacteristic
233C2×C61#188311yescharacteristic
234D61—162no—
235C22×SD16—1221no—
236C2×#127—611no—
237C2×C4×Q8C2×D6111yescharacteristic
238#460—614no—
239#460—1222no—
240C2×C4⋊C8—611no—
241C2×C4⋊C8—611no—
242C2×#102—1221no—
243C2×#102—1222no—
244C2×#102—1221no—
245C2×#102—1221no—
246C2×C4×C8—611no—
247#391—21no—
248#387—22no—
249#384C2×D6111yescharacteristic
250#382—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4C2×D6111yescharacteristic
254Q8⋊C8—1222no—
255C61#897811yescharacteristic, complemented, Sylow
256C22×Q8C22⋊D6111yescharacteristic
257C22×D4—1221no—
258C22×D4—1221no—
259C22×D4—1221no—
260C2×SD16—1224no—
261C2×SD16—2442no—
262C2×SD16—2442no—
263C22×C8—1221no—
264C22×C8—1221no—
265C22×C8—1221no—
266#127—612no—
267#127—1221no—
268#127—1221no—
269#127—1221no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8C4×D6112yes—
273C4×Q8D24412yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4C22⋊D6111yescharacteristic
278C2×C4⋊C4C22⋊D6111yescharacteristic
279C2×C4⋊C4C22⋊D6111yescharacteristic
280C2×C4⋊C4C22⋊D6112yes—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42C22×D6111yescharacteristic
285C4⋊C8—612no—
286C4⋊C8—1221no—
287C4⋊C8—1221no—
288C4⋊C8—1221no—
289#102—1224no—
290#102—2442no—
291#102—2444no—
292#102—2442no—
293C4×C8—1221no—
294C4×C8—1221no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—21no—
299C24—2441no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8#1175819812yes—
306C2×D4—1222no—
307C2×D4—1222no—
308C2×D4—1222no—
309C2×D4—2442no—
310C2×D4—2442no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4—2441no—
315C2×D4—2441no—
316C2×D4—2441no—
317C2×D4—2441no—
318C2×D4—2441no—
319C2×D4—2441no—
320C2×D4—2441no—
321C2×D4—2441no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4C2×C22⋊D6111yescharacteristic
328C22×C4C2×C22⋊D6111yescharacteristic
329C22×C4C2×C22⋊D6111yescharacteristic
330SD16—2448no—
331C2×C8—1222no—
332C2×C8—1222no—
333C2×C8—2441no—
334C2×C8—2441no—
335C2×C8—2441no—
336C2×C8—2441no—
337C2×C8—2441no—
338C2×C8—2441no—
339C2×C8—2441no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4#1175819812yes—
352C4⋊C4#1175820112yescomplemented
353C42—22no—
354C42—22no—
355C42—41no—
356C42—41no—
357C42—82no—
358C42C2×C4×D6111yescharacteristic
359C42C2×D24411yescharacteristic
360C42#1175822211yescharacteristic
361C42#1175822211yescharacteristic
362C23—4881no—
363C23—4881no—
364C23—4881no—
365C23—4881no—
366C23—4881no—
367C23—4881no—
368C23—2442no—
369C23#1451974611yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374D4—4882no—
375D4—4881no—
376D4—4881no—
377D4—4881no—
378D4—4881no—
379D4—2444no—
380D4—2442no—
381D4—2442no—
382D4—2442no—
383D4—2442no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4C2×#1175819811yescharacteristic
402C2×C4C4×C22⋊D6111yescharacteristic
403C2×C4C4×C22⋊D6111yescharacteristic
404C2×C4#1451975111yescharacteristic
405C2×C4#1451975111yescharacteristic
406C2×C4#1451976011yescharacteristic
407C2×C4C2×#1175820111yescharacteristic
408C2×C4#1451979711yescharacteristic
409C2×C4#1451979711yescharacteristic
410C2×C4#1451979711yescharacteristic
411C2×C4#1451980211yescharacteristic
412C2×C4#1451980211yescharacteristic
413C2×C4#1451980211yescharacteristic
414C2×C4#1451980811yescharacteristic
415C8—4881no—
416C8—2442no—
417C8—2442no—
418C22—9761no—
419C22—9761no—
420C22—9761no—
421C22—9761no—
422C22—4882no—
423C22—4882no—
424C22—4882no—
425C22#1683964111yescharacteristic
426C22#1683990111yescharacteristic
427C22#1683990711yescharacteristic
428C22#1683990711yescharacteristic
429C22#1683990711yescharacteristic
430C22#1683993011yescharacteristic
431C22#1683993511yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—81no—
439C4#1683975611yescharacteristic
440C4#1683976611yescharacteristic
441C4C4×#1175820111yescharacteristic
442C4#1683986011yescharacteristic
443C4#1683996211yescharacteristic
444C4#1683998411yescharacteristic
445C4#1684008611yescharacteristic
446C4#1684008611yescharacteristic
447C2—9762no—
448C2#2144487811yescharacteristic
449C2#2144489111yescharacteristic
450C2#2144491511yescharacteristic
451C2#2144492911yescharacteristic
452C2#2144614111yescharacteristic
453C2#2144617411yescharacteristic
454C2#2144619511yescharacteristic
subgroupG/H
centerC23#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2355C2
socleC23×C61#120
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#2355D61
hypocenterC61#8978
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2355

Complements

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

NH ≅ G/Ntype
4. C2×Q8⋊#92418C2semidirect
5. #21443702C2semidirect
6. #21443660C2semidirect
7. C61×#2355C2semidirect
11. #16839962C4semidirect
114. C4⋊C4×C61SD16semidirect
221. #2355D61semidirect
255. C61#8978semidirect
352. C4⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168399622
#214467462
#214466242
C2×Q8⋊#924182
#214437022
#214436602
C61×#23552
#897861