Finite Group Databasebeta

← #29665191

Subgroups — #29665191

Order 15616.

792 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456739C211yesmaximal, characteristic, complemented
2#21455283C211yesmaximal, characteristic, complemented
3C2×#16839957C211yesmaximal, characteristic, complemented
4#21445256C214yesmaximal, complemented
5#21445174C214yesmaximal, complemented
6#21444594C211yesmaximal, characteristic, complemented
7#21444486C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic, complemented
9C61×#2972C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11C22×#11758199—21no—
12#16840312C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14#16840139C2212yescomplemented
15#16840109C2212yescomplemented
16#16839977—22no—
17#16839967C2212yescomplemented
18#16839958—22no—
19#16839957—22no—
20#16839957C2214yes—
21#16839957C2212yescomplemented
22#16839898—22no—
23#16839786C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25#16839761—21no—
26C2×#14519697—21no—
27#16839515C2214yescomplemented
28#16839507C2212yescomplemented
29#16839505C2212yescomplemented
30#16839505C2214yescomplemented
31C2×C4×D4×C61C2211yescharacteristic
32C8×D4×C61—22no—
33C8×D4×C61C2214yescomplemented
34C61×#431C2211yescharacteristic
35C61×#431C2211yescharacteristic
36C2×C22⋊C8×C61—22no—
37C2×#11758222—21no—
38C22×D244—21no—
39C22×D244—21no—
40C22×D244—21no—
41#14519803—44no—
42C2×#11758200—22no—
43C2×D488—24no—
44C2×#11758199—24no—
45C2×#11758199—22no—
46C2×#11758199—44no—
47C2×#11758194—21no—
48C22×#92418—21no—
49#14519777C2311yescharacteristic
50#14519774—21no—
51#14519774—21no—
52#14519774—21no—
53#14519774C2312yes—
54#14519774C2311yescharacteristic
55#14519765—21no—
56#14519751—21no—
57#14519751—21no—
58C4×D244—21no—
59C4×D244—21no—
60C4⋊#92418—21no—
61C4⋊#92418—21no—
62C4⋊#92418C2312yes—
63C4⋊#92418C2312yes—
64#14519700—22no—
65#14519699—24no—
66#14519697—24no—
67#14519697—22no—
68C22×D4×C61D411yescharacteristic
69C23×C4×C61—21no—
70C22×C8×C61—24no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61D412yes—
74C4×D4×C61C2312yes—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61D412yes—
79C4⋊C8×C61C2312yes—
80C22⋊C8×C61—21no—
81C22⋊C8×C61—24no—
82C22⋊C8×C61D412yes—
83C4×C8×C61—21no—
84C4×C8×C61C2312yes—
85C24×C61—21no—
86C23×D61—41no—
87C2×D4×C61—21no—
88C2×D4×C61—22no—
89C2×D4×C61D814yescomplemented
90C2×D4×C61C2×D412yes—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—82no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C2×C22⋊D61—41no—
99#11758222—42no—
100#11758222—42no—
101C2×D244—22no—
102C2×D244—22no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244—22no—
106C2×D244—21no—
107C2×D244—42no—
108C2×D244—42no—
109C2×D244—41no—
110C2×D244—41no—
111C2×D244—41no—
112C2×D244—82no—
113C2×D244—82no—
114C2×C4×D61—41no—
115C2×Dic122—21no—
116C2×C8×C61—22no—
117C2×C8×C61—84no—
118C2×C8×C61C2×D414yes—
119C4⋊C4×C61—21no—
120C4⋊C4×C61C2×D412yes—
121C22⋊C4×C61—22no—
122C22⋊C4×C61—22no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126C42×C61C2411yescharacteristic
127#11758200—44no—
128#11758199—48no—
129#11758199—44no—
130#11758198—41no—
131#11758198—41no—
132C4⋊Dic61—21no—
133C4⋊Dic61—21no—
134#11758194—42no—
135C2×#92418—22no—
136C2×#92418—22no—
137C2×#92418—42no—
