Finite Group Databasebeta

← #29654139

Subgroups — #29654139

Order 15616.

1058 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 167 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×#16840388C211yesmaximal, characteristic
2#21456640C212yesmaximal, complemented
3#21444809C214yesmaximal, complemented
4#21444775C214yesmaximal, complemented
5#21444338C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic
7C61×#2561C211yesmaximal, characteristic
8#16840388—24no—
9#16840388—22no—
10#16840388C2214yes—
11C2×#14519756—21no—
12C2×#14519756C2212yes—
13C2×#14519754—22no—
14C2×#14519754—22no—
15C22×C22⋊Dic61—21no—
16#16840092C2214yescomplemented
17#16840090—22no—
18#16840018—24no—
19#16839918C2214yescomplemented
20#16839917—22no—
21#16839726C2212yes—
22#16839498C2218yescomplemented
23#16839486C2214yes—
24C2×C22⋊Q8×C61C2211yescharacteristic
25C22×C22⋊C4×C61C2211yescharacteristic
26C61×#397C2212yes—
27C61×#393C2212yes—
28C61×#388—22no—
29C61×#381—21no—
30C61×#379—21no—
31C4×C22⋊C4×C61C2211yescharacteristic
32C22×Dic122—21no—
33C22×Dic122D412yes—
34C23×Dic61—21no—
35C23×Dic61—21no—
36C2×#11758202—24no—
37C2×#11758194—22no—
38#14519756—24no—
39#14519756—24no—
40#14519756—22no—
41#14519756—22no—
42#14519756D414yes—
43#14519756C2314yes—
44#14519754—28no—
45#14519754—28no—
46#14519754—28no—
47#14519754—28no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61D412yes—
50C2×#11758196—22no—
51C2×#11758196—22no—
52C2×C22⋊Dic61—22no—
53C2×C22⋊Dic61—24no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—24no—
56#14519708—24no—
57#14519701—24no—
58#14519700—24no—
59C22⋊#92418—22no—
60C22⋊#92418C2314yes—
61C25×C61D411yescharacteristic
62C22×Q8×C61C2311yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65C22⋊Q8×C61—22no—
66C22⋊Q8×C61—22no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61C2311yescharacteristic
69C2×C22⋊C4×C61—22no—
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61D412yes—
76C2×C22⋊C4×C61C2312yes—
77C2×C42×C61—21no—
78#95×C61—22no—
79#95×C61—22no—
80#95×C61C2312yes—
81C24×C61—21no—
82C24×C61—21no—
83C24×C61—24no—
84C24×C61—42no—
85C24×C61—81no—
86C24×C61C2×D411yescharacteristic
87C24×C61C2×D412yes—
88C2×Q8×C61—22no—
89C2×Q8×C61—81no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—42no—
96C22×C4×C61—81no—
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—22no—
103C22×Dic61—22no—
104C22×Dic61—24no—
105C22×Dic61—24no—
106C22×Dic61—82no—
107C22×Dic61—82no—
108C2×Dic122—24no—
109C2×Dic122—22no—
110C2×Dic122—82no—
111C2×Dic122—82no—
112C2×Dic122C2×D414yes—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—42no—
115C22⋊C4×C61—24no—
116C22⋊C4×C61—24no—
117C22⋊C4×C61—42no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—44no—
121C22⋊C4×C61—82no—
122C42×C61—81no—
123C22⋊Dic61—28no—
124C22⋊Dic61—24no—
125C22⋊Dic61—24no—
126#11758202—84no—
127C4⋊Dic61—24no—
128C4⋊Dic61—22no—
129C4⋊Dic61C2×D414yes—
130#11758196—28no—
131#11758196—24no—
132#11758196—24no—
133#11758194—82no—
134C2×#92418—24no—
135C23×C61—24no—
136C23×C61—24no—
137C23×C61—22no—
138C23×C61—22no—
139C23×C61—22no—
140C23×C61—22no—
141C23×C61—41no—
142C23×C61—42no—
143C23×C61—42no—
144C23×C61—42no—
145C23×C61—42no—
146C23×C61—81no—
147C23×C61—81no—
148C23×C61—88no—
149C23×C61#12011yescharacteristic
150C23×C61#12011yescharacteristic
151C23×C61#13714yes—
152C23×C61C22×D411yescharacteristic
153Q8×C61—44no—
154C2×C4×C61—22no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61—81no—
169C2×C4×C61#12012yes—
170C2×C4×C61C22×D412yes—
171C2×Dic61—24no—
172C2×Dic61—24no—
173C2×Dic61—44no—
174C2×Dic61—44no—
175C2×Dic61—84no—
176C2×Dic61—84no—
177Dic122—44no—
178Dic122—82no—
179Dic122—84no—
180#92418—44no—
181#18319—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—24no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—44no—
191C22×C61—44no—
192C22×C61—48no—
193C22×C61—42no—
194C22×C61—42no—
195C22×C61—82no—
196C22×C61—84no—
197C22×C61#45512yes—
198C22×C61C2×#12011yescharacteristic
199C22×C61C2×#13712yes—
200C4×C61—22no—
201C4×C61—22no—
202C4×C61—42no—
203C4×C61—82no—
204C4×C61—81no—
205C4×C61—81no—
206Dic61—44no—
207Dic61—44no—
208C2×#543—611no—
209#3195—612no—
210#2561D6111yescharacteristic
