Finite Group Databasebeta

← #29595478

Subgroups — #29595478

Order 15616.

1220 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#14519758C211yesmaximal, characteristic
2C2×#16839553C214yesmaximal, complemented
3#21445734C211yesmaximal, characteristic
4C61×#1783C211yesmaximal, characteristic
5C24×Dic61C411yescharacteristic, complemented
6C2×#14519758—28no—
7C2×#14519758—21no—
8C2×#14519758—21no—
9C2×#14519758C412yescomplemented
10C2×#14519758C2212yes—
11C22×#11758196—21no—
12C22×C22⋊Dic61—21no—
13C22×C22⋊Dic61C411yescharacteristic, complemented
14C2×#14519692—21no—
15C2×#14519692C2212yes—
16#16839618—21no—
17#16839553—28no—
18C23×D4×C61C2211yescharacteristic
19C2×#100×C61—21no—
20C2×#100×C61C2212yes—
21C61×#407—21no—
22C23×Dic61—21no—
23C23×Dic61—21no—
24C23×Dic61—21no—
25C23×Dic61—24no—
26C23×Dic61—42no—
27C23×Dic61—42no—
28C23×Dic61C2×C412yes—
29#14519758—28no—
30#14519758—28no—
31#14519758—24no—
32#14519758—24no—
33#14519758—24no—
34#14519758D418yes—
35C2×#11758196—28no—
36C2×#11758196—22no—
37C2×#11758196—22no—
38C2×C22⋊Dic61—28no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—24no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—42no—
45C2×C22⋊Dic61C2×C412yes—
46#14519692—24no—
47#14519692—42no—
48C25×C61C2×C411yescharacteristic
49C25×C61D411yescharacteristic
50C22×D4×C61—24no—
51C22×D4×C61—42no—
52C22×D4×C61—42no—
53C22×D4×C61C2×C411yescharacteristic
54C22×D4×C61D411yescharacteristic
55C22×D4×C61D411yescharacteristic
56C22×D4×C61C2311yescharacteristic
57C23×C4×C61D411yescharacteristic
58C2×M16×C61—22no—
59#100×C61—24no—
60#100×C61—42no—
61C22⋊C8×C61—42no—
62C24×C61—22no—
63C24×C61—22no—
64C24×C61—22no—
65C24×C61—22no—
66C24×C61—22no—
67C24×C61—41no—
68C24×C61—44no—
69C24×C61—44no—
70C24×C61C22⋊C411yescharacteristic
71C24×C61C22⋊C411yescharacteristic
72C24×C61C22⋊C411yescharacteristic
73C24×C61C22×C411yescharacteristic
74C24×C61C2×D411yescharacteristic
75C2×D4×C61—22no—
76C2×D4×C61—22no—
77C2×D4×C61—22no—
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61—41no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—416no—
87C2×D4×C61C2×D414yes—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—41no—
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×Dic61—24no—
94C22×Dic61—24no—
95C22×Dic61—24no—
96C22×Dic61—22no—
97C22×Dic61—22no—
98C22×Dic61—22no—
99C22×Dic61—44no—
100C22×Dic61—416no—
101C22×Dic61—42no—
102C22×Dic61—42no—
103C22×Dic61—42no—
104C22×Dic61—42no—
105C22×Dic61—42no—
106C22×Dic61—42no—
107C22×Dic61—42no—
108C22×Dic61C22⋊C414yescomplemented
109M16×C61—42no—
110M16×C61—42no—
111C2×C8×C61—42no—
112C22⋊Dic61—24no—
113C22⋊Dic61—28no—
114C22⋊Dic61—22no—
115C22⋊Dic61—28no—
116C22⋊Dic61—42no—
117C22⋊Dic61—42no—
118C22⋊Dic61—84no—
119C22⋊Dic61C22⋊C414yescomplemented
120#11758196—28no—
121#11758196—28no—
122C23×C61—24no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—28no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—21no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—44no—
136C23×C61—48no—
137C23×C61—48no—
138C23×C61—44no—
139C23×C61—42no—
140C23×C61—42no—
141C23×C61—42no—
142C23×C61—42no—
143C23×C61—48no—
144C23×C61—44no—
145C23×C61—44no—
146C23×C61—82no—
147C23×C61—84no—
148C23×C61#9912yes—
149C23×C61#9912yes—
150C23×C61C2×C22⋊C411yescharacteristic
151C23×C61C2×C22⋊C412yes—
152C23×C61#12012yes—
153D4×C61—44no—
154D4×C61—44no—
155D4×C61—44no—
156D4×C61—44no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—44no—
163C2×C4×C61#12012yes—
164C2×Dic61—24no—
165C2×Dic61—24no—
166C2×Dic61—48no—
167C2×Dic61—44no—
168C2×Dic61—48no—
169C2×Dic61—48no—
170C2×Dic61—44no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—44no—
174C2×Dic61—82no—
175C2×Dic61—84no—
176C8×C61—82no—
177#12688—611nomaximal, Sylow
178C22×C61—24no—
179C22×C61—24no—
180C22×C61—22no—
181C22×C61—22no—
182C22×C61—22no—
183C22×C61—22no—
184C22×C61—22no—
185C22×C61—22no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61—42no—
190C22×C61—416no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61—41no—
198C22×C61—81no—
199C22×C61—88no—
200C22×C61—81no—
201C22×C61#35214yescomplemented
202C22×C61#37911yescharacteristic
203C22×C61C2×#9911yescharacteristic
204C22×C61C2×#9911yescharacteristic
205C4×C61—22no—
206C4×C61—41no—
207Dic61—44no—
208Dic61—44no—
209Dic61—162no—
210C22×#123—611no—
211C2×#352—614no—
212#1783D6111yescharacteristic
