Finite Group Databasebeta

← #29571445

Subgroups — #29571445

Order 15616.

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

normal subgroups only

This group has 748 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
1C2×#16840383C211yesmaximal, characteristic
2#21444213C212yesmaximal, complemented
3#21444211C212yesmaximal, complemented
4#21444133C211yesmaximal, characteristic, complemented
5C61×#1458C211yesmaximal, characteristic, complemented
6#16840383—24no—
7#16840383—24no—
8C2×#14519777C411yescharacteristic
9C2×#14519772—21no—
10C2×#14519765C411yescharacteristic
11C2×#14519763—21no—
12C2×#14519759—21no—
13C2×#14519756C411yescharacteristic
14C2×#14519755—21no—
15C2×#14519751C411yescharacteristic
16C2×C4×Dic122C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16839815C2212yes—
19#16839492—24no—
20#16839491—24no—
21#16839488—21no—
22#16839486—21no—
23C2×#117×C61C2211yescharacteristic
24C61×#431C2212yes—
25C61×#324—21no—
26C61×#323—21no—
27C22×Dic122—21no—
28C22×Dic122C2×C411yescharacteristic
29C22×D244C2×C411yescharacteristic
30C22×C22⋊D61—21no—
31C22×C4×D61—21no—
32#14519777—22no—
33#14519777—22no—
34#14519772—28no—
35#14519765—22no—
36#14519765—22no—
37#14519763—28no—
38#14519759—28no—
39#14519756—22no—
40#14519756—22no—
41#14519755—28no—
42#14519751—22no—
43#14519751—22no—
44C4×Dic122—22no—
45C4×Dic122D414yes—
46C4×D244—22no—
47C4×D244D414yes—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61C2×C411yescharacteristic
50C2×C4⋊Dic61C2×C411yescharacteristic
51C2×#11758196—21no—
52C2×#11758196—21no—
53C2×C22⋊Dic61—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×C4×Dic61—21no—
59C4⋊#92418—22no—
60C22⋊#92418—22no—
61C4×#92418—22no—
62C23×C4×C61C2×C411yescharacteristic
63#117×C61—22no—
64#117×C61—22no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61D411yescharacteristic
67C2×C22⋊C4×C61D411yescharacteristic
68C2×C22⋊C4×C61D411yescharacteristic
69C2×C42×C61C2×C411yescharacteristic
70C2×C42×C61C2311yescharacteristic
71C4⋊C8×C61—22no—
72C22⋊C8×C61—22no—
73C4×C8×C61—22no—
74C24×C61C22⋊C411yescharacteristic
75C23×D61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22×C411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61—24no—
87C2×C22⋊D61—82no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C2×D244—22no—
92C2×D244—22no—
93C2×D244—21no—
94C2×D244C22⋊C412yes—
95C2×C4×D61—24no—
96C2×C4×D61—82no—
97C2×Dic122—22no—
98C2×Dic122—22no—
99C2×Dic122—21no—
100C2×Dic122—24no—
101C2×Dic122—82no—
102C2×Dic122C22⋊C412yes—
103C2×C8×C61—42no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C22⋊C4×C61—22no—
107C22⋊C4×C61—22no—
108C42×C61—21no—
109C42×C61—21no—
110C42×C61C2×D412yes—
111C42×C61C2×D412yes—
112C22⋊Dic61—24no—
113#11758198—24no—
114#11758198—24no—
115#11758198—24no—
116#11758198—24no—
117C4⋊Dic61—24no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61C22⋊C412yes—
121C4⋊Dic61C22⋊C412yes—
122#11758196—24no—
123#11758196—24no—
124C4×Dic61—24no—
125C2×#92418—42no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61C2×C22⋊C411yescharacteristic
132C22×D61—22no—
133C22×D61—42no—
134C22×D61—42no—
135C22×D61—41no—
136C22×D61—41no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61C2×C22⋊C411yescharacteristic
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61#12012yes—
151C2×C4×C61#12012yes—
152C2×C4×C61#13112yes—
153C2×C4×C61#13112yes—
154C22⋊D61—84no—
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162D244—24no—
163D244—42no—
164C4×D61—84no—
165Dic122—24no—
166Dic122—42no—
167Dic122—84no—
168C8×C61—82no—
169#92418—82no—
170#8325—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—41no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61#37911yescharacteristic
179C22×C61#40811yescharacteristic
180C22×C61#40811yescharacteristic
181C2×D61—42no—
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—82no—
185C2×D61—82no—
186C4×C61—22no—
187C4×C61—22no—
188C4×C61—41no—
189C4×C61—41no—
190C4×C61—41no—
191C4×C61#40812yes—
192C4×C61#40812yes—
193Dic61—42no—
194Dic61—42no—
195Dic61—42no—
196C2×#542—611no—
197#1479—612no—
198#1478—612no—
199#1458—611no—
