Finite Group Databasebeta

← #29575907

Subgroups — #29575907

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455300C211yesmaximal, characteristic
2#21454388C211yesmaximal, characteristic, complemented
3#21444488C211yesmaximal, characteristic, complemented
4C2×#16839505C211yesmaximal, characteristic, complemented
5#21444204C211yesmaximal, characteristic, complemented
6#21443714C211yesmaximal, characteristic, complemented
7C61×#1292C211yesmaximal, characteristic, complemented
8C2×#14519805—21no—
9C2×#14519785—21no—
10C2×#14519774C2211yescharacteristic
11C4×#11758222C2211yescharacteristic
12#16839963—21no—
13#16839963C412yes—
14#16839958—21no—
15#16839958C412yes—
16#16839815C2211yescharacteristic
17#16839790—21no—
18#16839787—22no—
19#16839784—21no—
20C2×C4⋊#92418C2211yescharacteristic
21C4⋊C8×D61—21no—
22#16839768—21no—
23#16839505—21no—
24#16839505—21no—
25#16839505—22no—
26#16839492—21no—
27#16839489—21no—
28C61×#423C2211yescharacteristic
29C61×#423C2211yescharacteristic
30C2×C4⋊C8×C61C2211yescharacteristic
31C2×#11758222D411yescharacteristic
32C22×D244—21no—
33C22×D244—21no—
34C22×D244—21no—
35#14519805—42no—
36#14519805—42no—
37C2×#11758190—22no—
38C2×D488—22no—
39#14519785—42no—
40#14519785—42no—
41C2×#11758189—22no—
42C22×#92418—21no—
43C2×C8×D61—22no—
44#14519774—21no—
45#14519774—21no—
46#14519774—21no—
47#14519774C2×C412yes—
48C4×Dic122C2×C411yescharacteristic
49C4×Dic122D411yescharacteristic
50C4×D244C2×C411yescharacteristic
51C4×D244D411yescharacteristic
52C4×C22⋊D61—21no—
53C4×C22⋊D61—21no—
54#14519736D411yescharacteristic
55#14519734—21no—
56C42×D61—21no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418D411yescharacteristic
60C4⋊#92418D411yescharacteristic
61#14519706—22no—
62#14519706—22no—
63#14519699—22no—
64#14519699—22no—
65C22⋊#92418—21no—
66C4×#92418—21no—
67C2×M16×C61—21no—
68C2×M16×C61—21no—
69C22×C8×C61—21no—
70C2×C42×C61C2311yescharacteristic
71C4⋊C8×C61—21no—
72C4⋊C8×C61—21no—
73C4⋊C8×C61—21no—
74C4⋊C8×C61—21no—
75C4⋊C8×C61C2×C412yes—
76C4⋊C8×C61D412yes—
77C23×D61—41no—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61#4111yescharacteristic
80C22×C4×C61#4111yescharacteristic
81C2×C22⋊D61—21no—
82#11758222—21no—
83#11758222—21no—
84#11758222D812yes—
85#11758222SD1612yes—
86C2×D244—22no—
87C2×D244—22no—
88C2×D244—22no—
89C2×D244—41no—
90C2×D244—41no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—82no—
97C2×D244—81no—
98C2×D244—81no—
99C2×D244C22⋊C411yescharacteristic
100C2×C4×D61—21no—
101C2×C4×D61—21no—
102C2×C4×D61—21no—
103C2×Dic122C22⋊C411yescharacteristic
104M16×C61—41no—
105M16×C61—41no—
106M16×C61—41no—
107M16×C61—41no—
108C2×C8×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61—41no—
111C2×C8×C61—81no—
112C42×C61C22×C411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61C2×D411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C22⋊Dic61—21no—
117#11758198—21no—
118#11758198—21no—
119C4⋊Dic61C22⋊C411yescharacteristic
120C4⋊Dic61C22⋊C411yescharacteristic
121#11758196—21no—
122#11758196—21no—
123C4×Dic61—21no—
124C4×Dic61—21no—
125C4×Dic61—21no—
126C2×#92418—21no—
127C2×#92418—21no—
128C2×#92418—41no—
129C2×#92418—41no—
130C2×#92418—41no—
131D488—44no—
132#11758190—44no—
133#11758189—44no—
