Finite Group Databasebeta

← #29571659

Subgroups — #29571659

Order 15616.

788 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16840472C211yesmaximal, characteristic
2#21444420C212yesmaximal, complemented
3#21444263C212yesmaximal
4#21444163C211yesmaximal, characteristic, complemented
5C4×#101×C61C211yesmaximal, characteristic, complemented
6#16840472—24no—
7#16840472—24no—
8C2×#14519777C411yescharacteristic
9C2×#14519769C411yescharacteristic
10C2×#14519769C2211yescharacteristic
11C2×#14519755—21no—
12C2×#14519755—21no—
13C2×#14519753—21no—
14C2×C4×D244C411yescharacteristic
15C2×C4×D244C411yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×C4⋊C4×D61—21no—
18#16839819C2212yes—
19#16839503—24no—
20#16839491—24no—
21#16839488—21no—
22#16839488—21no—
23C2×C4×Q8×C61C2211yescharacteristic
24C2×#101×C61—21no—
25C4×M16×C61C2212yes—
26C61×#343—21no—
27C22×Dic122—21no—
28C22×D244C2×C411yescharacteristic
29C22×C4×D61—21no—
30C22×C4×D61—21no—
31C22×C4×D61—21no—
32#14519777—22no—
33#14519777—22no—
34#14519769—22no—
35#14519769—22no—
36#14519769—22no—
37#14519769D414yes—
38#14519755—28no—
39#14519755—28no—
40#14519753—28no—
41C4×D244—22no—
42C4×D244—22no—
43C4×D244—22no—
44C4×D244—22no—
45C4×D244—22no—
46C4×D244D414yes—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61C2×C411yescharacteristic
49C2×C4⋊Dic61C2×C411yescharacteristic
50C2×C4⋊Dic61C2×C411yescharacteristic
51C2×#11758196—21no—
52C2×#11758196—21no—
53C2×#11758196—21no—
54C2×#11758196—21no—
55C4⋊C4×D61—28no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C4⋊#92418—22no—
60C22⋊#92418—22no—
61#14519688—22no—
62C22×Q8×C61D411yescharacteristic
63C2×M16×C61—22no—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C2×C4⋊C4×C61D411yescharacteristic
67C2×C4⋊C4×C61D411yescharacteristic
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C42×C61C2×C411yescharacteristic
70C2×C42×C61C2×C411yescharacteristic
71C2×C42×C61C2311yescharacteristic
72#101×C61—24no—
73#97×C61—22no—
74C4×C8×C61—22no—
75C23×D61—21no—
76C2×Q8×C61—22no—
77C2×Q8×C61—22no—
78C2×Q8×C61—41no—
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22×C411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C2×D244—21no—
90C2×D244—24no—
91C2×D244C22⋊C412yes—
92C2×C4×D61—24no—
93C2×C4×D61—24no—
94C2×C4×D61—24no—
95C2×C4×D61—42no—
96C2×C4×D61—42no—
97C2×C4×D61—82no—
98C2×C4×D61—82no—
99C2×Dic122—24no—
100C2×Dic122—82no—
101M16×C61—24no—
102M16×C61—42no—
103C2×C8×C61—42no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—22no—
107C42×C61—21no—
108C42×C61—21no—
109C42×C61—21no—
110C42×C61—21no—
111C42×C61C2×D412yes—
112C42×C61C2×D412yes—
113#11758198—24no—
114#11758198—24no—
115#11758198—24no—
116C4⋊Dic61—24no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61C22⋊C412yes—
121C4⋊Dic61C22⋊C412yes—
122C4⋊Dic61C22⋊C412yes—
123#11758196—24no—
124#11758196—24no—
125#11758196—24no—
126#11758196—24no—
127C2×#92418—42no—
128C23×C61C2×C22⋊C411yescharacteristic
129C22×D61—24no—
130C22×D61—22no—
131C22×D61—42no—
132C22×D61—41no—
133C22×D61—41no—
134Q8×C61—44no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—22no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61#10412yes—
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C2×C22⋊C411yescharacteristic
151C2×C4×C61#12012yes—
152C2×C4×C61#12012yes—
153C2×Dic61—22no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160D244—24no—
161D244—22no—
162D244#10114yescomplemented
163C4×D61—48no—
164C4×D61—84no—
165C4×D61—84no—
166Dic122—84no—
167C8×C61—44no—
168#92418—82no—
169#8372—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61#37911yescharacteristic
173C22×C61C2×#10411yescharacteristic
174C22×C61#42111yescharacteristic
175C2×D61—24no—
176C2×D61—44no—
177C2×D61—44no—
178C2×D61—42no—
179C2×D61—42no—
180C4×C61—22no—
181C4×C61—21no—
182C4×C61—22no—
183C4×C61—22no—
184C4×C61—41no—
185C4×C61—41no—
186C4×C61—41no—
187C4×C61C2×#10112yes—
188Dic61—42no—
189Dic61—42no—
190Dic61—42no—
191C2×#554—611no—
192C4×#101D6111yescharacteristic, complemented
193#1528—612no—
194#1490—612no—