213#1781—611no—
214C2×C61—21no—
215C2×C61—24no—
216C2×C61—21no—
217C2×C61—42no—
218C2×C61—42no—
219C2×C61—42no—
220C2×C61—81no—
221C2×C61#178111yescharacteristic
222C2×C61C2×#35212yes—
223C23×D4C2×D6111yescharacteristic
224C24×C4—611no—
225C2×#123—612no—
226C2×#123—612no—
227C2×#123—1228no—
228C2×#123—1221no—
229C2×#123—1221no—
230C22×C4⋊C4—1221no—
231C22×C22⋊C4—611no—
232C22×C22⋊C4—1221no—
233C2×#100—21no—
234C2×#100C2×D6112yes—
235C2×#99—612no—
236C2×#99—1221no—
237#407—21no—
238#379—1221no—
239#352—1228no—
240C61#1268811yescharacteristic, complemented, Sylow
241C25C4×D6111yescharacteristic
242C25D24411yescharacteristic
243C22×D4—24no—
244C22×D4—42no—
245C22×D4—42no—
246C22×D4C4×D6111yescharacteristic
247C22×D4D24411yescharacteristic
248C22×D4C22⋊D6111yescharacteristic
249C22×D4C22×D6111yescharacteristic
250C23×C4—612no—
251C23×C4—1224no—
252C23×C4—1221no—
253C23×C4—1221no—
254C23×C4—1221no—
255C23×C4—2442no—
256C23×C4—2442no—
257C23×C4C22⋊D6111yescharacteristic
258C2×M16—22no—
259#123—618no—
260#123—1224no—
261#123—1224no—
262#123—1228no—
263#123—1228no—
264#123—1224no—
265C2×C4⋊C4—1228no—
266C2×C4⋊C4—1222no—
267C2×C4⋊C4—1222no—
268C2×C22⋊C4—612no—
269C2×C22⋊C4—1224no—
270C2×C22⋊C4—1228no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1222no—
273C2×C22⋊C4—1222no—
274C2×C22⋊C4—1222no—
275C2×C22⋊C4—2442no—
276#100—24no—
277#100—42no—
278#99—1224no—
279#99—2442no—
280C22⋊C8—42no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—41no—
287C24—44no—
288C24—44no—
289C24#1175819811yescharacteristic
290C24#1175819811yescharacteristic
291C24#1175819811yescharacteristic
292C24C2×C4×D6111yescharacteristic
293C24C2×D24411yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—416no—
306C2×D4D4×D6114yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—614no—
311C22×C4—1224no—
312C22×C4—1224no—
313C22×C4—1224no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4—1222no—
317C22×C4—2442no—
318C22×C4—24416no—
319C22×C4—2442no—
320C22×C4—2442no—
321C22×C4—2444no—
322C22×C4—2442no—
323C22×C4—2442no—
324C22×C4—2442no—
325C22×C4—2442no—
326C22×C4#1175819811yescharacteristic
327C22×C4C2×C22⋊D6111yescharacteristic
328M16—42no—
329M16—42no—
330C2×C8—42no—
331C4⋊C4—1228no—
332C4⋊C4—1228no—
333C22⋊C4—614no—
334C22⋊C4—1224no—
335C22⋊C4—1228no—
336C22⋊C4—1222no—
337C22⋊C4—1228no—
338C22⋊C4—4884no—
339C22⋊C4—2442no—
340C22⋊C4—2442no—
341C23—21no—
342C23—24no—
343C23—21no—
344C23—28no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—48no—
358C23—48no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—48no—
363C23—44no—
364C23—44no—
365C23—82no—
366C23—84no—
367C23#1451969112yes—
368C23#1451969112yes—
369C23C22⋊C4×D6112yes—
370C23C2×#1175819811yescharacteristic
371C23#1451974712yes—
372D4—44no—
373D4—44no—
374D4—44no—
375D4—44no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—44no—
382C2×C4—1224no—
383C2×C4—1224no—
384C2×C4—4882no—
385C2×C4—4884no—
386C2×C4—2448no—
387C2×C4—2444no—
388C2×C4—2448no—
389C2×C4—2448no—
390C2×C4—2444no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2444no—
394C2×C4#1451974512yes—
395C8—82no—
396C22—24no—
397C22—24no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—416no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—81no—
417C22—88no—
418C22—81no—
419C22#1683955314yescomplemented
420C22#1683961711yescharacteristic
421C22C2×#1451969111yescharacteristic
422C22C2×#1451969111yescharacteristic
423C4—22no—
424C4—41no—
425C4—9762no—
426C4—2444no—
427C4—2444no—
428C2—21no—
429C2—21no—
430C2—24no—
431C2—42no—
432C2—42no—
433C2—42no—
434C2—81no—
435C2#2144573311yescharacteristic
436C2C2×#1683955312yes—
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1783C2
socleC22×C61C2×#99
prime radicalC23×D4×C61C22
cosocleC22×D4×C61C23
hypercenter#1783D61
hypocenterC61#12688
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #1783

Complements

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

NH ≅ G/Ntype
2. C2×#16839553C2semidirect
5. C24×Dic61C4semidirect
9. C2×#14519758C4semidirect
13. C22×C22⋊Dic61C4semidirect
108. C22×Dic61C22⋊C4semidirect
119. C22⋊Dic61C22⋊C4semidirect
201. C22×C61#352semidirect
240. C61#12688semidirect
419. C22#16839553semidirect

Maximal subgroups

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

subgroupindex
C22×#145197582
C2×#168395532
#214457342
C61×#17832
#1268861