Finite Group Databasebeta

← #29577679

Subgroups — #29577679

Order 15616.

490 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 91 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#21454847C211yesmaximal, characteristic
2#21451020C211yesmaximal, characteristic
3#21444446C211yesmaximal, characteristic, complemented
4#21444159C211yesmaximal, characteristic
5#21443676C211yesmaximal, characteristic, complemented
6#21443674C211yesmaximal, characteristic, complemented
7C61×#1534C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16839835—22no—
10#16839832—22no—
11#16839819C2211yescharacteristic
12#16839819C2211yescharacteristic
13#16839724—21no—
14C22⋊C8×D61—21no—
15#16839684—21no—
16#16839684—21no—
17#16839663C411yescharacteristic
18#16839663C411yescharacteristic
19#16839633—21no—
20#16839633—21no—
21#16839631C411yescharacteristic
22#16839627C411yescharacteristic
23C4×C4⋊Dic61C2211yescharacteristic
24#16839504—22no—
25#16839485—21no—
26#16839485—21no—
27C2×#125×C61C2211yescharacteristic
28C61×#432C2211yescharacteristic
29C61×#432C2211yescharacteristic
30C61×#330—22no—
31C22×D244C2×C411yescharacteristic
32C22×C4×D61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34C2×#11758189—22no—
35C2×C8×D61—22no—
36C4×D244—21no—
37C4×D244—21no—
38C4×D244—21no—
39C4×D244—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×#11758196—21no—
43C2×#11758196—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198D411yescharacteristic
49C2×#11758198D411yescharacteristic
50C2×C4×Dic61C2×C411yescharacteristic
51C2×C4×Dic61C2×C411yescharacteristic
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56#14519689—22no—
57#14519689—22no—
58#14519688—21no—
59#14519688D412yes—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#125×C61—21no—
64#125×C61—21no—
65#125×C61—21no—
66#125×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61C2×C411yescharacteristic
70C2×C4⋊C4×C61C2×C411yescharacteristic
71C2×C42×C61C2311yescharacteristic
72C4⋊C8×C61—21no—
73C4⋊C8×C61—21no—
74C22⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76#97×C61—21no—
77#97×C61D412yes—
78C23×D61C22⋊C411yescharacteristic
79C23×D61C22⋊C411yescharacteristic
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61C22⋊C411yescharacteristic
88C22×Dic61C22⋊C411yescharacteristic
89C2×D244—22no—
90C2×D244—21no—
91C2×D244—21no—
92C2×D244—41no—
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—81no—
98C2×C4×D61—81no—
99C2×C8×C61—22no—
100C2×C8×C61—41no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61#4111yescharacteristic
112C42×C61#4111yescharacteristic
113#11758198—21no—
114#11758198—21no—
115#11758198—21no—
116#11758198—21no—
117#11758198—42no—
118#11758198—42no—
119#11758198—81no—
120#11758198—81no—
121C4⋊Dic61—21no—
122C4⋊Dic61—21no—
123C4⋊Dic61—42no—
124#11758196—81no—
125#11758196—81no—
126C4×Dic61—41no—
127C4×Dic61—41no—
128C2×#92418—22no—
129C2×#92418—41no—
130#11758189—82no—
131C8×D61—82no—
132C23×C61C2×C22⋊C411yescharacteristic
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—21no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61#10112yes—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12311yescharacteristic
159C2×Dic61—22no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169D244—42no—
170D244—42no—
171C4×D61—42no—
172C4×D61—42no—
173C4×D61—81no—
174C4×D61—81no—
175C8×C61—42no—
176C8×C61—81no—
177#92418—42no—
178#92418—81no—
179#9449—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61#38611yescharacteristic
183C22×C61C2×#10111yescharacteristic
184C22×C61#41311yescharacteristic
185C2×D61—22no—
186C2×D61—41no—
187C2×D61—41no—
188C2×D61—42no—
189C2×D61—42no—
190C2×D61—41no—
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—81no—
194C2×D61—81no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—81no—
202C4×C61—81no—
203Dic61—42no—
204Dic61—41no—
205Dic61—41no—
206Dic61—81no—
207#2973—611no—
208#2566—611no—
209#1534—611no—
210#1534D6111yescharacteristic, complemented
211#1466—611no—
212#1285—611no—
213#1285—611no—
214C2×C61—21no—
215C2×C61—21no—
216C2×C61#188411yescharacteristic
217C2×C61#190111yescharacteristic
