Finite Group Databasebeta

← #29580813

Subgroups — #29580813

Order 15616.

572 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 85 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#21456806C211yesmaximal, characteristic
2#21455941C211yesmaximal, characteristic, complemented
3#21446737C211yesmaximal, characteristic, complemented
4#21444417C211yesmaximal, characteristic, complemented
5#21444266C211yesmaximal, characteristic, complemented
6#21443675C211yesmaximal, characteristic, complemented
7C61×#2592C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840152—21no—
10#16840121—21no—
11#16840075—21no—
12#16840057—21no—
13#16839974—22no—
14#16839962—21no—
15#16839901—21no—
16#16839901—21no—
17#16839896—21no—
18#16839860—22no—
19#16839819C2211yescharacteristic
20#16839819C2211yescharacteristic
21C2×#14519705—22no—
22C4×#11758194C2211yescharacteristic
23#16839502—22no—
24Q8⋊#92418—22no—
25C2×#124×C61C2211yescharacteristic
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#397—21no—
28C61×#393—21no—
29C61×#391—21no—
30C61×#390—21no—
31C61×#388—22no—
32C4×C4⋊C4×C61C2211yescharacteristic
33C2×#11758222—21no—
34C2×#11758201—22no—
35C2×#11758201—22no—
36C2×#11758199—22no—
37C2×#11758194—21no—
38#14519777D411yescharacteristic
39#14519774D411yescharacteristic
40#14519765—21no—
41#14519751—21no—
42#14519751—21no—
43C4×D244—21no—
44C4×D244—21no—
45#14519710—22no—
46C4⋊#92418—21no—
47C4⋊#92418D412yes—
48#14519705—44no—
49#14519701—22no—
50#14519701—22no—
51#14519698—22no—
52#14519697—22no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55#14519688—21no—
56#14519688—21no—
57#14519688D412yes—
58C4×#92418—21no—
59C22×Q8×C61D411yescharacteristic
60C22×Q8×C61D411yescharacteristic
61C22×D4×C61D411yescharacteristic
62#124×C61—21no—
63#124×C61—21no—
64#124×C61—21no—
65#124×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—21no—
76C2×C22⋊C4×C61—22no—
77C2×C42×C61—21no—
78C2×C42×C61D412yes—
79C2×C42×C61C2311yescharacteristic
80#95×C61—22no—
81#95×C61—22no—
82#95×C61—21no—
83C24×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×Q8×C61—42no—
89C2×Q8×C61—42no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—42no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C2×C22⋊D61—41no—
101#11758222—42no—
102#11758222—42no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244—21no—
106C2×C4×D61—41no—
107C2×Dic122—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—42no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C22⋊C4×C61—44no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—82no—
119C42×C61—42no—
120C42×C61—41no—
121C42×C61—41no—
122C42×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126#11758201—44no—
127#11758201—82no—
128#11758199—82no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133#11758194—22no—
134#11758194—41no—
135C2×#92418—22no—
136C2×#92418—22no—
137C2×#92418—41no—
138C23×C61—42no—
139C23×C61—42no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61#12011yescharacteristic
144C22×D61—41no—
145Q8×C61—22no—
146Q8×C61—22no—
147Q8×C61—41no—
148Q8×C61—41no—
149Q8×C61—42no—
150Q8×C61—81no—
151D4×C61—42no—
152D4×C61—81no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61#12011yescharacteristic
169C2×C4×C61#12011yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12711yescharacteristic
174C22⋊D61—81no—
175C22⋊D61—81no—
176C2×Dic61—41no—
177D244—42no—
178D244—42no—
179D244—42no—
180C4×D61—82no—
181Dic122—42no—
182#92418—42no—
183#92418—42no—
184#92418—42no—
185#10006—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—82no—
189C22×C61—82no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61#39211yescharacteristic
194C22×C61#45311yescharacteristic
195C22×C61#45411yescharacteristic
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—42no—
205C4×C61—82no—
206C4×C61—81no—
207C4×C61—81no—
208C4×C61#45512yes—
209Dic61—81no—
210#3213—611no—
211#3112—611no—
212#2592D6111yescharacteristic, complemented
213#1980—611no—
214#1527—611no—
215#1491—611no—
216#1285—611no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—161no—
220C2×C61#200411yescharacteristic
