Finite Group Databasebeta

← #29577656

Subgroups — #29577656

Order 15616.

474 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 89 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#21454454C211yesmaximal, characteristic
2#21450272C211yesmaximal, characteristic
3#21445894C211yesmaximal, characteristic
4#21444353C211yesmaximal, characteristic
5#21444351C211yesmaximal, characteristic
6#21443672C211yesmaximal, characteristic
7C61×#1284C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9C2×#14519736C2211yescharacteristic
10#16839820—22no—
11#16839820—22no—
12#16839785—21no—
13#16839726C2211yescharacteristic
14M16×Dic61—21no—
15#16839687—21no—
16#16839687—21no—
17#16839668C411yescharacteristic, complemented
18#16839668C411yescharacteristic, complemented
19#16839647—21no—
20#16839647—21no—
21#16839632C411yescharacteristic, complemented
22#16839628C411yescharacteristic, complemented
23#16839625C2211yescharacteristic
24#16839528—21no—
25#16839528—21no—
26#16839500—22no—
27#16839498—22no—
28C61×#407C2211yescharacteristic
29C61×#381C2211yescharacteristic
30C4×C22⋊C4×C61C2211yescharacteristic
31C22×Dic122C2×C411yescharacteristic
32C2×#11758194—21no—
33#14519756—21no—
34#14519756—21no—
35#14519756—21no—
36#14519756—21no—
37#14519736—21no—
38#14519736—21no—
39#14519736—21no—
40#14519736—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61D411yescharacteristic
50C2×C22⋊Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53#14519706—24no—
54C22⋊#92418—21no—
55C22⋊#92418D412yes—
56#14519687—22no—
57C8×Dic61—22no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65#14519684—21no—
66C23×C4×C61C2311yescharacteristic
67C2×M16×C61—21no—
68C2×C22⋊C4×C61—21no—
69C2×C22⋊C4×C61—21no—
70C2×C42×C61C2×C411yescharacteristic
71C2×C42×C61C2×C411yescharacteristic
72C22⋊C8×C61—21no—
73C22⋊C8×C61D412yes—
74#95×C61—21no—
75#95×C61—21no—
76#95×C61—21no—
77C24×C61C2×D411yescharacteristic
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—41no—
81C22×C4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61C22×C411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—41no—
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C22⋊C411yescharacteristic
96C2×Dic122—21no—
97C2×Dic122—21no—
98C2×Dic122—41no—
99C2×Dic122—41no—
100M16×C61—41no—
101M16×C61—41no—
102C2×C8×C61—22no—
103C22⋊C4×C61—42no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C42×C61—41no—
107C42×C61—41no—
108C22⋊Dic61—21no—
109C22⋊Dic61—21no—
110C22⋊Dic61—21no—
111C22⋊Dic61—21no—
112C22⋊Dic61—41no—
113C22⋊Dic61—41no—
114C22⋊Dic61—41no—
115C22⋊Dic61—41no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—42no—
123C4⋊Dic61—42no—
124#11758196—41no—
125#11758196—41no—
126C4×Dic61—21no—
127C4×Dic61—21no—
128C4×Dic61—21no—
129C4×Dic61—21no—
130#11758194—41no—
131#11758194—41no—
132C2×#92418—22no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61C2×C22⋊C411yescharacteristic
138C23×C61#12111yescharacteristic
139C23×C61#12111yescharacteristic
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—22no—
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×C61—41no—
154C2×C4×C61—81no—
155C2×C4×C61—81no—
156C2×C4×C61—81no—
157C2×C4×C61C4×D411yescharacteristic
158C2×C4×C61C4×D411yescharacteristic
159C2×C4×C61#12011yescharacteristic
160C2×C4×C61#12311yescharacteristic
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—81no—
176C2×Dic61—81no—
177C2×Dic61—81no—
178Dic122—81no—
179Dic122—81no—
180C8×C61—42no—
181#92418—42no—
182#9444—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61#38611yescharacteristic
194C22×C61#41011yescharacteristic
195C22×C61#41011yescharacteristic
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—42no—
201C4×C61—81no—
202C4×C61—161no—
203Dic61—41no—
204Dic61—41no—
205Dic61—41no—
206Dic61—41no—
207Dic61—81no—
208Dic61—81no—
209Dic61—161no—
210#2921—611no—
211#2499—611no—
212#1820—611no—
213#1511—611no—
214#1511—611no—
215#1284—611no—
216#1284D6111yescharacteristic
