Finite Group Databasebeta

← #29568267

Subgroups — #29568267

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519756C211yesmaximal, characteristic
2#21445513C211yesmaximal, characteristic
3#21443644C212yesmaximal, complemented
4#21443641C212yesmaximal, complemented
5C61×#1749C211yesmaximal, characteristic
6C22×#11758194—21no—
7C2×#14519756C411yescharacteristic
8C2×C4×Dic122C411yescharacteristic
9#16839668C2211yescharacteristic
10#16839660C2211yescharacteristic
11#16839646—21no—
12#16839635—21no—
13#16839622C411yescharacteristic
14C4×#11758196—21no—
15C4×C4⋊Dic61C411yescharacteristic
16C4×C22⋊Dic61—21no—
17#16839517—21no—
18#16839517C2212yes—
19C22×C42×C61C2211yescharacteristic
20C2×C22⋊C8×C61—21no—
21C2×#97×C61—21no—
22C61×#336C2212yes—
23C22×Dic122C2×C411yescharacteristic
24C2×#11758194—24no—
25C2×#11758194—42no—
26C2×#11758194—42no—
27C22×#92418—22no—
28#14519756—22no—
29#14519756—22no—
30C4×Dic122—42no—
31C2×C4⋊Dic61C2×C411yescharacteristic
32C2×C4⋊Dic61C2×C411yescharacteristic
33C2×C4⋊Dic61C2×C411yescharacteristic
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C4×Dic61—21no—
40C2×C4×Dic61—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46C23×C4×C61C2×C411yescharacteristic
47C23×C4×C61D411yescharacteristic
48C23×C4×C61D411yescharacteristic
49C22×C8×C61—22no—
50C2×C42×C61—42no—
51C2×C42×C61C2×C411yescharacteristic
52C2×C42×C61D411yescharacteristic
53C2×C42×C61D411yescharacteristic
54C2×C42×C61C2311yescharacteristic
55C22⋊C8×C61—24no—
56#97×C61—42no—
57C24×C61C22⋊C411yescharacteristic
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—22no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—22no—
66C22×C4×C61—41no—
67C22×C4×C61—41no—
68C22×C4×C61—41no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C2×Dic122—21no—
80C2×Dic122—41no—
81C2×Dic122—41no—
82C2×Dic122M1612yescomplemented
83C2×C8×C61—24no—
84C2×C8×C61—42no—
85C2×C8×C61—42no—
86C42×C61—22no—
87C42×C61—22no—
88C42×C61—41no—
89C42×C61—41no—
90C22⋊Dic61—24no—
91C22⋊Dic61—42no—
92C4⋊Dic61—21no—
93C4⋊Dic61—41no—
94C4⋊Dic61—41no—
95C4⋊Dic61M1612yescomplemented
96#11758196—24no—
97#11758196—42no—
98#11758196—41no—
99#11758196—41no—
100C4×Dic61—42no—
101C4×Dic61—42no—
102#11758194—48no—
103C2×#92418—24no—
104C2×#92418—42no—
105C2×#92418—42no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61#10412yes—
113C23×C61C2×C22⋊C411yescharacteristic
114C2×C4×C61—24no—
115C2×C4×C61—24no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—44no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61#9912yes—
133C2×C4×C61#10412yes—
134C2×C4×C61C2×M1611yescharacteristic
135C2×C4×C61C2×M1611yescharacteristic
136C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C61#13711yescharacteristic
138C2×C4×C61#13712yes—
139C2×Dic61—22no—
140C2×Dic61—22no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151Dic122—42no—
152Dic122—42no—
153C8×C61—44no—
154#92418—44no—
155#7634—611nomaximal, Sylow
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—22no—
159C22×C61—41no—
160C22×C61—41no—
161C22×C61—41no—
162C22×C61—41no—
163C22×C61#40711yescharacteristic
164C22×C61C2×#9911yescharacteristic
165C22×C61#41311yescharacteristic
166C22×C61#41711yescharacteristic
167C22×C61#41911yescharacteristic
168C22×C61C2×#10411yescharacteristic
169C22×C61C2×#10411yescharacteristic
170C4×C61—21no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—22no—
174C4×C61—21no—
175C4×C61—22no—
176C4×C61—22no—
177C4×C61—41no—
178Dic61—42no—
179Dic61—42no—
180Dic61—81no—
181Dic61—81no—
182C4×C22⋊Q8—611no—
183#1749D6111yescharacteristic
184#1728—611no—
185#1272—612no—
186#1271—612no—
187C2×C61—42no—
188C2×C61#146511yescharacteristic
189C2×C61#148911yescharacteristic
