Finite Group Databasebeta

← #29582392

Subgroups — #29582392

Order 15616.

590 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 103 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#21455279C211yesmaximal, characteristic
2#21450993C211yesmaximal, characteristic, complemented
3#21444504C211yesmaximal, characteristic, complemented
4#21444314C211yesmaximal, characteristic, complemented
5#21444209C211yesmaximal, characteristic, complemented
6#21443721C211yesmaximal, characteristic, complemented
7C61×#3227C211yesmaximal, characteristic, complemented
8C2×#14519802—21no—
9C2×C4×D244C2211yescharacteristic
10#16839974—22no—
11#16839964—21no—
12#16839954—21no—
13#16839815C2211yescharacteristic
14#16839786C2211yescharacteristic
15#16839763—21no—
16#16839673—22no—
17#16839663—21no—
18#16839627—21no—
19#16839619—22no—
20C4×C4⋊Dic61C2211yescharacteristic
21#16839508—22no—
22#16839496—22no—
23#16839489—22no—
24C61×#532C2211yescharacteristic
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#498—21no—
27C61×#486—21no—
28C61×#481—22no—
29C61×#476—22no—
30C61×#431C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C4×D61—21no—
33#14519802—42no—
34#14519802—42no—
35C2×#11758201—22no—
36C2×#11758199—22no—
37C2×#11758194—21no—
38C4×D244—21no—
39C4×D244—21no—
40C4×D244—21no—
41C4×D244—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61D411yescharacteristic
44C2×#11758198—21no—
45C2×#11758198—24no—
46C2×C4×Dic61—21no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49C4⋊#92418D412yes—
50#14519701—22no—
51#14519698—22no—
52C22⋊#92418—21no—
53C4×#92418—21no—
54#14519684—24no—
55#14519684D412yes—
56C2×#41×C61—21no—
57C22×Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61D411yescharacteristic
63#127×C61D411yescharacteristic
64#124×C61—21no—
65#121×C61—21no—
66#121×C61—21no—
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61D412yes—
71C2×C42×C61C2311yescharacteristic
72#106×C61—24no—
73C4⋊C8×C61D412yes—
74#102×C61—24no—
75C22⋊C8×C61—21no—
76C4×C8×C61—21no—
77C23×D61—21no—
78#41×C61—41no—
79#41×C61—41no—
80#41×C61—41no—
81#41×C61—41no—
82C2×Q8×C61—21no—
83C2×Q8×C61—41no—
84C2×Q8×C61—41no—
85C2×Q8×C61—82no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—41no—
90C22×C4×C61—22no—
91C22×C4×C61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×Dic61—21no—
96C22×Dic61—22no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244SD1612yes—
103C2×C4×D61—44no—
104C2×C4×D61—41no—
105C2×C4×D61—41no—
106C2×C8×C61—22no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119#11758201—44no—
120#11758199—44no—
121#11758198—41no—
122#11758198—41no—
123#11758198—84no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—42no—
127C4×Dic61—41no—
128C4×Dic61—41no—
129#11758194—41no—
130#11758194—41no—
131C2×#92418—22no—
132C2×#92418—41no—
133C23×C61—41no—
134C23×C61#12011yescharacteristic
135C22×D61—22no—
136C22×D61—42no—
137C22×D61—42no—
138C22×D61—41no—
139C22×D61—41no—
140Q8×C61—41no—
141Q8×C61—41no—
142Q8×C61—82no—
143Q8×C61—82no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—42no—
147D4×C61—81no—
148D4×C61—81no—
149D4×C61—81no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—44no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—81no—
156C2×C4×C61—81no—
157C2×C4×C61#12011yescharacteristic
158C2×C4×C61#12011yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12711yescharacteristic
163C2×C4×C61C2×SD1611yescharacteristic
164C2×C4×C61#13511yescharacteristic
165C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C61#13711yescharacteristic
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—41no—
171C2×Dic61—42no—
172D244—22no—
173D244—22no—
174D244—41no—
175D244—41no—
176C4×D61—82no—
177C4×D61—82no—
178C8×C61—42no—
179#92418—42no—
180#92418—81no—
181#10241—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61#39211yescharacteristic
188C22×C61#44811yescharacteristic
189C22×C61#44911yescharacteristic
190C2×D61—44no—
191C2×D61—42no—
192C2×D61—82no—
193C2×D61—81no—
194C2×D61—81no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—81no—
199C4×C61—82no—
200C4×C61#45011yescharacteristic
201C4×C61#45211yescharacteristic
202C4×C61#45412yes—
203C4×C61#46511yescharacteristic
204C4×C61#46811yescharacteristic
205Dic61—82no—
206Dic61—81no—
207#3227D6111yescharacteristic, complemented
208#3036—611no—
209#2563—611no—
210#1549—611no—
211#1502—611no—
212#1478—611no—
213#1303—611no—
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—161no—
