Finite Group Databasebeta

← #29574879

Subgroups — #29574879

Order 15616.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455850C211yesmaximal, characteristic
2#21446738C211yesmaximal, characteristic
3C4×#14519701C211yesmaximal, characteristic, complemented
4#21444269C211yesmaximal, characteristic
5#21443720C211yesmaximal, characteristic, complemented
6#21443657C211yesmaximal, characteristic, complemented
7Q8×M16×C61C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840080—21no—
10#16840047C411yescharacteristic, complemented
11#16840047C411yescharacteristic, complemented
12#16839962C411yescharacteristic, complemented
13#16839962C411yescharacteristic, complemented
14#16839901—21no—
15C4×#11758201—21no—
16C4×#11758201—21no—
17#16839815C2211yescharacteristic
18C2×#14519703—21no—
19C2×#14519703—21no—
20C2×#14519701—21no—
21C2×C4×#92418C2211yescharacteristic
22C4×C4⋊Dic61C2211yescharacteristic
23#16839524—21no—
24#16839489—22no—
25C2×C4×Q8×C61C2211yescharacteristic
26C61×#446—21no—
27C61×#446—22no—
28C8×Q8×C61—21no—
29C61×#423—21no—
30C61×#423C2211yescharacteristic
31C4×M16×C61—21no—
32C4×M16×C61C2211yescharacteristic
33C2×#11758222—21no—
34C2×#11758201—21no—
35C2×#11758201—21no—
36C22×#92418—21no—
37#14519777C2×C411yescharacteristic
38#14519774C2×C411yescharacteristic
39#14519765—21no—
40#14519751—21no—
41#14519751—21no—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4⋊Dic61—21no—
45C2×C4×Dic61—21no—
46#14519708—21no—
47#14519708—21no—
48C4⋊#92418C2×C411yescharacteristic
49C4⋊#92418C2×C411yescharacteristic
50#14519703—42no—
51#14519703—42no—
52#14519701—24no—
53#14519701—21no—
54#14519701—21no—
55#14519697—21no—
56#14519697—21no—
57C22⋊#92418—21no—
58C4×#92418—21no—
59C4×#92418—21no—
60C4×#92418—21no—
61#14519684—21no—
62C22×Q8×C61D411yescharacteristic
63C2×M16×C61—21no—
64C2×M16×C61D411yescharacteristic
65C2×M16×C61D411yescharacteristic
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×Q8×C61C2×C411yescharacteristic
69C4×Q8×C61C2×C411yescharacteristic
70C4×Q8×C61D411yescharacteristic
71C4×Q8×C61D411yescharacteristic
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C42×C61—21no—
75C2×C42×C61C2311yescharacteristic
76C4⋊C8×C61—22no—
77C4⋊C8×C61—22no—
78C4⋊C8×C61—21no—
79C4⋊C8×C61D412yes—
80#97×C61—22no—
81#97×C61—21no—
82C4×C8×C61—22no—
83C4×C8×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—22no—
86C2×Q8×C61—41no—
87C2×Q8×C61C22⋊C411yescharacteristic
88C2×Q8×C61C22⋊C411yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C2×C22⋊D61—41no—
95C22×Dic61—41no—
96#11758222—41no—
97#11758222—41no—
98#11758222—41no—
99#11758222—41no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—21no—
103C2×C4×D61—41no—
104C2×Dic122—21no—
105M16×C61—21no—
106M16×C61—21no—
107M16×C61—21no—
108M16×C61—41no—
109M16×C61—41no—
110M16×C61SD1612yescomplemented
111C2×C8×C61—21no—
112C2×C8×C61—21no—
113C2×C8×C61—22no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61C22⋊C411yescharacteristic
120C4⋊C4×C61C22⋊C411yescharacteristic
121C42×C61—21no—
122C42×C61—21no—
123C42×C61—21no—
124C42×C61—21no—
125C42×C61C22×C411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61C2×D411yescharacteristic
128C42×C61#4111yescharacteristic
129#11758201—42no—
130#11758201—42no—
131#11758198—41no—
132#11758198—41no—
133C4⋊Dic61—22no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—41no—
137C4×Dic61—41no—
138C4×Dic61—41no—