218C2×C61#191111yescharacteristic
219D61—42no—
220D61—81no—
221C2×#125C2×D6111yescharacteristic
222C2×C4×D4—611no—
223#435—1222no—
224#435—1222no—
225#432—611no—
226#432—611no—
227#432C2×D6111yescharacteristic
228#432C2×D6111yescharacteristic
229#407—1221no—
230C2×C22⋊C8—1221no—
231#396—1221no—
232#396—1221no—
233#390—611no—
234#390—611no—
235#385—1221no—
236#385—1221no—
237#384—611no—
238#382—611no—
239C4×C4⋊C4—611no—
240#330—22no—
241#330—1222no—
242#323—1221no—
243#323—1221no—
244C61#944911yescharacteristic, complemented, Sylow
245C22×D4—611no—
246C23×C4—611no—
247C23×C4—611no—
248C2×M16—1222no—
249C22×C8—1222no—
250#125—21no—
251#125—21no—
252#125—21no—
253#125—21no—
254C4×D4—1221no—
255C4×D4—1221no—
256C4×D4—1221no—
257C4×D4—1221no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1221no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4C4×D6111yescharacteristic
265C2×C4⋊C4C4×D6111yescharacteristic
266C2×C22⋊C4—611no—
267C2×C22⋊C4—611no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C42—611no—
273C2×C42—611no—
274C2×C42C22×D6111yescharacteristic
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—1221no—
278C4⋊C8—1221no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—1222no—
282C22⋊C8—1222no—
283C22⋊C8—1221no—
284C22⋊C8—1221no—
285#97—21no—
286#97—612no—
287#97—1221no—
288#97C22⋊D6112yes—
289#95—1221no—
290#95—1221no—
291#95—1221no—
292C24—611no—
293C24—611no—
294C2×D4—1222no—
295C2×D4—1221no—
296C2×D4—1221no—
297C2×D4—2441no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—611no—
301C22×C4—611no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—4881no—
309C22×C4—4881no—
310C22×C4C2×C4×D6111yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C22×C4D4×D6111yescharacteristic
313M16—4882no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—1222no—
317C2×C8—4882no—
318C2×C8—2441no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—1221no—
328C4⋊C4—1221no—
329C4⋊C4—4881no—
330C4⋊C4—4881no—
331C4⋊C4—2442no—
332C22⋊C4—1221no—
333C22⋊C4—1221no—
334C22⋊C4—1221no—
335C22⋊C4—1221no—
336C22⋊C4—4881no—
337C22⋊C4—4881no—
338C22⋊C4—2442no—
339C22⋊C4—2442no—
340C42—2441no—
341C42—2441no—
342C42#1175822211yescharacteristic
343C42D4×D6111yescharacteristic
344C42#1175822411yescharacteristic
345C42C2×C22⋊D6111yescharacteristic
346C23—612no—
347C23—1221no—
348C23—1221no—
349C23—1221no—
350C23—1221no—
351C23—1221no—
352C23—2441no—
353C23—2441no—
354C23—2441no—
355C23—2441no—
356C23C22⋊C4×D6111yescharacteristic
357D4—2442no—
358D4—2442no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—1222no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—4881no—
373C2×C4—4881no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4#1451973711yescharacteristic
384C2×C4C4×C22⋊D6111yescharacteristic
385C2×C4#1451974511yescharacteristic
386C2×C4#1451974811yescharacteristic
387C2×C4#1451975011yescharacteristic
388C2×C4#1451975911yescharacteristic
389C8—42no—
390C8—81no—
391C8—4881no—
392C8—2442no—
393C22—21no—
394C22—21no—
395C22—1222no—
396C22—4881no—
397C22—4881no—
398C22—2442no—
399C22—2442no—
400C22—2441no—
401C22—2441no—
402C22—2441no—
403C22—2441no—
404C22—2441no—
405C22#1683963811yescharacteristic
406C22#101×D6111yescharacteristic
407C22#1683974511yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—81no—
416C4—4881no—
417C4—2442no—
418C4—2441no—
419C4—2441no—
420C2—21no—
421C2—21no—
422C2—4881no—
423C2—2442no—
424C2#2144619911yescharacteristic
425C2#2144628411yescharacteristic
426C2#2144634811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1534C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1534D61
hypocenterC61#9449
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1534

Complements

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

NH ≅ G/Ntype
3. #21444446C2semidirect
5. #21443676C2semidirect
6. #21443674C2semidirect
7. C61×#1534C2semidirect
210. #1534D61semidirect
244. C61#9449semidirect

Maximal subgroups

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

subgroupindex
#214548472
#214510202
#214444462
#214441592
#214436762
#214436742
C61×#15342
#944961