Finite Group Databasebeta

← #29571612

Subgroups — #29571612

Order 15616.

852 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#16840458C211yesmaximal, characteristic
2#21445916C212yesmaximal, complemented
3C4×#14519692C211yesmaximal, characteristic, complemented
4#21444336C212yesmaximal, complemented
5C61×#1465C211yesmaximal, characteristic, complemented
6#16840458—24no—
7#16840458—24no—
8C2×#14519772—21no—
9C2×#14519769C411yescharacteristic, complemented
10C2×#14519759—21no—
11C2×#14519751C411yescharacteristic, complemented
12C2×#14519751C2211yescharacteristic
13C2×#14519749—21no—
14C2×C4×D244C411yescharacteristic, complemented
15C2×#14519737—21no—
16C2×#14519736C2211yescharacteristic
17C22×C4⋊Dic61C411yescharacteristic, complemented
18C2×#14519692—21no—
19C4×C22⋊Dic61C2212yes—
20#16839529—21no—
21#16839523—24no—
22#16839497—24no—
23C2×C4×D4×C61C2211yescharacteristic
24C61×#407C2212yes—
25C61×#324—21no—
26C61×#323—21no—
27C22×D244C2×C411yescharacteristic
28C22×C22⋊D61—21no—
29C22×C22⋊D61—21no—
30C23×Dic61—21no—
31C22×C4×D61—21no—
32#14519772—28no—
33#14519769—22no—
34#14519769—22no—
35#14519759—28no—
36#14519751—22no—
37#14519751—22no—
38#14519751—22no—
39#14519751D414yes—
40#14519749—28no—
41C4×D244—22no—
42C4×D244—22no—
43#14519737—28no—
44#14519736—22no—
45#14519736D414yes—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61C2×C411yescharacteristic
50C2×C4⋊Dic61C2×C411yescharacteristic
51C2×C4⋊Dic61C2×C411yescharacteristic
52C2×#11758196—21no—
53C2×#11758196—21no—
54C2×C22⋊Dic61—21no—
55C2×C22⋊Dic61—22no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×C4×Dic61—21no—
60C2×C4×Dic61—22no—
61#14519692—24no—
62#14519684—22no—
63C22×D4×C61D411yescharacteristic
64C23×C4×C61C2×C411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C2×M16×C61—22no—
67C4×D4×C61—22no—
68C4×D4×C61—22no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C22⋊C4×C61D411yescharacteristic
71C2×C22⋊C4×C61D411yescharacteristic
72C2×C42×C61C2×C411yescharacteristic
73C22⋊C8×C61—22no—
74C22⋊C8×C61—22no—
75C24×C61C22⋊C411yescharacteristic
76C24×C61C2×D411yescharacteristic
77C23×D61—21no—
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61—41no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—42no—
84C22×C4×C61—42no—
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22×C411yescharacteristic
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D411yescharacteristic
92C2×C22⋊D61—24no—
93C2×C22⋊D61—24no—
94C2×C22⋊D61—42no—
95C2×C22⋊D61—42no—
96C2×C22⋊D61—82no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—24no—
101C22×Dic61—42no—
102C22×Dic61—82no—
103C2×D244—21no—
104C2×D244—42no—
105C2×D244C22⋊C412yes—
106C2×C4×D61—24no—
107C2×C4×D61—82no—
108M16×C61—44no—
109C2×C8×C61—42no—
110C4⋊C4×C61—22no—
111C22⋊C4×C61—22no—
112C22⋊C4×C61—22no—
113C42×C61—21no—
114C42×C61—21no—
115C22⋊Dic61—24no—
116C22⋊Dic61—24no—
117C22⋊Dic61—42no—
118#11758198—24no—
119#11758198—24no—
120#11758198—24no—
121C4⋊Dic61—24no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61C22⋊C412yes—
126C4⋊Dic61C22⋊C412yes—
127C4⋊Dic61C22⋊C412yes—
128#11758196—24no—
129#11758196—24no—
130C4×Dic61—24no—
131C4×Dic61—44no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61#10412yes—
141C23×C61C2×C22⋊C411yescharacteristic
142C23×C61#12012yes—
143C22×D61—22no—
144C22×D61—42no—
145C22×D61—42no—
146C22×D61—41no—
147C22×D61—41no—
148D4×C61—44no—
149C2×C4×C61—24no—
150C2×C4×C61—22no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×C4×C61#12012yes—
165C22⋊D61—48no—
166C22⋊D61—84no—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—22no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—44no—
175C2×Dic61—84no—
176C2×Dic61—82no—
177C2×Dic61—82no—
178D244—42no—
179D244—42no—
180C4×D61—84no—
181C8×C61—82no—
182#8362—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—22no—
186C22×C61—22no—
187C22×C61—22no—
188C22×C61—22no—
189C22×C61—42no—
190C22×C61—42no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61#37911yescharacteristic
196C22×C61C2×#10411yescharacteristic
197C22×C61#42111yescharacteristic
198C2×D61—42no—
199C2×D61—42no—
200C2×D61—42no—
201C2×D61—82no—
202C2×D61—82no—
203C4×C61—22no—
204C4×C61—21no—
205C4×C61—41no—
206C4×C61—41no—
207C4×C61—41no—
208C4×C61#41012yes—
209Dic61—42no—
210Dic61—42no—
211Dic61—42no—
212Dic61—84no—
213C2×#553—611no—
214#1826—612no—
215C4×#99—611no—
216#1507—612no—
217#1465D6111yescharacteristic, complemented
218C2×C61—21no—
219C2×C61—21no—
220C2×C61—42no—
221C2×C61—42no—
222C2×C61#178211yescharacteristic