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—41no—
220C2×C61—41no—
221C2×C61#188411yescharacteristic
222C2×C61#189911yescharacteristic
223C2×C61#190911yescharacteristic
224C2×C22⋊Q8—611no—
225C2×#117—611no—
226#433—1222no—
227#433—1222no—
228#423—1221no—
229#407—611no—
230#407C2×D6111yescharacteristic
231C4×M16—1221no—
232#397—1221no—
233#397—1221no—
234#391—611no—
235#391—611no—
236#387—1221no—
237#387—1221no—
238#384—611no—
239#382—611no—
240#381—611no—
241#381C2×D6111yescharacteristic
242C4×C22⋊C4C2×D6111yescharacteristic
243#342—1221no—
244#342—1221no—
245#328—1222no—
246#328—1222no—
247C61#944411yescharacteristic, complemented, Sylow
248C22×Q8—611no—
249C23×C4C22×D6111yescharacteristic
250C2×M16—21no—
251C2×M16—1221no—
252C22⋊Q8—1221no—
253C22⋊Q8—1221no—
254C22⋊Q8—1221no—
255C22⋊Q8—1221no—
256#117—1221no—
257#117—1221no—
258#117—1221no—
259#117—1221no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—611no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4—1221no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—611no—
269C2×C22⋊C4—611no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4—1221no—
272C2×C42—611no—
273C2×C42—611no—
274C2×C42C4×D6111yescharacteristic
275C2×C42C4×D6111yescharacteristic
276C4⋊C8—1224no—
277C22⋊C8—21no—
278C22⋊C8—612no—
279C22⋊C8—1221no—
280C22⋊C8D24412yes—
281#97—1222no—
282C4×C8—1222no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—1221no—
287#95—1221no—
288#95—1221no—
289#95—1221no—
290#95—1221no—
291#95—1221no—
292#95—1221no—
293#95—1221no—
294C24D4×D6111yescharacteristic
295C2×Q8—1221no—
296C2×Q8—1221no—
297C2×Q8—2441no—
298C2×Q8—2441no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—611no—
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—2441no—
311C22×C4C2×C4×D6111yescharacteristic
312C22×C4C2×D24411yescharacteristic
313C22×C4#1175822211yescharacteristic
314C22×C4D4×D6111yescharacteristic
315C22×C4D4×D6111yescharacteristic
316C22×C4#1175822611yescharacteristic
317M16—41no—
318M16—41no—
319M16—2441no—
320M16—2441no—
321C2×C8—22no—
322C2×C8—1222no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C4⋊C4—1221no—
328C4⋊C4—1221no—
329C4⋊C4—2442no—
330C4⋊C4—2442no—
331C4⋊C4—2441no—
332C4⋊C4—2441no—
333C22⋊C4—42no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—1221no—
337C22⋊C4—1221no—
338C22⋊C4—1221no—
339C22⋊C4—1221no—
340C22⋊C4—2441no—
341C22⋊C4—2441no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C42—41no—
345C42—41no—
346C42—1221no—
347C42—1221no—
348C42—1221no—
349C42—1221no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23C22⋊C4×D6111yescharacteristic
355C23#1451975011yescharacteristic
356C23#1451975211yescharacteristic
357Q8—4881no—
358Q8—4881no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—4881no—
381C2×C4—4881no—
382C2×C4—4881no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4#1451973911yescharacteristic
394C2×C4C4×D24411yescharacteristic
395C2×C4#1451974711yescharacteristic
396C2×C4#1451976211yescharacteristic
397C8—42no—
398C8—2442no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22#1683964011yescharacteristic
410C22#1683973411yescharacteristic
411C22#1683973411yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—81no—
418C4—161no—
419C4—9761no—
420C4—4881no—
421C4—4881no—
422C4—2441no—
423C4—2441no—
424C4—2441no—
425C4—2441no—
426C2—21no—
427C2—21no—
428C2—41no—
429C2—41no—
430C2#2144620111yescharacteristic
431C2#2144627611yescharacteristic
432C2#2144634011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1284C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1284D61
hypocenterC61#9444
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
17. #16839668C4semidirect
18. #16839668C4semidirect
21. #16839632C4semidirect
22. #16839628C4semidirect
247. C61#9444semidirect

Maximal subgroups

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

subgroupindex
#214544542
#214502722
#214458942
#214443532
#214443512
#214436722
C61×#12842
#944461