139C2×#92418—22no—
140C2×#92418—21no—
141C2×#92418—21no—
142C2×#92418—41no—
143C2×#92418—41no—
144C23×C61#12111yescharacteristic
145C22×D61—41no—
146Q8×C61—21no—
147Q8×C61—21no—
148Q8×C61—21no—
149Q8×C61—42no—
150Q8×C61#10412yescomplemented
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—81no—
160C2×C4×C61C2×C22⋊C411yescharacteristic
161C2×C4×C61C4×D411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61#12011yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12311yescharacteristic
166C2×C4×C61C2×SD1611yescharacteristic
167C2×C4×C61#13511yescharacteristic
168C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C61#13711yescharacteristic
170C22⋊D61—81no—
171C22⋊D61—81no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—81no—
175C2×Dic61—81no—
176D244—42no—
177D244—41no—
178D244—41no—
179D244—41no—
180D244—41no—
181C4×D61—81no—
182C4×D61—81no—
183Dic122—41no—
184Dic122—41no—
185C8×C61—22no—
186C8×C61—41no—
187C8×C61—42no—
188#92418—42no—
189#92418—41no—
190#92418—41no—
191#8992—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61#38611yescharacteristic
195C22×C61#46011yescharacteristic
196C22×C61#46411yescharacteristic
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C4×C61—21no—
201C4×C61—22no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—41no—
205C4×C61—41no—
206C4×C61—41no—
207C4×C61C2×#10411yescharacteristic
208C4×C61#42111yescharacteristic
209C4×C61C4×SD1611yescharacteristic
210C4×C61#44011yescharacteristic
211C4×C61#44811yescharacteristic
212C4×C61#44911yescharacteristic
213Dic61—82no—
214Dic61—81no—
215#3103—611no—
216Q8×M16D6111yescharacteristic, complemented
217#1980—611no—
218C4×#103—611no—
219#1492—611no—
220#1303—611no—
221#1278—611no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61#187111yescharacteristic
225C2×C61#188011yescharacteristic
226C2×C61#188111yescharacteristic
227D61—161no—
228#532—611no—
229C2×C4×Q8C2×D6111yescharacteristic
230#481—1221no—
231#475—611no—
232#475—611no—
233#460—611no—
234#460—611no—
235#448—1221no—
236#446—21no—
237#446—22no—
238C8×Q8—21no—
239C4×SD16—1221no—
240C4×SD16—1221no—
241#431—611no—
242#423—21no—
243#423C2×D6111yescharacteristic
244C2×#104—1221no—
245C2×#104—1221no—
246C2×#103—1221no—
247C4×M16—21no—
248C4×M16C2×D6111yescharacteristic
249C2×C4×C8—611no—
250C4×C4⋊C4—611no—
251#340—1221no—
252D4⋊C8—1222no—
253C61#899211yescharacteristic, complemented, Sylow
254C2×#41—1221no—
255C22×Q8D24411yescharacteristic
256C2×SD16—1221no—
257C2×SD16—1221no—
258C2×M16—21no—
259C2×M16C22⋊D6111yescharacteristic
260C2×M16C22⋊D6111yescharacteristic
261C22×C8—1221no—
262C4⋊Q8—611no—
263#127—611no—
264#124—1221no—
265#121—1221no—
266#121—1221no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C4×D6111yescharacteristic
270C4×Q8C4×D6111yescharacteristic
271C4×Q8C22⋊D6111yescharacteristic
272C4×Q8C22⋊D6111yescharacteristic
273C4×D4—1221no—
274C4×D4—1221no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4D24411yescharacteristic
278C2×C42—21no—
279C2×C42—1221no—
280C2×C42C22×D6111yescharacteristic
281#106—1221no—
282#106—1221no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—21no—
286C4⋊C8—611no—
287C4⋊C8—611no—
288C4⋊C8C22⋊D6112yes—
289#104—2442no—
290#104—2442no—
291#103—1224no—
292#103—1221no—
293#103—1221no—
294#102—1221no—
295#102—1221no—
296C22⋊C8—1221no—
297#97—22no—
298#97—21no—
299C4×C8—22no—
300C4×C8—21no—
301C4×C8—1221no—