211#1608—614no—
212#1602—614no—
213#1508—612no—
214#1462—611no—
215C2×C61—22no—
216C2×C61—22no—
217C2×C61—41no—
218C2×C61—44no—
219C2×C61#300411yescharacteristic
220C2×C61C2×#45511yescharacteristic
221C2×C61#302611yescharacteristic
222#543—614no—
223#543—1224no—
224#543—1222no—
225C2×C22⋊Q8—612no—
226C2×C22⋊Q8—1222no—
227C2×C22⋊Q8—1222no—
228C2×C22⋊Q8—1221no—
229C2×C22⋊Q8C2×D6111yescharacteristic
230C22×C22⋊C4—1221no—
231C22×C22⋊C4C2×D6111yescharacteristic
232#483—614no—
233#483—1222no—
234#470—1224no—
235#451—614no—
236#451—1222no—
237#407—612no—
238#397C2×D6112yes—
239#393C2×D6112yes—
240#388—22no—
241#381—21no—
242#379—21no—
243C4×C22⋊C4C2×D6111yescharacteristic
244#328—618no—
245#323—614no—
246C61#1831911yescharacteristic, complemented, Sylow
247C25C22⋊D6111yescharacteristic
248C22×Q8—612no—
249C22×Q8—1221no—
250C22×Q8C22×D6111yescharacteristic
251C23×C4—1221no—
252C23×C4—1221no—
253C23×C4C22⋊D6111yescharacteristic
254C23×C4C22×D6111yescharacteristic
255C2×Q16—1224no—
256C2×M16—1222no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—614no—
260C22⋊Q8—614no—
261C22⋊Q8—1228no—
262C22⋊Q8—1228no—
263C22⋊Q8—1228no—
264C22⋊Q8—1228no—
265C22⋊Q8—1224no—
266C22⋊Q8—1222no—
267C22⋊Q8—1222no—
268C22⋊Q8—1224no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—612no—
271C2×C4⋊C4—1222no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1222no—
274C2×C4⋊C4C22×D6111yescharacteristic
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—1222no—
282C2×C22⋊C4—1224no—
283C2×C22⋊C4—1222no—
284C2×C22⋊C4—1224no—
285C2×C22⋊C4C22⋊D6112yes—
286C2×C22⋊C4C22×D6112yes—
287C2×C42—21no—
288#106—1224no—
289#103—1224no—
290#103—1224no—
291C22⋊C8—614no—
292C22⋊C8—1222no—
293#95—22no—
294#95—22no—
295#95C22×D6112yes—
296C24—21no—
297C24—21no—
298C24—24no—
299C24—42no—
300C24—81no—
301C24C2×C22⋊D6111yescharacteristic
302C24C2×C22⋊D6112yes—
303C2×Q8—22no—
304C2×Q8—81no—
305C2×Q8—614no—
306C2×Q8—1224no—
307C2×Q8—1222no—
308C2×Q8—4882no—
309C2×Q8—4882no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—81no—
317C22×C4—1224no—
318C22×C4—1224no—
319C22×C4—1222no—
320C22×C4—1222no—
321C22×C4—4882no—
322C22×C4—4882no—
323C22×C4D4×D6112yes—
324C22×C4D4×D6112yes—
325C22×C4C2×C22⋊D6112yes—
326C22×C4C2×C22⋊D6111yescharacteristic
327C22×C4C23×D6111yescharacteristic
328Q16—4884no—
329M16—4882no—
330C2×C8—1224no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—614no—
334C4⋊C4—1228no—
335C4⋊C4—1224no—
336C4⋊C4—1224no—
337C4⋊C4—1222no—
338C4⋊C4—1224no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—44no—
345C22⋊C4—82no—
346C22⋊C4—1228no—
347C22⋊C4—1224no—
348C22⋊C4—1224no—
349C42—81no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—81no—
362C23—81no—
363C23—88no—
364C23#1451974511yescharacteristic
365C23#1451974511yescharacteristic
366C23#1451980714yes—
367C23C22×C22⋊D6111yescharacteristic
368Q8—44no—
369Q8—4882no—
370Q8—4884no—
371Q8—2444no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—1224no—
388C2×C4—1224no—
389C2×C4—4884no—
390C2×C4—4884no—
391C2×C4—2444no—
392C2×C4—2444no—
393C2×C4#1451974512yes—
394C2×C4C2×D4×D6112yes—
395C8—2444no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—24no—
404C22—44no—
405C22—44no—
406C22—48no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—84no—
411C22#1683993912yes—
412C22C2×#1451974511yescharacteristic
413C22C2×#1451980712yes—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—82no—
418C4—81no—
419C4—81no—
420C4—2444no—
421C4—2444no—
422C2—22no—
423C2—22no—
424C2—41no—
425C2—44no—
426C2#2145505211yescharacteristic
427C2C2×#1683993911yescharacteristic
428C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2561C2
socleC22×C61C2×#120
prime radicalC25×C61D4
hypercenter#2561D61
hypocenterC61#18319
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
2. #21456640C2semidirect
3. #21444809C2semidirect
4. #21444775C2semidirect
5. #21444338C2semidirect
16. #16840092C22semidirect
19. #16839918C22semidirect
22. #16839498C22semidirect
246. C61#18319semidirect

Maximal subgroups

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

subgroupindex
C2×#168403882
#214566402
#214448092
#214447752
#214443382
#214441462
C61×#25612
#1831961