Finite Group Databasebeta

← #29574713

Subgroups — #29574713

Order 15616.

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

normal subgroups only

This group has 600 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
1C4×#14519758C211yesmaximal, characteristic
2#21445885C212yesmaximal, complemented
3#21444137C211yesmaximal, characteristic
4#21443666C212yesmaximal, complemented
5C61×#2973C211yesmaximal, characteristic
6C2×#14519758C411yescharacteristic
7C22×C4×Dic61C2211yescharacteristic
8C2×C22⋊#92418C2212yes—
9#16839650—21no—
10#16839643—21no—
11#16839642C411yescharacteristic
12#16839634C411yescharacteristic
13#16839623C411yescharacteristic
14C4×#11758196—21no—
15C4×C22⋊Dic61—21no—
16C4×C22⋊Dic61C2211yescharacteristic
17#16839516—24no—
18#16839488—21no—
19#16839486—21no—
20C2×C4×D4×C61C2211yescharacteristic
21C61×#435—24no—
22C61×#432—21no—
23C61×#407—21no—
24C2×C22⋊C8×C61C2212yes—
25C23×Dic61C2×C411yescharacteristic
26C23×Dic61C2×C411yescharacteristic
27C22×#92418—22no—
28#14519758—22no—
29#14519758—22no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61C2×C411yescharacteristic
35C2×C22⋊Dic61C2×C411yescharacteristic
36C2×C4×Dic61—22no—
37C2×C4×Dic61—21no—
38C2×C4×Dic61—21no—
39C2×C4×Dic61—21no—
40C2×C4×Dic61—22no—
41C22⋊#92418—22no—
42C22⋊#92418—22no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48C22×D4×C61C2×C411yescharacteristic
49C23×C4×C61D411yescharacteristic
50C23×C4×C61C2311yescharacteristic
51C2×M16×C61—22no—
52C22×C8×C61D414yes—
53C4×D4×C61—22no—
54C4×D4×C61—22no—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61D411yescharacteristic
58C2×C42×C61D411yescharacteristic
59C4⋊C8×C61—22no—
60C22⋊C8×C61—22no—
61C22⋊C8×C61—22no—
62C22⋊C8×C61—22no—
63#97×C61—22no—
64C24×C61C22⋊C411yescharacteristic
65C24×C61C22×C411yescharacteristic
66C2×D4×C61—22no—
67C2×D4×C61—21no—
68C2×D4×C61—21no—
69C2×D4×C61—22no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61#4112yes—
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—22no—
85C22×Dic61—22no—
86C22×Dic61—21no—
87C22×Dic61—22no—
88C22×Dic61—21no—
89C22×Dic61—42no—
90C22×Dic61M1612yescomplemented
91M16×C61—82no—
92C2×C8×C61—24no—
93C2×C8×C61—44no—
94C4⋊C4×C61—22no—
95C22⋊C4×C61—22no—
96C22⋊C4×C61—22no—
97C42×C61—21no—
98C42×C61—21no—
99C22⋊Dic61—24no—
100C22⋊Dic61—21no—
101C22⋊Dic61—22no—
102C22⋊Dic61—42no—
103C22⋊Dic61M1612yescomplemented
104#11758196—24no—
105#11758196—42no—
106C4×Dic61—42no—
107C4×Dic61—41no—
108C4×Dic61—42no—
109C4×Dic61—41no—
110C2×#92418—44no—
111C2×#92418—42no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61C2×C22⋊C411yescharacteristic
120C23×C61C4×D412yes—
121C23×C61C2×M1611yescharacteristic
122C23×C61C2×M1611yescharacteristic
123C23×C61#13112yes—
124D4×C61—42no—
125D4×C61—42no—
126C2×C4×C61—24no—
127C2×C4×C61—22no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—41no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—81no—
139C2×C4×C61#12011yescharacteristic
140C2×C4×C61#12112yes—
141C2×C4×C61#12311yescharacteristic
142C2×Dic61—22no—
143C2×Dic61—24no—
144C2×Dic61—22no—
145C2×Dic61—22no—
146C2×Dic61—24no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—42no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—42no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C8×C61—44no—
159#92418—82no—
160#8968—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—22no—
163C22×C61—21no—
164C22×C61—22no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—81no—
173C22×C61#38611yescharacteristic
174C22×C61#40711yescharacteristic
175C22×C61#40811yescharacteristic
176C22×C61#43512yes—
177C22×C61#43612yes—
178C4×C61—22no—
179C4×C61—21no—
180C4×C61—41no—
181C4×C61—41no—
182C4×C61—41no—
183C4×C61#45812yes—
184Dic61—42no—
185Dic61—42no—
186Dic61—42no—
187Dic61—81no—
188#2973D6111yescharacteristic
189C4×#123—611no—
190#1817—612no—
191#1459—611no—
192#1282—612no—