138D488—84no—
139C23×C61—22no—
140C23×C61—24no—
141C23×C61—81no—
142C23×C61—81no—
143C23×C61#12111yescharacteristic
144C22×D61—41no—
145C22×D61—42no—
146C22×D61—82no—
147C22×D61—82no—
148C22×D61—82no—
149D4×C61—22no—
150D4×C61—24no—
151D4×C61C2×D814yes—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—44no—
156C2×C4×C61—82no—
157C2×C4×C61—81no—
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61C2×D812yes—
162C2×C4×C61#13512yes—
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C22⋊D61—82no—
167C2×Dic61—41no—
168D244—42no—
169D244—41no—
170D244—42no—
171D244—41no—
172D244—42no—
173D244—44no—
174D244—81no—
175D244—82no—
176D244—82no—
177D244—82no—
178D244—82no—
179C4×D61—82no—
180Dic122—41no—
181Dic122—41no—
182C8×C61—24no—
183C8×C61—42no—
184#92418—42no—
185#92418—42no—
186#19386—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—24no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—84no—
193C22×C61#46612yes—
194C22×C61C2×#12111yescharacteristic
195C2×D61—82no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—82no—
199C2×D61—82no—
200C2×D61—81no—
201C2×D61—164no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—42no—
206C4×C61#46612yes—
207C4×C61C22×D811yescharacteristic
208C4×C61C2×#13511yescharacteristic
209C4×C61#57612yes—
210Dic61—81no—
211#3205—611no—
212#3036—611no—
213C2×#459—611no—
214#2972D6111yescharacteristic, complemented
215#1681—614no—
216#1667—614no—
217#1571—611no—
218#1545—611no—
219#1540—611no—
220C2×C61—22no—
221C2×C61—44no—
222C2×C61#304111yescharacteristic
223C2×C61C2×#46611yescharacteristic
224C2×C61#306811yescharacteristic
225D61—161no—
226D61—162no—
227C2×AGL(1,ℤ/8)—1221no—
228C22×D8—1221no—
229#532—611no—
230C2×#127—611no—
231C2×C4×D4C2×D6111yescharacteristic
232#493—612no—
233#486—612no—
234#463—1222no—
235#461—612no—
236#459—614no—
237#459—612no—
238#459—1222no—
239#459—1222no—
240#447—1222no—
241C8×D4—22no—
242C8×D4C2×D6114yescomplemented
243#431C2×D6111yescharacteristic
244#431C2×D6111yescharacteristic
245#423—611no—
246C2×C4⋊C8—611no—
247#417—1221no—
248C2×#102—1221no—
249C2×C22⋊C8—22no—
250#335—614no—
251#332—612no—
252#331—612no—
253#331—614no—
254C61#1938611yescharacteristic, complemented, Sylow
255C2×#41—1221no—
256C22×D4—1221no—
257C22×D4—1221no—
258C22×D4—1221no—
259C22×D4D24411yescharacteristic
260C23×C4—21no—
261AGL(1,ℤ/8)—2444no—
262C2×SD16—1222no—
263C2×D8—1224no—
264C2×D8—1222no—
265C2×D8—1224no—
266C2×D8—2444no—
267C2×M16—1221no—
268C22×C8—24no—
269C22×C8—1221no—
270C4⋊Q8—611no—
271#127—612no—
272#127—611no—
273#127—1221no—
274#127—1221no—
275#127—1221no—
276#124—1221no—
277#121—1221no—
278#121—1221no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—1221no—
282C4×D4—1221no—
283C4×D4D24412yes—
284C4×D4C22×D6112yes—
285C2×C4⋊C4D24411yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42C22×D6111yescharacteristic
288C4⋊C8—612no—
289C4⋊C8—612no—
290C4⋊C8—1221no—
291C4⋊C8—1221no—
292C4⋊C8C22⋊D6112yes—
293C4⋊C8C22×D6112yes—
294#103—1222no—
295#102—1224no—
296#102—1224no—
297#102—1222no—
298C22⋊C8—21no—
299C22⋊C8—24no—
300C22⋊C8C22⋊D6112yes—
301C4×C8—21no—
302C4×C8C22×D6112yes—
303C24—21no—
304C24—2441no—
305#41—2442no—
306#41—2442no—
307C2×Q8—1221no—
308C2×D4—21no—
309C2×D4—22no—
310C2×D4—1222no—
