Finite Group Databasebeta

← #29582386

Subgroups — #29582386

Order 15616.

658 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456741C211yesmaximal, characteristic, complemented
2#21455279C211yesmaximal, characteristic
3#21450768C211yesmaximal, characteristic, complemented
4#21444499C211yesmaximal, characteristic, complemented
5#21444211C211yesmaximal, characteristic, complemented
6#21443722C211yesmaximal, characteristic, complemented
7C61×#1502C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×#14519777C2211yescharacteristic
10C2×#14519757—22no—
11C2×C4×D244C2211yescharacteristic
12#16840134—21no—
13#16840114—21no—
14#16839974—22no—
15#16839967—22no—
16#16839964—21no—
17#16839954—21no—
18#16839900—22no—
19#16839815C2211yescharacteristic
20#16839786C2211yescharacteristic
21#16839763—21no—
22#16839699—22no—
23#16839664—21no—
24#16839632—21no—
25#16839510—22no—
26#16839491—22no—
27C61×#532C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C4×C4⋊C4×C61C2211yescharacteristic
30C61×#327—22no—
31C22×Dic122D412yes—
32C22×D244D411yescharacteristic
33C22×C4×D61—21no—
34#14519802—42no—
35#14519802—42no—
36C2×#11758200—24no—
37C2×#11758201—22no—
38C2×#11758199—22no—
39C2×#11758194—21no—
40#14519777—21no—
41#14519777—21no—
42#14519777—21no—
43#14519777—21no—
44#14519757—48no—
45C4×D244—21no—
46C4×D244—21no—
47C4×D244—21no—
48C4×D244—21no—
49C2×C4⋊Dic61—24no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61D411yescharacteristic
53C2×#11758198—24no—
54C2×#11758198—21no—
55C4⋊#92418—21no—
56C4⋊#92418D412yes—
57C4⋊#92418D412yes—
58#14519701—22no—
59#14519700—24no—
60#14519698—22no—
61C22⋊#92418—21no—
62C4×#92418—21no—
63C2×#41×C61—21no—
64C4⋊Q8×C61D411yescharacteristic
65#127×C61D411yescharacteristic
66#124×C61—21no—
67#121×C61—21no—
68#121×C61—21no—
69C4×D4×C61—21no—
70C4×D4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C42×C61—21no—
73C2×C42×C61C2311yescharacteristic
74C4⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76C4×C8×C61—21no—
77#95×C61D412yes—
78C23×D61—21no—
79#41×C61—41no—
80#41×C61—41no—
81#41×C61—41no—
82#41×C61—41no—
83C2×Q8×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61—21no—
87C2×D4×C61—41no—
88C22×C4×C61—22no—
89C22×C4×C61—41no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×Dic61—21no—
94C22×Dic61—22no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244SD1612yescomplemented
101C2×C4×D61—44no—
102C2×C4×D61—41no—
103C2×C4×D61—41no—
104C2×Dic122—22no—
105C2×Dic122—22no—
106C2×Dic122—44no—
107C2×C8×C61—41no—
108C4⋊C4×C61—24no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C42×C61—41no—
114C42×C61—41no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119#11758201—44no—
120#11758200—84no—
121#11758199—44no—
122#11758198—48no—
123#11758198—41no—
124#11758198—41no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127C4⋊Dic61—48no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130C4⋊Dic61—41no—
131C4⋊Dic61—41no—
132#11758194—41no—
133#11758194—41no—
134C2×#92418—22no—
135C2×#92418—22no—
136C23×C61—41no—
137C23×C61#12011yescharacteristic
138C22×D61—22no—
139C22×D61—42no—
140C22×D61—42no—
141C22×D61—41no—
142C22×D61—41no—
143Q8×C61—41no—
144Q8×C61—41no—
145D4×C61—41no—
146D4×C61—41no—
147D4×C61—42no—
148D4×C61—81no—
149D4×C61—81no—
150D4×C61—81no—
151C2×C4×C61—22no—
152C2×C4×C61—24no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12711yescharacteristic
165C2×C4×C61C2×SD1611yescharacteristic
166C2×C4×C61#13511yescharacteristic
167C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C61#13711yescharacteristic
169C2×Dic61—42no—
170C2×Dic61—44no—
171C2×Dic61—41no—
172C2×Dic61—42no—
173D244—22no—
174D244—22no—
175D244—41no—
176D244—41no—
177C4×D61—82no—
178C4×D61—82no—
179Dic122—44no—
180Dic122—82no—
181C8×C61—81no—
182#92418—42no—
183#92418—42no—
184#10240—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61#39211yescharacteristic
191C22×C61#44811yescharacteristic
192C22×C61#44911yescharacteristic
193C2×D61—44no—
194C2×D61—42no—
195C2×D61—82no—
196C2×D61—81no—
197C2×D61—81no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—44no—
202C4×C61—81no—
203C4×C61#45011yescharacteristic
204C4×C61#45211yescharacteristic
205C4×C61#45312yes—
206C4×C61#46511yescharacteristic
207C4×C61#46811yescharacteristic
208Dic61—82no—
209Dic61—81no—
210#3206—611no—
211#3036—611no—
212#2542—611no—
213#1548—611no—
214#1502D6111yescharacteristic, complemented
215#1478—611no—
216#1303—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—161no—
220C2×C61#200011yescharacteristic
221C2×C61#200211yescharacteristic
222C2×C61#200911yescharacteristic
