Finite Group Databasebeta

← #29578000

Subgroups — #29578000

Order 15616.

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

normal subgroups only

This group has 716 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#21454264C211yesmaximal, characteristic
2#21449150C211yesmaximal, characteristic
3#21444148C212yesmaximal, complemented
4#21443728C212yesmaximal, complemented
5C61×#1785C211yesmaximal, characteristic
6C22×#11758194—21no—
7C2×#14519758C411yescharacteristic
8C2×#14519758C411yescharacteristic
9C22×#11758196C2211yescharacteristic
10#16839729—24no—
11#16839726—21no—
12#16839726C2212yes—
13#16839681—21no—
14#16839676—21no—
15#16839642C411yescharacteristic
16#16839642C411yescharacteristic
17#16839635—21no—
18#16839635—21no—
19#16839625C2211yescharacteristic
20#16839487—24no—
21C22×#41×C61C2211yescharacteristic
22C2×#101×C61—21no—
23C2×#100×C61—21no—
24C61×#407C2212yes—
25C23×Dic61C2×C411yescharacteristic
26C23×Dic61C2×C411yescharacteristic
27C2×#11758194—22no—
28C2×#11758194—22no—
29C2×#11758194—42no—
30C2×#11758194—42no—
31C22×#92418—22no—
32#14519758—42no—
33#14519758—42no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×#11758196—22no—
38C2×#11758196—21no—
39C2×#11758196—22no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61C2×C411yescharacteristic
45C2×C22⋊Dic61C2×C411yescharacteristic
46C22⋊#92418—24no—
47C22⋊#92418—22no—
48C22⋊#92418D414yes—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C2×#41×C61—22no—
54C2×#41×C61—22no—
55C2×#41×C61—42no—
56C2×#41×C61—42no—
57C22×Q8×C61D411yescharacteristic
58C22×D4×C61C2×C411yescharacteristic
59C22×D4×C61C2×C411yescharacteristic
60C22×D4×C61D411yescharacteristic
61C23×C4×C61D411yescharacteristic
62C23×C4×C61D411yescharacteristic
63C23×C4×C61C2311yescharacteristic
64C2×M16×C61—22no—
65#101×C61—42no—
66#100×C61—42no—
67C22⋊C8×C61—22no—
68C22⋊C8×C61—22no—
69C24×C61C22⋊C411yescharacteristic
70C24×C61C22⋊C411yescharacteristic
71C24×C61C22×C411yescharacteristic
72#41×C61—44no—
73#41×C61—44no—
74#41×C61—48no—
75C2×Q8×C61—22no—
76C2×Q8×C61—22no—
77C2×Q8×C61—41no—
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61—22no—
81C2×D4×C61—22no—
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C22⋊C411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D412yes—
102C22×C4×C61#4112yes—
103C22×Dic61—21no—
104C22×Dic61—22no—
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—22no—
109C22×Dic61—42no—
110C22×Dic61—41no—
111C22×Dic61—41no—
112M16×C61—42no—
113M16×C61—42no—
114C2×C8×C61—42no—
115C22⋊Dic61—42no—
116C22⋊Dic61—42no—
117C22⋊Dic61—42no—
118C22⋊Dic61—42no—
119C22⋊Dic61—41no—
120C22⋊Dic61—41no—
121#11758196—42no—
122#11758196—42no—
123#11758196—41no—
124#11758196—42no—
125#11758196—41no—
126#11758194—44no—
127#11758194—42no—
128#11758194—42no—
129C2×#92418—24no—
130C2×#92418—44no—
131C23×C61—21no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61#9912yes—
145C23×C61C2×C22⋊C411yescharacteristic
146C23×C61C4×D412yes—
147Q8×C61—44no—
148D4×C61—44no—
149D4×C61—44no—
150D4×C61—44no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—24no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61#9912yes—
168C2×C4×C61#12011yescharacteristic
169C2×C4×C61#12112yes—
170C2×C4×C61#12311yescharacteristic
171C2×Dic61—22no—
172C2×Dic61—24no—
173C2×Dic61—41no—
174C2×Dic61—42no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—82no—
186C8×C61—82no—
187#92418—44no—
188#9513—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—22no—
192C22×C61—22no—
193C22×C61—22no—
194C22×C61—22no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61—42no—
198C22×C61—42no—
199C22×C61—41no—
200C22×C61—41no—
201C22×C61—41no—
202C22×C61—41no—
203C22×C61—41no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61#38611yescharacteristic
207C22×C61C2×#9911yescharacteristic
208C22×C61C2×#9911yescharacteristic
209C4×C61—21no—
210C4×C61—22no—
211C4×C61—21no—
212C4×C61—22no—
213C4×C61—41no—
214Dic61—42no—