302C4×C8—1221no—
303C4×C8—1221no—
304#95—1221no—
305#41—2441no—
306#41—2441no—
307#41—2441no—
308#41—2441no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—1221no—
313C2×Q8#1175819811yescharacteristic
314C2×Q8#1175819811yescharacteristic
315C2×D4—1221no—
316C2×D4—1221no—
317C2×D4—1221no—
318C2×D4—2441no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—2441no—
322C22×C4—2441no—
323C22×C4C2×D24411yescharacteristic
324C22×C4#1175822211yescharacteristic
325C22×C4C2×C22⋊D6111yescharacteristic
326SD16—2442no—
327SD16—2442no—
328M16—21no—
329M16—21no—
330M16—21no—
331M16—41no—
332M16—41no—
333M16#1175820112yescomplemented
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—1222no—
338C2×C8—1221no—
339C2×C8—1221no—
340C2×C8—2441no—
341C2×C8—2441no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—1222no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—2441no—
351C4⋊C4#1175819811yescharacteristic
352C4⋊C4#1175819811yescharacteristic
353C22⋊C4—2441no—
354C22⋊C4—2441no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—2441no—
360C42—2441no—
361C42C2×C4×D6111yescharacteristic
362C42#1175822211yescharacteristic
363C42C2×C22⋊D6111yescharacteristic
364C42C2×C22⋊D6111yescharacteristic
365C23—2441no—
366C23#1451975111yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—2441no—
372Q8—2441no—
373Q8#1451970312yescomplemented
374D4—4881no—
375D4—4881no—
376D4—2442no—
377D4—2441no—
378D4—2441no—
379D4—2441no—
380D4—2441no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—4881no—
391C2×C4—4881no—
392C2×C4—4881no—
393C2×C4—4881no—
394C2×C4—2442no—
395C2×C4—2441no—
396C2×C4C2×#1175819811yescharacteristic
397C2×C4C4×C22⋊D6111yescharacteristic
398C2×C4C4×C22⋊D6111yescharacteristic
399C2×C4#1451974611yescharacteristic
400C2×C4#1451975111yescharacteristic
401C2×C4#1451976011yescharacteristic
402C2×C4C2×#1175820111yescharacteristic
403C2×C4#1451979711yescharacteristic
404C2×C4#1451980211yescharacteristic
405C2×C4#1451980811yescharacteristic
406C8—22no—
407C8—41no—
408C8—42no—
409C8—2442no—
410C8—2441no—
411C8—2441no—
412C22—21no—
413C22—21no—
414C22—4881no—
415C22—4881no—
416C22—4881no—
417C22#1683964111yescharacteristic
418C22#1683996211yescharacteristic
419C22#1683998411yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—4882no—
428C4—4881no—
429C4C2×#1451970311yescharacteristic
430C4#1683977811yescharacteristic
431C4C4×#1175820111yescharacteristic
432C4#1683986011yescharacteristic
433C4#1683990111yescharacteristic
434C4#1683990711yescharacteristic
435C2—21no—
436C2—21no—
437C2—9761no—
438C2#2144612111yescharacteristic
439C2#2144617411yescharacteristic
440C2#2144618111yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingQ8×M16×C61C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenterQ8×M16D61
hypocenterC61#8992
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. C4×#14519701C2semidirect
5. #21443720C2semidirect
6. #21443657C2semidirect
7. Q8×M16×C61C2semidirect
10. #16840047C4semidirect
11. #16840047C4semidirect
12. #16839962C4semidirect
13. #16839962C4semidirect
110. M16×C61SD16semidirect
150. Q8×C61#104semidirect
216. Q8×M16D61semidirect
253. C61#8992semidirect
333. M16#11758201semidirect
373. Q8#14519703semidirect

Maximal subgroups

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

subgroupindex
#214558502
#214467382
C4×#145197012
#214442692
#214437202
#214436572
Q8×M16×C612
#899261