134C8×D61—44no—
135C23×C61#12311yescharacteristic
136C22×D61—21no—
137C22×D61—42no—
138C22×D61—82no—
139C22×D61—82no—
140C22×D61—81no—
141C22×D61—81no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—41no—
145C2×C4×C61C2×C22⋊C411yescharacteristic
146C2×C4×C61C4×D411yescharacteristic
147C2×C4×C61C4×D411yescharacteristic
148C2×C4×C61#12011yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C2×SD1611yescharacteristic
153C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C61#13711yescharacteristic
155C22⋊D61—22no—
156C22⋊D61—22no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160D244—21no—
161D244—42no—
162D244—42no—
163D244—82no—
164D244—82no—
165D244—82no—
166D244—81no—
167D244—81no—
168D244—81no—
169D244—81no—
170D244#10212yes—
171C4×D61—22no—
172C4×D61—22no—
173C4×D61—41no—
174C4×D61—41no—
175C4×D61—81no—
176Dic122—21no—
177Dic122#10212yes—
178C8×C61—42no—
179C8×C61—42no—
180C8×C61—81no—
181#92418—41no—
182#92418—41no—
183#92418—81no—
184#9173—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61#38611yescharacteristic
188C22×C61#48011yescharacteristic
189C22×C61#48111yescharacteristic
190C2×D61—22no—
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—82no—
194C2×D61—82no—
195C2×D61—82no—
196C2×D61—162no—
197C2×D61—162no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61C2×#10211yescharacteristic
202C4×C61#41711yescharacteristic
203C4×C61#44711yescharacteristic
204C4×C61#44811yescharacteristic
205C4×C61#47111yescharacteristic
206C4×C61#47211yescharacteristic
207Dic61—22no—
208Dic61—41no—
209Dic61—41no—
210#3038—611no—
211#2906—611no—
212#1545—611no—
213C2×#331—611no—
214#1477—611no—
215#1301—611no—
216#1292D6111yescharacteristic, complemented
217C2×C61—21no—
218C2×C61—21no—
219C2×C61#187411yescharacteristic
220C2×C61#190311yescharacteristic
221C2×C61#190711yescharacteristic
222D61—42no—
223D61—162no—
224C2×AGL(1,ℤ/8)—1221no—
225C2×#131—1221no—
226C2×#127—611no—
227C4×#41—611no—
228#460—612no—
229#460—1221no—
230#459—612no—
231#459—1221no—
232#431—611no—
233#424—1221no—
234#424—1222no—
235#423—1221no—
236#423C2×D6111yescharacteristic
237#423C2×D6111yescharacteristic
238C2×C4⋊C8—611no—
239C2×C4⋊C8—1221no—
240C2×C4⋊C8C2×D6111yescharacteristic
241#418—1221no—
242#331—1221no—
243#331—1221no—
244#331—1222no—
245Q8⋊C8—1221no—
246D4⋊C8—1221no—
247C61#917311yescharacteristic, complemented, Sylow
248C2×#41—611no—
249C22×D4—1221no—
250C22×D4—1221no—
251C22×D4—1221no—
252AGL(1,ℤ/8)—2442no—
253AGL(1,ℤ/8)—2442no—
254C2×SD16—1222no—
255C2×D8—1222no—
256#131—2442no—
257#131—2442no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—1222no—
261C22×C8—21no—
262C22×C8—1222no—
263C22×C8—1221no—
264#127—612no—
265#127—1221no—
266#127—1221no—
267#127—1221no—
268C4×Q8—611no—
269C4×Q8—611no—
270C4×D4—611no—
271C4×D4—611no—
272C4×D4—1221no—
273C4×D4—1221no—
274#117—611no—
275#117—1221no—
276C2×C42—1221no—
277C2×C42C22×D6111yescharacteristic
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—611no—
283C4⋊C8—611no—
284C4⋊C8—1222no—
285C4⋊C8—1222no—
286C4⋊C8—1221no—
287C4⋊C8—1221no—
288C4⋊C8C4×D6112yes—
289C4⋊C8D24412yes—
290#102—1222no—
291#102—1222no—
292C22⋊C8—1221no—
293C4×C8—1221no—
294C24—2441no—
295#41—612no—
296#41—612no—
297#41—1221no—