193C2×C61—21no—
194C2×C61—21no—
195C2×C61—22no—
196C2×C61—22no—
197C2×C61—81no—
198C2×C61#187011yescharacteristic
199C2×C61#189011yescharacteristic
200C2×C61#189111yescharacteristic
201C2×#123—611no—
202C2×C4×D4C2×D6111yescharacteristic
203C22×C42—611no—
204#435—24no—
205#432—21no—
206#407—21no—
207C2×C22⋊C8—612no—
208C2×C22⋊C8C2×D6112yes—
209#388—1221no—
210#387—1221no—
211#386—611no—
212#385—611no—
213#381—611no—
214C4×C4⋊C4—1221no—
215C4×C22⋊C4—611no—
216C4×C22⋊C4—1221no—
217#336—1224no—
218#324—1221no—
219#323—1221no—
220C61#896811yescharacteristic, complemented, Sylow
221C22×D4C4×D6111yescharacteristic
222C23×C4—611no—
223C23×C4—611no—
224C23×C4C22⋊D6111yescharacteristic
225C23×C4C22×D6111yescharacteristic
226C2×M16—22no—
227C22×C8—1222no—
228C22×C8C22⋊D6114yes—
229#123—1222no—
230#123—1222no—
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4D24411yescharacteristic
236C2×C22⋊C4—611no—
237C2×C22⋊C4—611no—
238C2×C22⋊C4—1221no—
239C2×C22⋊C4—1221no—
240C2×C22⋊C4C4×D6111yescharacteristic
241C2×C22⋊C4D24411yescharacteristic
242C2×C42—1221no—
243C2×C42—1222no—
244C2×C42—1222no—
245C2×C42—1221no—
246C2×C42—1221no—
247C2×C42C22⋊D6111yescharacteristic
248C4⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—1222no—
253C22⋊C8—1222no—
254#97—22no—
255#95—1221no—
256#95—1221no—
257#95—1221no—
258#95—1221no—
259#95—1221no—
260C24#1175819811yescharacteristic
261C24C2×C4×D6111yescharacteristic
262C2×D4—22no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—612no—
271C22×C4—1221no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—1222no—
275C22×C4—1222no—
276C22×C4—1221no—
277C22×C4—1222no—
278C22×C4—1221no—
279C22×C4—2442no—
280C22×C4#1175819811yescharacteristic
281C22×C4#1175819811yescharacteristic
282C22×C4#1175819811yescharacteristic
283C22×C4C2×D24411yescharacteristic
284C22×C4#1175822212yes—
285C22×C4C2×C22⋊D6112yes—
286C22×C4C2×C22⋊D6111yescharacteristic
287M16—82no—
288C2×C8—24no—
289C2×C8—44no—
290C2×C8—2444no—
291C2×C8—2442no—
292C4⋊C4—22no—
293C4⋊C4—1224no—
294C4⋊C4—2442no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—612no—
298C22⋊C4—1224no—
299C22⋊C4—1221no—
300C22⋊C4—1222no—
301C22⋊C4—2442no—
302C42—21no—
303C42—21no—
304C42—2442no—
305C42—2441no—
306C42—2442no—
307C42—2441no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23C2×#1175819811yescharacteristic
316C23C4×C22⋊D6112yes—
317C23M16×D6111yescharacteristic
318C23M16×D6111yescharacteristic
319C23#1451978512yes—
320D4—42no—
321D4—42no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—1222no—
336C2×C4—1224no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—1224no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2442no—
347C2×C4—2441no—
348C2×C4—2442no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4#1451974611yescharacteristic
352C2×C4#1451975112yes—
353C2×C4#1451976011yescharacteristic
354C8—44no—
355C8—4882no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—81no—
368C22#1683964111yescharacteristic
369C22#1683972511yescharacteristic
370C22#1683973011yescharacteristic
371C22#1683983112yes—
372C22#1683983812yes—
373C4—22no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—4881no—
379C4—2442no—
380C4—2442no—
381C4—2442no—
382C4#1683995212yes—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—22no—
387C2—81no—
388C2#2144611611yescharacteristic
389C2#2144623211yescharacteristic
390C2#2144623711yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2973C2
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#2973D61
hypocenterC61#8968
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #21445885C2semidirect
4. #21443666C2semidirect
90. C22×Dic61M16semidirect
103. C22⋊Dic61M16semidirect
220. C61#8968semidirect

Maximal subgroups

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

subgroupindex
C4×#145197582
#214458852
#214441372
#214436662
C61×#29732
#896861