223C2×C61#178911yescharacteristic
224C2×C61#179511yescharacteristic
225D61—82no—
226#553—1224no—
227#553—1224no—
228C2×#126—1221no—
229C2×#125—611no—
230C2×#123—1221no—
231C2×#121—611no—
232C2×#121—611no—
233C2×#121—1221no—
234C2×C4×D4—611no—
235C2×C4×D4—1221no—
236C2×C4×D4C2×D6111yescharacteristic
237C2×#117—611no—
238C22×C4⋊C4—611no—
239C2×#99—1221no—
240#407C2×D6112yes—
241C4×C22⋊C4—612no—
242#342—1221no—
243#339—1224no—
244#327—1224no—
245#324—21no—
246#323—21no—
247C61#836211yescharacteristic, complemented, Sylow
248C22×D4—611no—
249C22×D4—1221no—
250C22×D4—1221no—
251C22×D4D24411yescharacteristic
252C23×C4—1221no—
253C23×C4—1221no—
254C23×C4C4×D6111yescharacteristic
255C23×C4C22×D6111yescharacteristic
256C2×M16—22no—
257#126—1228no—
258#125—1222no—
259#125—1222no—
260#123—1228no—
261#121—614no—
262#121—1222no—
263#121—1222no—
264#121—1228no—
265#121—1222no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—1222no—
269C4×D4—1222no—
270C4×D4—1228no—
271#117—614no—
272#117—1222no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—611no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—1222no—
277C2×C4⋊C4—1222no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4—1221no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4C22⋊D6111yescharacteristic
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1222no—
287C2×C22⋊C4D24411yescharacteristic
288C2×C22⋊C4C22⋊D6111yescharacteristic
289C2×C42—1221no—
290C2×C42—1222no—
291C2×C42C4×D6111yescharacteristic
292#99—1224no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295#95—1222no—
296C24—1221no—
297C24#1175819811yescharacteristic
298C24C2×D24411yescharacteristic
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—612no—
303C2×D4—1224no—
304C2×D4—1224no—
305C2×D4—1221no—
306C2×D4—4882no—
307C2×D4—2442no—
308C2×D4—2442no—
309C2×D4—2442no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—1224no—
315C22×C4—1224no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—1221no—
319C22×C4—4882no—
320C22×C4—4882no—
321C22×C4—2442no—
322C22×C4#1175819811yescharacteristic
323C22×C4#1175819811yescharacteristic
324C22×C4#1175819811yescharacteristic
325C22×C4C2×C4×D6111yescharacteristic
326C22×C4D4×D6112yes—
327C22×C4D4×D6112yes—
328C22×C4C2×C22⋊D6111yescharacteristic
329M16—44no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—612no—
333C4⋊C4—612no—
334C4⋊C4—612no—
335C4⋊C4—1224no—
336C4⋊C4—1224no—
337C4⋊C4—1224no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—1224no—
344C22⋊C4—1224no—
345C22⋊C4—1224no—
346C22⋊C4—1224no—
347C22⋊C4—1224no—
348C22⋊C4—2442no—
349C42—21no—
350C42—21no—
351C42—1224no—
352C42—2444no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—1222no—
362C23—2442no—
363C23—2442no—
364C23—2441no—
365C23—2441no—
366C23#1451970312yes—
367C23C2×#1175819811yescharacteristic
368C23#1451974712yes—
369D4—44no—
370D4—4884no—
371D4—2442no—
372D4—2448no—
373D4—2442no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—1222no—
388C2×C4—1222no—
389C2×C4—1222no—
390C2×C4—4884no—
391C2×C4—4882no—
392C2×C4—4882no—
393C2×C4—4884no—
394C2×C4—2444no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4C22⋊C4×D6111yescharacteristic
400C2×C4C22⋊C4×D6111yescharacteristic
401C2×C4#1451974512yes—
402C8—82no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—42no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—81no—
414C22—81no—
415C22—4882no—
416C22—4882no—
417C22—2442no—
418C22—2442no—
419C22—2442no—
420C22#1683961711yescharacteristic
421C22C2×#1451970311yescharacteristic
422C22#1683977811yescharacteristic
423C4—22no—
424C4—21no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—4884no—
429C4—2442no—
430C4—2442no—
431C4—2442no—
432C4#1683973412yes—
433C2—21no—
434C2—21no—
435C2—42no—
436C2—42no—
437C2—4882no—
438C2#2144573511yescharacteristic
439C2#2144576211yescharacteristic
440C2#2144579011yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1465C2
socleC22×C61#379
prime radicalC2×#16840458C2
cosocleC23×C4×C61C23
hypercenter#1465D61
hypocenterC61#8362
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #21445916C2semidirect
3. C4×#14519692C2semidirect
4. #21444336C2semidirect
5. C61×#1465C2semidirect
9. C2×#14519769C4semidirect
11. C2×#14519751C4semidirect
14. C2×C4×D244C4semidirect
17. C22×C4⋊Dic61C4semidirect
217. #1465D61semidirect
247. C61#8362semidirect

Maximal subgroups

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

subgroupindex
C2×#168404582
#214459162
C4×#145196922
#214443362
C61×#14652
#836261