298#41—1221no—
299C2×Q8—611no—
300C2×D4—611no—
301C2×D4—1221no—
302C2×D4—1222no—
303C2×D4—1222no—
304C2×D4—1222no—
305C2×D4—4881no—
306C2×D4—4881no—
307C2×D4—4882no—
308C2×D4—2441no—
309C2×D4—2441no—
310C2×D4—2441no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4—2441no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4#1175822211yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4#1175822611yescharacteristic
321SD16—2444no—
322D8—2444no—
323M16—41no—
324M16—41no—
325M16—41no—
326M16—41no—
327M16—2444no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—81no—
332C2×C8—1221no—
333C2×C8—1221no—
334C2×C8—2444no—
335C2×C8—2441no—
336C2×C8—2441no—
337C2×C8—2441no—
338C4⋊C4—611no—
339C4⋊C4—611no—
340C4⋊C4—1221no—
341C4⋊C4—1221no—
342C22⋊C4—1221no—
343C22⋊C4—1221no—
344C22⋊C4—1221no—
345C42—1221no—
346C42—1221no—
347C42—1221no—
348C42C2×C4×D6111yescharacteristic
349C42C2×D24411yescharacteristic
350C42D4×D6111yescharacteristic
351C42D4×D6111yescharacteristic
352C23—1221no—
353C23—4882no—
354C23—4882no—
355C23—4881no—
356C23—4881no—
357C23—2442no—
358C23#1451976211yescharacteristic
359Q8—612no—
360Q8—1221no—
361D4—612no—
362D4—1222no—
363D4—1222no—
364D4—1221no—
365D4—4882no—
366D4—4882no—
367D4—4882no—
368D4—4881no—
369D4—4881no—
370D4—4881no—
371D4—4881no—
372D4—2442no—
373D4—2442no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—1221no—
378C2×C4—1222no—
379C2×C4—1222no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—4881no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4C22⋊C4×D6111yescharacteristic
386C2×C4#1451973911yescharacteristic
387C2×C4C4×D24411yescharacteristic
388C2×C4#1451974711yescharacteristic
389C2×C4#1451975011yescharacteristic
390C2×C4#1451975211yescharacteristic
391C2×C4D8×D6111yescharacteristic
392C2×C4SD16×D6111yescharacteristic
393C2×C4#1451980411yescharacteristic
394C2×C4#1451980911yescharacteristic
395C8—42no—
396C8—42no—
397C8—81no—
398C8—4881no—
399C8—2441no—
400C8—2441no—
401C22—21no—
402C22—21no—
403C22—1222no—
404C22—9762no—
405C22—9762no—
406C22—4882no—
407C22—4882no—
408C22—4882no—
409C22—2441no—
410C22—2441no—
411C22#1683964011yescharacteristic
412C22#1684007711yescharacteristic
413C22#1684008211yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—1222no—
418C4—2441no—
419C4—2441no—
420C4#102×D6111yescharacteristic
421C4#1683976211yescharacteristic
422C4#1683989711yescharacteristic
423C4#1683990211yescharacteristic
424C4#1684002811yescharacteristic
425C4#1684003111yescharacteristic
426C2—21no—
427C2—21no—
428C2—9762no—
429C2—2442no—
430C2#2144613511yescharacteristic
431C2#2144630011yescharacteristic
432C2#2144632811yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1292C2
socleC22×C61#386
prime radical#21455300C2
cosocleC2×C42×C61C23
hypercenter#1292D61
hypocenterC61#9173
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
2. #21454388C2semidirect
3. #21444488C2semidirect
4. C2×#16839505C2semidirect
5. #21444204C2semidirect
6. #21443714C2semidirect
7. C61×#1292C2semidirect
216. #1292D61semidirect
247. C61#9173semidirect

Maximal subgroups

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

subgroupindex
#214553002
#214543882
#214444882
C2×#168395052
#214442042
#214437142
C61×#12922
#917361