195#1467—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61#178411yescharacteristic
199C2×C61#178911yescharacteristic
200C2×C61#179511yescharacteristic
201D61—44no—
202#554—1224no—
203#554—1224no—
204C2×C4⋊Q8—611no—
205C2×#125—611no—
206C2×#125—611no—
207C2×C22⋊Q8—1221no—
208C2×C22⋊Q8—1221no—
209C2×C22⋊Q8—1221no—
210C2×C4×Q8C2×D6111yescharacteristic
211C2×C4×D4—611no—
212C2×C4×D4—611no—
213C2×C4×D4—611no—
214C22×C4⋊C4—1221no—
215#432—612no—
216C2×#101—21no—
217C4×M16C2×D6112yes—
218#343—21no—
219#330—1224no—
220D4⋊C8—1224no—
221#324—1221no—
222#324—1221no—
223C61#837211yescharacteristic, complemented, Sylow
224C22×Q8—1221no—
225C22×Q8C22⋊D6111yescharacteristic
226C22×D4—611no—
227C23×C4—1221no—
228C23×C4—1221no—
229C23×C4—1221no—
230C2×M16—22no—
231C4⋊Q8—1222no—
232C4⋊Q8—1222no—
233#125—614no—
234#125—1222no—
235#125—1222no—
236#125—1222no—
237C22⋊Q8—1228no—
238C22⋊Q8—1228no—
239C22⋊Q8—1228no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×D4—614no—
243C4×D4—1222no—
244C4×D4—1222no—
245C4×D4—1222no—
246C4×D4—1222no—
247C4×D4—1222no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—611no—
250C2×C4⋊C4—611no—
251C2×C4⋊C4—1228no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4D24411yescharacteristic
258C2×C4⋊C4D24411yescharacteristic
259C2×C4⋊C4C22⋊D6111yescharacteristic
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C42C4×D6111yescharacteristic
264C2×C42C4×D6111yescharacteristic
265C2×C42C22×D6111yescharacteristic
266C4⋊C8—1222no—
267#101—24no—
268C22⋊C8—1222no—
269#97—22no—
270#97—1222no—
271C4×C8—22no—
272C24—1221no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—1224no—
277C2×Q8—4882no—
278C2×D4—612no—
279C2×D4—1224no—
280C2×D4—1221no—
281C22×C4—1224no—
282C22×C4—1224no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1224no—
287C22×C4—4882no—
288C22×C4—4882no—
289C22×C4—2442no—
290C22×C4—2442no—
291C22×C4#1175819811yescharacteristic
292C22×C4#1175819811yescharacteristic
293C22×C4#1175819811yescharacteristic
294C22×C4#1175819811yescharacteristic
295C22×C4C2×C4×D6111yescharacteristic
296C22×C4C2×D24411yescharacteristic
297C22×C4C2×C22⋊D6111yescharacteristic
298M16—24no—
299M16—42no—
300C2×C8—42no—
301C2×C8—2442no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—612no—
306C4⋊C4—612no—
307C4⋊C4—612no—
308C4⋊C4—1224no—
309C4⋊C4—1224no—
310C4⋊C4—1224no—
311C4⋊C4—1221no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C4⋊C4—1224no—
315C4⋊C4—1224no—
316C22⋊C4—1224no—
317C22⋊C4—1224no—
318C22⋊C4—1224no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42D4×D6112yes—
324C42D4×D6112yes—
325C23—1224no—
326C23—1222no—
327C23—2442no—
328C23—2441no—
329C23—2441no—
330C23C2×#1175819811yescharacteristic
331Q8—44no—
332Q8—4884no—
333D4—614no—
334D4—1224no—
335D4—1222no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—1222no—
350C2×C4—1222no—
351C2×C4—1222no—
352C2×C4—4884no—
353C2×C4—4884no—
354C2×C4—2448no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4#1451970512yes—
360C2×C4C22⋊C4×D6111yescharacteristic
361C2×C4C22⋊C4×D6111yescharacteristic
362C2×C4#1451974512yes—
363C2×C4#1451974712yes—
364C8—44no—
365C8—4882no—
366C22—21no—
367C22—21no—
368C22—1224no—
369C22—2444no—
370C22—2444no—
371C22—2442no—
372C22—2442no—
373C22#1683961711yescharacteristic
374C22C2×#1451970511yescharacteristic
375C22#1683978011yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—2442no—
384C4—2442no—
385C4—2442no—
386C4#101×D6112yes—
387C2—21no—
388C2—21no—
389C2—2444no—
390C2#2144574111yescharacteristic
391C2#2144576411yescharacteristic
392C2#2144579011yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC4×#101×C61C2
socleC22×C61#379
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenterC4×#101D61
hypocenterC61#8372
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. #21444420C2semidirect
4. #21444163C2semidirect
5. C4×#101×C61C2semidirect
162. D244#101semidirect
192. C4×#101D61semidirect
223. C61#8372semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214444202
#214442632
#214441632
C4×#101×C612
#837261