215Dic61—81no—
216Dic61—81no—
217Dic61—81no—
218#2879—611no—
219#2406—611no—
220#1785D6111yescharacteristic
221#1463—612no—
222#1306—612no—
223C2×C61—21no—
224C2×C61—21no—
225C2×C61—42no—
226C2×C61—42no—
227C2×C61—81no—
228C2×C61#189111yescharacteristic
229C2×C61#189811yescharacteristic
230C2×C61#191511yescharacteristic
231C22×#41C2×D6111yescharacteristic
232C22×M16—1221no—
233C2×#123—611no—
234C2×#123—611no—
235C22×C4⋊C4—611no—
236C2×#101—21no—
237C2×#100—21no—
238#408—1224no—
239#407—612no—
240#407—1221no—
241#407C2×D6112yes—
242#395—1221no—
243#394—1221no—
244#386—611no—
245#386—611no—
246#385—1221no—
247#385—1221no—
248#381—611no—
249#324—1224no—
250C61#951311yescharacteristic, complemented, Sylow
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—42no—
254C2×#41—42no—
255C22×Q8C22⋊D6111yescharacteristic
256C22×D4C4×D6111yescharacteristic
257C22×D4C4×D6111yescharacteristic
258C22×D4C22⋊D6111yescharacteristic
259C23×C4—611no—
260C23×C4—611no—
261C23×C4D24411yescharacteristic
262C23×C4D24411yescharacteristic
263C23×C4C22×D6111yescharacteristic
264C2×M16—22no—
265C2×M16—1222no—
266C2×M16—1222no—
267C2×M16—2442no—
268C2×M16—2442no—
269C22×C8—1222no—
270#123—2442no—
271#123—2442no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1222no—
276C2×C4⋊C4—1222no—
277C2×C4⋊C4—1221no—
278C2×C22⋊C4—611no—
279C2×C22⋊C4—611no—
280C2×C22⋊C4—1221no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1221no—
284#101—42no—
285#100—42no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—614no—
289C22⋊C8—1224no—
290C22⋊C8—1222no—
291#95—1221no—
292#95—1221no—
293#95—1221no—
294#95—1221no—
295C24#1175819811yescharacteristic
296C24#1175819811yescharacteristic
297C24C2×C4×D6111yescharacteristic
298#41—44no—
299#41—44no—
300#41—48no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—41no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—1221no—
324C22×C4—1222no—
325C22×C4—1222no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—2442no—
330C22×C4—2441no—
331C22×C4—2441no—
332C22×C4#1175819811yescharacteristic
333C22×C4#1175819811yescharacteristic
334C22×C4C2×D24411yescharacteristic
335C22×C4D4×D6112yes—
336C22×C4#1175822412yes—
337C22×C4C2×C22⋊D6111yescharacteristic
338M16—42no—
339M16—42no—
340M16—2444no—
341M16—2442no—
342M16—2442no—
343C2×C8—42no—
344C2×C8—1224no—
345C2×C8—2444no—
346C4⋊C4—2442no—
347C4⋊C4—2442no—
348C4⋊C4—2441no—
349C4⋊C4—2442no—
350C4⋊C4—2441no—
351C22⋊C4—2442no—
352C22⋊C4—2442no—
353C22⋊C4—2442no—
354C22⋊C4—2442no—
355C22⋊C4—2441no—
356C22⋊C4—2441no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23#1451969112yes—
371C23C2×#1175819811yescharacteristic
372C23#1451973712yes—
373Q8—44no—
374D4—44no—
375D4—44no—
376D4—44no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—1222no—
394C2×C4—1224no—
395C2×C4—4882no—
396C2×C4—2441no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4—2441no—
408C2×C4#1451969112yes—
409C2×C4#1451974711yescharacteristic
410C2×C4#1451974912yes—
411C2×C4#1451976111yescharacteristic
412C8—82no—
413C8—2444no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—81no—
430C22—81no—
431C22#1683963911yescharacteristic
432C22C2×#1451969111yescharacteristic
433C22C2×#1451969111yescharacteristic
434C4—22no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—41no—
439C4—4881no—
440C4—4881no—
441C4—4881no—
442C4—2442no—
443C2—21no—
444C2—21no—
445C2—42no—
446C2—42no—
447C2—81no—
448C2#2144623511yescharacteristic
449C2#2144627011yescharacteristic
450C2#2144636911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1785C2
socleC22×C61#386
prime radicalC22×#41×C61C22
cosocleC23×C4×C61C23
hypercenter#1785D61
hypocenterC61#9513
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
3. #21444148C2semidirect
4. #21443728C2semidirect
250. C61#9513semidirect

Maximal subgroups

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

subgroupindex
#214542642
#214491502
#214441482
#214437282
C61×#17852
#951361