200#1458D6111yescharacteristic, complemented
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—81no—
204C2×C61#178011yescharacteristic
205C2×C61#179511yescharacteristic
206C2×C61#179611yescharacteristic
207D61—82no—
208#542—1224no—
209#542—1224no—
210C2×C4⋊Q8—611no—
211C2×#126—1221no—
212C2×#124—611no—
213C2×#124—1221no—
214C2×#123—1221no—
215C2×C22⋊Q8—611no—
216C2×C22⋊Q8—1221no—
217C2×#121—611no—
218C2×C4×Q8—611no—
219C2×C4×D4—611no—
220C2×#117C2×D6111yescharacteristic
221#431—612no—
222#431C2×D6112yes—
223Q8⋊C8—1224no—
224D4⋊C8—1224no—
225#324—21no—
226#324—1221no—
227#323—21no—
228#323—1221no—
229C61#832511yescharacteristic, complemented, Sylow
230C22×Q8—611no—
231C22×Q8—1221no—
232C22×D4—611no—
233C22×D4—1221no—
234C23×C4—1221no—
235C23×C4C4×D6111yescharacteristic
236C4⋊Q8—1222no—
237C4⋊Q8—1222no—
238#126—1228no—
239#124—1222no—
240#124—1222no—
241#124—1228no—
242#123—1228no—
243C22⋊Q8—1228no—
244C22⋊Q8—1222no—
245C22⋊Q8—1222no—
246#121—1222no—
247#121—1222no—
248C4×Q8—614no—
249C4×Q8—1222no—
250C4×D4—614no—
251C4×D4—1222no—
252#117—22no—
253#117—22no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4D24411yescharacteristic
260C2×C4⋊C4C22⋊D6111yescharacteristic
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4D24411yescharacteristic
267C2×C22⋊C4C22⋊D6111yescharacteristic
268C2×C42—1221no—
269C2×C42C4×D6111yescharacteristic
270C2×C42C22×D6111yescharacteristic
271C4⋊C8—22no—
272C4⋊C8—1222no—
273C22⋊C8—22no—
274C22⋊C8—1222no—
275C4×C8—22no—
276C4×C8—1222no—
277C24—1221no—
278C24#1175819811yescharacteristic
279C2×Q8—612no—
280C2×Q8—1224no—
281C2×Q8—1221no—
282C2×Q8—1222no—
283C2×Q8—1222no—
284C2×Q8—4882no—
285C2×D4—612no—
286C2×D4—1224no—
287C2×D4—1221no—
288C2×D4—1222no—
289C2×D4—1222no—
290C2×D4—4882no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—1224no—
299C22×C4—4882no—
300C22×C4#1175819811yescharacteristic
301C22×C4#1175819811yescharacteristic
302C22×C4#1175819811yescharacteristic
303C22×C4C2×C4×D6111yescharacteristic
304C22×C4C2×D24411yescharacteristic
305C22×C4C2×C22⋊D6111yescharacteristic
306C2×C8—42no—
307C2×C8—2442no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—612no—
311C4⋊C4—612no—
312C4⋊C4—1224no—
313C4⋊C4—1224no—
314C4⋊C4—1221no—
315C4⋊C4—1221no—
316C4⋊C4—1224no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—1224no—
320C22⋊C4—1224no—
321C22⋊C4—1224no—
322C22⋊C4—1224no—
323C22⋊C4—1224no—
324C42—21no—
325C42—21no—
326C42—1224no—
327C42D4×D6112yes—
328C42D4×D6112yes—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—1222no—
335C23—2442no—
336C23—2442no—
337C23—2441no—
338C23—2441no—
339C23C2×#1175819811yescharacteristic
340Q8—1224no—
341Q8—4884no—
342Q8—2442no—
343D4—1224no—
344D4—4884no—
345D4—2442no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—1222no—
358C2×C4—1222no—
359C2×C4—1222no—
360C2×C4—4884no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4C22⋊C4×D6111yescharacteristic
366C2×C4C22⋊C4×D6111yescharacteristic
367C2×C4#1451974512yes—
368C2×C4#1451974712yes—
369C2×C4#1451978312yes—
370C2×C4#1451978512yes—
371C8—82no—
372C8—4882no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—81no—
380C22—4882no—
381C22—4882no—
382C22—2442no—
383C22—2442no—
384C22—2442no—
385C22#1683961711yescharacteristic
386C22#1683972811yescharacteristic
387C22#1683973011yescharacteristic
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—2442no—
394C4—2442no—
395C4—2442no—
396C4#1683972712yes—
397C4#1683972712yes—
398C2—21no—
399C2—21no—
400C2—81no—
401C2—4882no—
402C2#2144573011yescharacteristic
403C2#2144579011yescharacteristic
404C2#2144579211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1458C2
socleC22×C61#379
prime radicalC2×#14519751C4
cosocleC2×C42×C61C23
hypercenter#1458D61
hypocenterC61#8325
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #21444213C2semidirect
3. #21444211C2semidirect
4. #21444133C2semidirect
5. C61×#1458C2semidirect
200. #1458D61semidirect
229. C61#8325semidirect

Maximal subgroups

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

subgroupindex
C2×#168403832
#214442132
#214442112
#214441332
C61×#14582
#832561