Finite Group Databasebeta

← #29572538

Subgroups — #29572538

Order 15616.

720 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 107 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#21444535C212yesmaximal, complemented
3#21444442C212yesmaximal, complemented
4#21444153C211yesmaximal, characteristic, complemented
5C61×#1464C211yesmaximal, characteristic, complemented
6#16840383—24no—
7#16840383—24no—
8C2×#14519777C2211yescharacteristic
9C2×#14519772—21no—
10C2×#14519765C2211yescharacteristic
11C2×#14519763—21no—
12C2×#14519759—21no—
13C2×#14519756C411yescharacteristic
14C2×#14519755—21no—
15C2×#14519751C411yescharacteristic
16C2×C4×Dic122C411yescharacteristic
17C2×C4×D244C411yescharacteristic
18#16839819C2212yes—
19#16839507—24no—
20#16839501—24no—
21#16839488—21no—
22#16839486—21no—
23C2×#117×C61C2211yescharacteristic
24C61×#432C2212yes—
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#14519777D414yes—
34#14519772—28no—
35#14519765—22no—
36#14519765D414yes—
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×Dic122—22no—
46C4×D244—22no—
47C4×D244—22no—
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—
61#14519688—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—
73#97×C61—22no—
74C24×C61C22⋊C411yescharacteristic
75C23×D61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—42no—
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C2×C22⋊D61—24no—
86C2×C22⋊D61—82no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C2×D244—21no—
91C2×D244—42no—
92C2×D244C22⋊C412yes—
93C2×C4×D61—24no—
94C2×C4×D61—82no—
95C2×Dic122—21no—
96C2×Dic122—24no—
97C2×Dic122—42no—
98C2×Dic122—82no—
99C2×Dic122C22⋊C412yes—
100C2×C8×C61—42no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C22⋊C4×C61—22no—
104C22⋊C4×C61—22no—
105C42×C61—21no—
106C42×C61—21no—
107C42×C61C2×D412yes—
108C42×C61C2×D412yes—
109C22⋊Dic61—24no—
110#11758198—24no—
111#11758198—24no—
112#11758198—24no—
113#11758198—24no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61—24no—
117C4⋊Dic61C22⋊C412yes—
118C4⋊Dic61C22⋊C412yes—
119#11758196—24no—
120#11758196—24no—
121C4×Dic61—24no—
122C2×#92418—42no—
123C23×C61—21no—
124C23×C61—22no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61C2×C22⋊C411yescharacteristic
128C22×D61—22no—
129C22×D61—42no—
130C22×D61—42no—
131C22×D61—42no—
132C2×C4×C61—22no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—22no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61C2×C22⋊C411yescharacteristic
144C2×C4×C61C2×C22⋊C411yescharacteristic
145C2×C4×C61#12012yes—
146C2×C4×C61#12012yes—
147C22⋊D61—84no—
148C2×Dic61—22no—
149C2×Dic61—22no—
150C2×Dic61—22no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—42no—
154D244—42no—
155D244—42no—
156C4×D61—84no—
157Dic122—42no—
158Dic122—42no—
159Dic122—84no—
160C8×C61—82no—
161#92418—82no—
162#8523—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—82no—
169C22×C61#37911yescharacteristic
170C22×C61#41312yes—
171C2×D61—42no—
172C2×D61—42no—
173C2×D61—42no—
174C2×D61—84no—
175C4×C61—22no—
176C4×C61—22no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—41no—
180C4×C61—41no—
181C4×C61—41no—
182Dic61—42no—
183Dic61—42no—
184Dic61—42no—
185C2×#542—611no—
186#1556—612no—
187#1533—612no—
188#1464—611no—
189#1464D6111yescharacteristic, complemented
190C2×C61—22no—
191C2×C61—81no—
192C2×C61#178511yescharacteristic
193C2×C61#179511yescharacteristic
194C2×C61#179611yescharacteristic
195D61—82no—
196#542—1224no—
197#542—1224no—
198C2×C4⋊Q8—611no—
199C2×#126—1221no—
200C2×#124—611no—
201C2×#124—1221no—
202C2×#123—1221no—
203C2×C22⋊Q8—611no—
204C2×C22⋊Q8—1221no—
205C2×#121—611no—
206C2×C4×Q8—611no—