190C2×C61#149111yescharacteristic
191C2×C61#150411yescharacteristic
192C2×C61#150811yescharacteristic
193C2×C61#152411yescharacteristic
194C2×C61#155911yescharacteristic
195C22×M16—1221no—
196C2×C22⋊Q8—611no—
197C2×C4×Q8—611no—
198C22×C42C2×D6111yescharacteristic
199C2×C22⋊C8—21no—
200C2×#97—21no—
201#391—611no—
202#389—611no—
203#387—1221no—
204#385—1221no—
205#380—611no—
206C4×C4⋊C4—611no—
207C4×C4⋊C4—1221no—
208C4×C22⋊C4—1221no—
209#336—612no—
210#336—1221no—
211#336C2×D6112yes—
212C61#763411yescharacteristic, complemented, Sylow
213C22×Q8—611no—
214C23×C4C4×D6111yescharacteristic
215C23×C4D24411yescharacteristic
216C23×C4C22⋊D6111yescharacteristic
217C2×M16—1224no—
218C2×M16—2442no—
219C2×M16—2442no—
220C22×C8—22no—
221C22×C8—1222no—
222C22⋊Q8—1222no—
223C22⋊Q8—1222no—
224C4×Q8—2442no—
225C2×C4⋊C4—611no—
226C2×C4⋊C4—611no—
227C2×C4⋊C4—611no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4—1221no—
231C2×C22⋊C4—1221no—
232C2×C22⋊C4—1221no—
233C2×C42—42no—
234C2×C42—1221no—
235C2×C42—1221no—
236C2×C42C4×D6111yescharacteristic
237C2×C42D24411yescharacteristic
238C2×C42C22⋊D6111yescharacteristic
239C2×C42C22×D6111yescharacteristic
240C22⋊C8—24no—
241#97—42no—
242#95—1221no—
243#95—1221no—
244#95—1221no—
245#95—1221no—
246#95—1221no—
247C24#1175819811yescharacteristic
248C2×Q8—612no—
249C2×Q8—1221no—
250C2×Q8—2441no—
251C2×Q8—2441no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4#1175819811yescharacteristic
268C22×C4#1175819811yescharacteristic
269C22×C4#1175819811yescharacteristic
270C22×C4C2×C4×D6111yescharacteristic
271C22×C4C2×D24411yescharacteristic
272C22×C4C2×C22⋊D6111yescharacteristic
273M16—2448no—
274C2×C8—24no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—1224no—
278C2×C8—2442no—
279C2×C8—2442no—
280C4⋊C4—612no—
281C4⋊C4—1224no—
282C4⋊C4—1221no—
283C4⋊C4—2441no—
284C4⋊C4—2441no—
285C4⋊C4—2441no—
286C4⋊C4—2442no—
287C4⋊C4—2441no—
288C22⋊C4—1224no—
289C22⋊C4—2442no—
290C42—22no—
291C42—22no—
292C42—41no—
293C42—41no—
294C42—2442no—
295C42—2442no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#1451970312yes—
303C23C2×#1175819811yescharacteristic
304Q8—2442no—
305Q8—2442no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—1222no—
325C2×C4—1222no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4#1451969112yes—
337C2×C4#1451970512yes—
338C2×C4M16×D6111yescharacteristic
339C2×C4M16×D6111yescharacteristic
340C2×C4#1451980511yescharacteristic
341C2×C4#1451980712yes—
342C2×C4#1451981011yescharacteristic
343C8—44no—
344C8—2444no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#1683972511yescharacteristic
353C22C2×#1451969111yescharacteristic
354C22#1683974411yescharacteristic
355C22#1683976411yescharacteristic
356C22#1683977011yescharacteristic
357C22C2×#1451970311yescharacteristic
358C22C2×#1451970511yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—22no—
364C4—22no—
365C4—22no—
366C4—41no—
367C4—4881no—
368C4—4881no—
369C4—2442no—
370C4—2442no—
371C2—42no—
372C2#2144415511yescharacteristic
373C2#2144425911yescharacteristic
374C2#2144426511yescharacteristic
375C2#2144432411yescharacteristic
376C2#2144433811yescharacteristic
377C2#2144440511yescharacteristic
378C2#2144455011yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1749C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1749D61
hypocenterC61#7634
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #21443644C2semidirect
4. #21443641C2semidirect
82. C2×Dic122M16semidirect
95. C4⋊Dic61M16semidirect
212. C61#7634semidirect

Maximal subgroups

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

subgroupindex
C4×#145197562
#214455132
#214436442
#214436412
C61×#17492
#763461