217C2×C61#200011yescharacteristic
218C2×C61#200311yescharacteristic
219C2×C61#200811yescharacteristic
220D61—82no—
221C2×AGL(1,ℤ/8)—1221no—
222#532C2×D6111yescharacteristic
223C2×C4⋊Q8C2×D6111yescharacteristic
224C2×C4×D4—611no—
225#498—21no—
226#486—21no—
227#481—22no—
228#476—22no—
229#463—1222no—
230#461—1221no—
231#459—1221no—
232#431—611no—
233#431C2×D6111yescharacteristic
234#423—611no—
235#417—1221no—
236#393—1222no—
237#390—1221no—
238#382—1221no—
239#380—1222no—
240C4×C4⋊C4—611no—
241#332—1222no—
242#327—1222no—
243D4⋊C8—1222no—
244C61#1024111yescharacteristic, complemented, Sylow
245C2×#41—21no—
246C22×Q8—21no—
247C22×D4—611no—
248C23×C4—1221no—
249AGL(1,ℤ/8)—2442no—
250AGL(1,ℤ/8)—2442no—
251C2×SD16—1222no—
252C2×D8—1222no—
253C2×M16—1221no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C22⋊D6111yescharacteristic
259#127C22⋊D6111yescharacteristic
260#124—21no—
261#121—21no—
262#121—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—1221no—
266C4×D4—1221no—
267C4×D4—1221no—
268C4×D4—1221no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—611no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4C22⋊D6112yes—
273C2×C22⋊C4—1224no—
274C2×C22⋊C4—1221no—
275C2×C42—1221no—
276C2×C42C22×D6111yescharacteristic
277#106—24no—
278C4⋊C8—612no—
279C4⋊C8—1221no—
280C4⋊C8—1221no—
281C4⋊C8C22⋊D6112yes—
282#103—1222no—
283#102—24no—
284#102—1222no—
285C22⋊C8—21no—
286C22⋊C8—1221no—
287C4×C8—21no—
288C4×C8—1221no—
289#95—612no—
290#95—1224no—
291C24—1221no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—82no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4—612no—
305C2×D4—1221no—
306C2×D4—1221no—
307C2×D4—1221no—
308C2×D4—1221no—
309C2×D4—1221no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—1222no—
313C22×C4—1221no—
314C22×C4—2444no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4D4×D6111yescharacteristic
318C22×C4D4×D6111yescharacteristic
319C22×C4C2×C22⋊D6111yescharacteristic
320SD16—2444no—
321D8—2444no—
322M16—2441no—
323M16—2441no—
324C2×C8—22no—
325C2×C8—1222no—
326C2×C8—2441no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—2442no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—4884no—
339C22⋊C4—2441no—
340C22⋊C4—2441no—
341C42—2441no—
342C42—2441no—
343C42D4×D6111yescharacteristic
344C42#1175822411yescharacteristic
345C42C2×C22⋊D6111yescharacteristic
346C42C2×C22⋊D6111yescharacteristic
347C23—41no—
348C23—1222no—
349C23—2442no—
350C23—2442no—
351C23—2441no—
352C23—2441no—
353C23#1451974511yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—82no—
357Q8—82no—
358D4—41no—
359D4—41no—
360D4—42no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—1222no—
365D4—1222no—
366D4—2441no—
367D4—2441no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—4882no—
376C2×C4—4882no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2441no—
382C2×C4#1451974511yescharacteristic
383C2×C4#1451974611yescharacteristic
384C2×C4#1451974811yescharacteristic
385C2×C4#1451974911yescharacteristic
386C2×C4#1451974911yescharacteristic
387C2×C4#1451977311yescharacteristic
388C2×C4SD16×D6111yescharacteristic
389C2×C4#1451979811yescharacteristic
390C2×C4#1451980411yescharacteristic
391C2×C4#1451980611yescharacteristic
392C8—42no—
393C8—4881no—
394C8—2442no—
395C22—21no—
396C22—21no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—4882no—
401C22—4881no—
402C22—4881no—
403C22—2444no—
404C22—2442no—
405C22#1683967011yescharacteristic
406C22#1683989911yescharacteristic
407C22#1683990611yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—81no—
412C4—82no—
413C4—4882no—
414C4—4881no—
415C4#1683991111yescharacteristic
416C4#1683992111yescharacteristic
417C4#1683993512yes—
418C4#1683998811yescharacteristic
419C4#1684000511yescharacteristic
420C2—21no—
421C2—21no—
422C2—161no—
423C2—4882no—
424C2#2144681911yescharacteristic
425C2#2144683911yescharacteristic
426C2#2144687011yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3227C2
socleC22×C61#392
prime radical#21455279C2
hypercenter#3227D61
hypocenterC61#10241
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
2. #21450993C2semidirect
3. #21444504C2semidirect
4. #21444314C2semidirect
5. #21444209C2semidirect
6. #21443721C2semidirect
7. C61×#3227C2semidirect
207. #3227D61semidirect
244. C61#10241semidirect

Maximal subgroups

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

subgroupindex
#214552792
#214509932
#214445042
#214443142
#214442092
#214437212
C61×#32272
#1024161