207C2×C4×D4—611no—
208C2×#117C2×D6111yescharacteristic
209#432—612no—
210#432C2×D6112yes—
211#332—1224no—
212#329—1224no—
213#324—21no—
214#324—1221no—
215#323—21no—
216#323—1221no—
217C61#852311yescharacteristic, complemented, Sylow
218C22×Q8—611no—
219C22×Q8—1221no—
220C22×D4—611no—
221C22×D4—1221no—
222C23×C4—1221no—
223C23×C4C4×D6111yescharacteristic
224C4⋊Q8—614no—
225C4⋊Q8—1222no—
226#126—1228no—
227#124—614no—
228#124—1228no—
229#124—1222no—
230#123—1228no—
231C22⋊Q8—1228no—
232C22⋊Q8—1222no—
233C22⋊Q8—1222no—
234#121—1222no—
235#121—1222no—
236C4×Q8—1222no—
237C4×Q8—1222no—
238C4×D4—1222no—
239C4×D4—1222no—
240#117—22no—
241#117—22no—
242C2×C4⋊C4—611no—
243C2×C4⋊C4—611no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4D24411yescharacteristic
248C2×C4⋊C4C22⋊D6111yescharacteristic
249C2×C22⋊C4—1221no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4D24411yescharacteristic
255C2×C22⋊C4C22⋊D6111yescharacteristic
256C2×C42—1221no—
257C2×C42C4×D6111yescharacteristic
258C2×C42C22×D6111yescharacteristic
259C4⋊C8—22no—
260C4⋊C8—1222no—
261C22⋊C8—22no—
262C22⋊C8—1222no—
263#97—22no—
264#97—1222no—
265C24—1221no—
266C24#1175819811yescharacteristic
267C2×Q8—612no—
268C2×Q8—1221no—
269C2×Q8—1224no—
270C2×Q8—4882no—
271C2×Q8—2442no—
272C2×D4—612no—
273C2×D4—1224no—
274C2×D4—1221no—
275C2×D4—4882no—
276C2×D4—2442no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—42no—
280C22×C4—1224no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—4882no—
285C22×C4#1175819811yescharacteristic
286C22×C4#1175819811yescharacteristic
287C22×C4#1175819811yescharacteristic
288C22×C4C2×C4×D6111yescharacteristic
289C22×C4C2×D24411yescharacteristic
290C22×C4C2×C22⋊D6111yescharacteristic
291C2×C8—42no—
292C2×C8—2442no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—612no—
296C4⋊C4—612no—
297C4⋊C4—1224no—
298C4⋊C4—1224no—
299C4⋊C4—1221no—
300C4⋊C4—1221no—
301C4⋊C4—1224no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—1224no—
305C22⋊C4—1224no—
306C22⋊C4—1224no—
307C22⋊C4—1224no—
308C22⋊C4—1224no—
309C42—21no—
310C42—21no—
311C42—1224no—
312C42D4×D6112yes—
313C42D4×D6112yes—
314C23—21no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23—1222no—
319C23—2442no—
320C23—2442no—
321C23—2442no—
322C23C2×#1175819811yescharacteristic
323Q8—4884no—
324Q8—2442no—
325Q8—2442no—
326D4—4884no—
327D4—2442no—
328D4—2442no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—1222no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—4884no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2442no—
347C2×C4C22⋊C4×D6111yescharacteristic
348C2×C4C22⋊C4×D6111yescharacteristic
349C2×C4#1451974512yes—
350C2×C4#1451974712yes—
351C8—82no—
352C8—4882no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—82no—
359C22—4884no—
360C22—2442no—
361C22—2442no—
362C22—2442no—
363C22#1683961711yescharacteristic
364C22#1683974412yes—
365C4—21no—
366C4—21no—
367C4—22no—
368C4—22no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—2442no—
373C4—2442no—
374C4—2442no—
375C2—22no—
376C2—81no—
377C2—4882no—
378C2#2144574511yescharacteristic
379C2#2144579011yescharacteristic
380C2#2144579211yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1464C2
socleC22×C61#379
prime radicalC2×#14519751C4
cosocleC2×C42×C61C23
hypercenter#1464D61
hypocenterC61#8523
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #21444535C2semidirect
3. #21444442C2semidirect
4. #21444153C2semidirect
5. C61×#1464C2semidirect
189. #1464D61semidirect
217. C61#8523semidirect

Maximal subgroups

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

subgroupindex
C2×#168403832
#214445352
#214444422
#214441532
C61×#14642
#852361