311C2×D4—1221no—
312C2×D4—1221no—
313C2×D4—1222no—
314C2×D4—1221no—
315C2×D4—1222no—
316C2×D4—4882no—
317C2×D4—4882no—
318C2×D4—2441no—
319C2×D4—2441no—
320C2×D4—2441no—
321C2×D4—2441no—
322C2×D4—2442no—
323C2×D4—2442no—
324C2×D4D48814yescomplemented
325C2×D4C2×D24412yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—82no—
330C22×C4—2441no—
331C22×C4C2×D24411yescharacteristic
332C22×C4#1175822211yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334SD16—2444no—
335D8—4884no—
336D8—2448no—
337D8—2444no—
338M16—2442no—
339C2×C8—22no—
340C2×C8—84no—
341C2×C8—1222no—
342C2×C8—1222no—
343C2×C8—2442no—
344C2×C8C2×C22⋊D6114yes—
345C4⋊C4—21no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4C2×D24412yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—2441no—
352C22⋊C4—2441no—
353C42C2×D24411yescharacteristic
354C42#1175822211yescharacteristic
355C42C2×C22⋊D6111yescharacteristic
356C42C23×D6111yescharacteristic
357C23—22no—
358C23—24no—
359C23—81no—
360C23—81no—
361C23—4882no—
362C23—4882no—
363C23—4882no—
364C23—2441no—
365C23—2442no—
366C23#1451975111yescharacteristic
367Q8—2441no—
368Q8—2441no—
369D4—24no—
370D4—22no—
371D4—4881no—
372D4—4882no—
373D4—4882no—
374D4—4882no—
375D4—4882no—
376D4—4882no—
377D4—2442no—
378D4—2441no—
379D4—2442no—
380D4—2441no—
381D4—2442no—
382D4—2444no—
383D4C2×D48814yes—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—4882no—
391C2×C4—2441no—
392C2×C4#1451975111yescharacteristic
393C2×C4#1451975111yescharacteristic
394C2×C4#1451975111yescharacteristic
395C2×C4C2×D48812yes—
396C2×C4#1451979912yes—
397C2×C4C22×C22⋊D6111yescharacteristic
398C2×C4C22×D24411yescharacteristic
399C2×C4C2×#1175822211yescharacteristic
400C8—24no—
401C8—42no—
402C8—2442no—
403C8—2442no—
404C22—21no—
405C22—21no—
406C22—24no—
407C22—42no—
408C22—42no—
409C22—84no—
410C22—9764no—
411C22—4881no—
412C22—4882no—
413C22—4882no—
414C22—4881no—
415C22—4882no—
416C22—4881no—
417C22#1683999812yes—
418C22C2×#1451975111yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—4881no—
424C4#1683999812yes—
425C4C22×D48811yescharacteristic
426C4C2×#1451979911yescharacteristic
427C4#1684056512yes—
428C2—22no—
429C2—44no—
430C2—9762no—
431C2—9761no—
432C2#2145532211yescharacteristic
433C2C2×#1683999811yescharacteristic
434C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2972C2
socleC22×C61C2×#121
hypercenter#2972D61
hypocenterC61#19386
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #21456739C2semidirect
2. #21455283C2semidirect
3. C2×#16839957C2semidirect
4. #21445256C2semidirect
5. #21445174C2semidirect
6. #21444594C2semidirect
7. #21444486C2semidirect
8. #21444466C2semidirect
9. C61×#2972C2semidirect
14. #16840139C22semidirect
15. #16840109C22semidirect
17. #16839967C22semidirect
21. #16839957C22semidirect
27. #16839515C22semidirect
28. #16839507C22semidirect
29. #16839505C22semidirect
30. #16839505C22semidirect
33. C8×D4×C61C22semidirect
89. C2×D4×C61D8semidirect
214. #2972D61semidirect
242. C8×D4C2×D61semidirect
254. C61#19386semidirect
324. C2×D4D488semidirect

Maximal subgroups

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

subgroupindex
#214567392
#214552832
C2×#168399572
#214452562
#214451742
#214445942
#214444862
#214444662
C61×#29722
#1938661