221C2×C61#200611yescharacteristic
222C2×C61#201011yescharacteristic
223D61—161no—
224#532—611no—
225C2×#124C2×D6111yescharacteristic
226C2×C4×Q8C2×D6111yescharacteristic
227#496—1221no—
228#489—1221no—
229#480—1221no—
230#477—1221no—
231#463—1222no—
232#460—1221no—
233#448—1221no—
234#448—1221no—
235#447—1221no—
236#440—1222no—
237#432—611no—
238#432—611no—
239C2×#104—1222no—
240C4×M16—611no—
241#397—21no—
242#393—21no—
243#391—21no—
244#390—21no—
245#388—22no—
246C4×C4⋊C4C2×D6111yescharacteristic
247#329—1222no—
248Q8⋊C8—1222no—
249C61#1000611yescharacteristic, complemented, Sylow
250C2×#41—1221no—
251C22×Q8C22⋊D6111yescharacteristic
252C22×Q8C22⋊D6111yescharacteristic
253C22×D4C22⋊D6111yescharacteristic
254C2×SD16—1222no—
255C2×SD16—1222no—
256C2×D8—1222no—
257C2×M16—1221no—
258C4⋊Q8—611no—
259#127—611no—
260#124—21no—
261#124—21no—
262#124—21no—
263#124—21no—
264#124—1221no—
265#121—1221no—
266#121—1221no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×D4—1221no—
272C4×D4—1221no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4C22⋊D6111yescharacteristic
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C42—21no—
281C2×C42C22⋊D6112yes—
282C2×C42C22×D6111yescharacteristic
283#107—1222no—
284C4⋊C8—612no—
285C4⋊C8—1221no—
286#104—2444no—
287#103—1222no—
288#103—1222no—
289#102—1222no—
290#102—1222no—
291C22⋊C8—1221no—
292C22⋊C8—1221no—
293#97—612no—
294#97—1221no—
295#97—1221no—
296C4×C8—1221no—
297#95—22no—
298#95—22no—
299#95—21no—
300C24—21no—
301#41—2442no—
302#41—2442no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—1221no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—1221no—
314C2×D4—1221no—
315C2×D4—1221no—
316C2×D4—2441no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—2441no—
322C22×C4C2×C22⋊D6111yescharacteristic
323C22×C4C2×C22⋊D6111yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325SD16—4882no—
326SD16—2444no—
327D8—4882no—
328M16—1222no—
329M16—2441no—
330C2×C8—1222no—
331C2×C8—1222no—
332C2×C8—2441no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C22⋊C4—44no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—82no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C42—42no—
349C42—41no—
350C42—41no—
351C42D4×D6111yescharacteristic
352C42D4×D6111yescharacteristic
353C42D4×D6111yescharacteristic
354C42#1175822411yescharacteristic
355C23—42no—
356C23—42no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—2441no—
361C23#1451974611yescharacteristic
362Q8—22no—
363Q8—22no—
364Q8—41no—
365Q8—41no—
366Q8—42no—
367Q8—81no—
368Q8—2442no—
369D4—42no—
370D4—81no—
371D4—4881no—
372D4—4881no—
373D4—2442no—
374D4—2442no—
375D4—2442no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—4882no—
392C2×C4—2441no—
393C2×C4#1451974511yescharacteristic
394C2×C4#1451974511yescharacteristic
395C2×C4#1451974811yescharacteristic
396C2×C4#1451974911yescharacteristic
397C2×C4#1451974911yescharacteristic
398C2×C4#1451977311yescharacteristic
399C8—2442no—
400C8—2442no—
401C8—2442no—
402C22—21no—
403C22—21no—
404C22—82no—
405C22—82no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22#1683967011yescharacteristic
413C22#1683993011yescharacteristic
414C22#1683993511yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—42no—
421C4—82no—
422C4—81no—
423C4—81no—
424C4—4881no—
425C4#1683993912yes—
426C2—21no—
427C2—21no—
428C2—161no—
429C2—9761no—
430C2#2144684811yescharacteristic
431C2#2144686111yescharacteristic
432C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2592C2
socleC22×C61#392
prime radical#21456806C2
hypercenter#2592D61
hypocenterC61#10006
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2592

Complements

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

NH ≅ G/Ntype
2. #21455941C2semidirect
3. #21446737C2semidirect
4. #21444417C2semidirect
5. #21444266C2semidirect
6. #21443675C2semidirect
7. C61×#2592C2semidirect
212. #2592D61semidirect
249. C61#10006semidirect

Maximal subgroups

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

subgroupindex
#214568062
#214559412
#214467372
#214444172
#214442662
#214436752
C61×#25922
#1000661