223D61—82no—
224C2×AGL(1,ℤ/8)—1221no—
225#532C2×D6111yescharacteristic
226C2×C4⋊Q8—611no—
227C2×C22⋊Q8—1222no—
228C2×C4×D4—611no—
229#492—1221no—
230#487—1221no—
231#463—1222no—
232#461—1222no—
233#461—1221no—
234#459—1221no—
235#448—1222no—
236#431—611no—
237#431C2×D6111yescharacteristic
238#423—611no—
239#417—1221no—
240#399—1222no—
241#390—1221no—
242#384—1221no—
243C4×C4⋊C4C2×D6111yescharacteristic
244#333—1222no—
245#327—22no—
246D4⋊C8—1222no—
247C61#1024011yescharacteristic, complemented, Sylow
248C2×#41—21no—
249C22×Q8—612no—
250C22×D4—611no—
251C23×C4—1221no—
252AGL(1,ℤ/8)—2442no—
253AGL(1,ℤ/8)—2442no—
254C2×SD16—1224no—
255C2×SD16—1222no—
256C2×D8—1222no—
257C2×M16—1221no—
258C4⋊Q8—1221no—
259C4⋊Q8—1221no—
260C4⋊Q8—1221no—
261C4⋊Q8—1221no—
262C4⋊Q8D24411yescharacteristic
263#127D24411yescharacteristic
264#124—21no—
265C22⋊Q8—2448no—
266#121—21no—
267#121—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—1221no—
271C4×D4—1221no—
272C4×D4—1221no—
273C4×D4—1221no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—1224no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4—1221no—
279C2×C22⋊C4—1224no—
280C2×C22⋊C4—1221no—
281C2×C42—21no—
282C2×C42C22×D6111yescharacteristic
283C4⋊C8—21no—
284C4⋊C8—612no—
285C4⋊C8—612no—
286C4⋊C8—1221no—
287#103—1224no—
288#103—1222no—
289#102—1222no—
290C22⋊C8—21no—
291C22⋊C8—1221no—
292C4×C8—21no—
293C4×C8—1221no—
294#95C22⋊D6112yes—
295C24—1221no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300C2×Q8—21no—
301C2×Q8—1222no—
302C2×Q8—1222no—
303C2×Q8—2444no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C2×D4—612no—
309C2×D4—1221no—
310C2×D4—1221no—
311C2×D4—1221no—
312C2×D4—1221no—
313C2×D4—1221no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—1222no—
317C22×C4—1221no—
318C22×C4—2444no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4D4×D6111yescharacteristic
322C22×C4D4×D6111yescharacteristic
323C22×C4C2×C22⋊D6111yescharacteristic
324SD16—4884no—
325SD16—2444no—
326D8—2444no—
327M16—2441no—
328M16—2441no—
329C2×C8—41no—
330C2×C8—1222no—
331C2×C8—1222no—
332C4⋊C4—24no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—2448no—
338C4⋊C4—2441no—
339C4⋊C4—2441no—
340C4⋊C4—2441no—
341C4⋊C4—2441no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—2448no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C42—41no—
348C42—41no—
349C42C2×D24411yescharacteristic
350C42#1175822211yescharacteristic
351C42D4×D6111yescharacteristic
352C42D4×D6111yescharacteristic
353C23—41no—
354C23—1222no—
355C23—2442no—
356C23—2442no—
357C23—2441no—
358C23—2441no—
359C23#1451974511yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—4882no—
363Q8—2444no—
364D4—41no—
365D4—41no—
366D4—42no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—1222no—
371D4—1222no—
372D4—2441no—
373D4—2441no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—4882no—
383C2×C4—4882no—
384C2×C4—2444no—
385C2×C4—2442no—
386C2×C4—2441no—
387C2×C4—2442no—
388C2×C4#1451974711yescharacteristic
389C2×C4#1451974711yescharacteristic
390C2×C4#1451975011yescharacteristic
391C2×C4#1451975011yescharacteristic
392C2×C4#1451975111yescharacteristic
393C2×C4#1451977311yescharacteristic
394C2×C4SD16×D6111yescharacteristic
395C2×C4#1451979611yescharacteristic
396C2×C4#1451980111yescharacteristic
397C2×C4#1451980611yescharacteristic
398C8—81no—
399C8—2442no—
400C8—2442no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—4882no—
407C22—4881no—
408C22—4881no—
409C22—2444no—
410C22—2442no—
411C22#1683967111yescharacteristic
412C22#1683989911yescharacteristic
413C22#1683990411yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—44no—
418C4—81no—
419C4—4881no—
420C4—4882no—
421C4#1683991411yescharacteristic
422C4#1683992511yescharacteristic
423C4#1683993112yes—
424C4#1683999011yescharacteristic
425C4#1684000711yescharacteristic
426C2—21no—
427C2—21no—
428C2—161no—
429C2—4882no—
430C2#2144682311yescharacteristic
431C2#2144683311yescharacteristic
432C2#2144687811yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1502C2
socleC22×C61#392
prime radical#21455279C2
hypercenter#1502D61
hypocenterC61#10240
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
1. #21456741C2semidirect
3. #21450768C2semidirect
4. #21444499C2semidirect
5. #21444211C2semidirect
6. #21443722C2semidirect
7. C61×#1502C2semidirect
100. C2×D244SD16semidirect
214. #1502D61semidirect
247. C61#10240semidirect

Maximal subgroups

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

subgroupindex
#214567412
#214552792
#214507682
#214444992
#214442112
#214437222
C61×#15022
#1024061