Finite Group Databasebeta

← #29571475

Subgroups — #29571475

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840319C211yesmaximal, characteristic
2#21445887C211yesmaximal, characteristic
3C2×#16839498C214yesmaximal
4C61×#1434C211yesmaximal, characteristic
5#16840319—28no—
6C2×#14519777C411yescharacteristic, complemented
7C2×#14519777C411yescharacteristic, complemented
8C2×#14519769—21no—
9C2×#14519756—21no—
10C2×#14519756C2212yes—
11C2×C4×Dic122—21no—
12C2×C4×Dic122C412yescomplemented
13C2×#14519736—21no—
14C2×C22⋊#92418C2212yes—
15#16839517—21no—
16#16839517—21no—
17#16839498—28no—
18C22×C42×C61C2211yescharacteristic
19C2×#95×C61—21no—
20C2×#95×C61—21no—
21C2×#95×C61C2212yes—
22C22×Dic122—21no—
23C22×Dic122C2×C412yes—
24C22×#92418—22no—
25#14519777—24no—
26#14519777—24no—
27#14519769—28no—
28#14519756—28no—
29#14519756—24no—
30#14519756D418yes—
31C4×Dic122—28no—
32C4×Dic122—24no—
33C4×Dic122—24no—
34#14519736—28no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61C2×C412yes—
39C2×#11758196—22no—
40C2×#11758196—22no—
41C2×C22⋊Dic61—22no—
42C2×C4×Dic61—22no—
43C22⋊#92418—24no—
44C23×C4×C61—22no—
45C23×C4×C61D411yescharacteristic
46C23×C4×C61D411yescharacteristic
47C23×C4×C61C2311yescharacteristic
48C2×C42×C61—24no—
49C2×C42×C61C2×C411yescharacteristic
50C2×C42×C61C2×C411yescharacteristic
51C2×C42×C61D411yescharacteristic
52C2×C42×C61D411yescharacteristic
53#95×C61—24no—
54#95×C61—24no—
55#95×C61—42no—
56C24×C61C2×D411yescharacteristic
57C22×C4×C61—22no—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—22no—
61C22×C4×C61—24no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61—42no—
66C22×C4×C61—42no—
67C22×C4×C61—42no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61—42no—
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61SD1612yes—
76C22×C4×C61Q1612yes—
77C22×C4×C61C22×C411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D414yes—
80C22×Dic61—22no—
81C22×Dic61—22no—
82C2×Dic122—24no—
83C2×Dic122—22no—
84C2×Dic122—42no—
85C2×Dic122—42no—
86C2×Dic122—42no—
87C2×Dic122—42no—
88C2×Dic122C22⋊C414yes—
89C42×C61—22no—
90C42×C61—22no—
91C42×C61—22no—
92C42×C61—22no—
93C22⋊Dic61—28no—
94C4⋊Dic61—24no—
95C4⋊Dic61—24no—
96C4⋊Dic61—24no—
97C4⋊Dic61—22no—
98C4⋊Dic61C22⋊C414yes—
99#11758196—28no—
100#11758196—28no—
101C4×Dic61—28no—
102C2×#92418—42no—
103C2×#92418—42no—
104C2×#92418—42no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61C2×C22⋊C411yescharacteristic
110C23×C61#12012yes—
111C23×C61C2×SD1611yescharacteristic
112C23×C61C2×Q1611yescharacteristic
113C23×C61#13712yes—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
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—24no—
125C2×C4×C61—24no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—44no—
130C2×C4×C61—44no—
131C2×C4×C61—44no—
132C2×C4×C61—44no—
133C2×C4×C61—44no—
134C2×C4×C61—82no—
135C2×C4×C61—82no—
136C2×C4×C61—82no—
137C2×C4×C61—82no—
138C2×C4×C61#10312yes—
139C2×C4×C61#10312yes—
140C2×C4×C61#10412yes—
141C2×C4×C61C2×C22⋊C412yes—
142C2×C4×C61#12012yes—
143C2×Dic61—24no—
144C2×Dic61—24no—
145C2×Dic61—42no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149Dic122—44no—
150Dic122—48no—
151Dic122—44no—
152#92418—82no—
153#8333—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—22no—
157C22×C61—22no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—22no—
161C22×C61—22no—
162C22×C61#37911yescharacteristic
163C22×C61#41011yescharacteristic
164C22×C61#41011yescharacteristic
165C22×C61C2×#10311yescharacteristic
166C22×C61#41911yescharacteristic
167C22×C61C2×#10411yescharacteristic
168C22×C61#42111yescharacteristic
169C22×C61#44812yes—
170C22×C61#45112yes—
171C4×C61—22no—
172C4×C61—22no—
173C4×C61—22no—
174C4×C61—22no—
175C4×C61—22no—
176C4×C61—41no—
177C4×C61—84no—
178Dic61—44no—
179Dic61—44no—
180C2×#533—611no—
181#1817—611no—
182C2×#328—614no—
183#1434D6111yescharacteristic
184C2×C61—22no—
185C2×C61—22no—
186C2×C61#150311yescharacteristic
187C2×C61#150611yescharacteristic
188C2×C61#151511yescharacteristic
189C2×C61#151811yescharacteristic
190C2×C61#178211yescharacteristic
191C2×C61#178711yescharacteristic
192C2×C61#178911yescharacteristic
193#533—1228no—
194C2×C4⋊Q8—611no—
195C2×C4⋊Q8—611no—
196C2×#125—1221no—
197C2×C22⋊Q8—612no—
198C2×C22⋊Q8—1221no—
199C2×C4×Q8—612no—
200C2×C4×Q8—1221no—
201C2×#117—1221no—
202C22×C42C2×D6111yescharacteristic
203C2×C22⋊C8—612no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95C2×D6112yes—
207#336—1221no—
208#336—1221no—
209#328—1228no—
210C61#833311yescharacteristic, complemented, Sylow
211C22×Q8—612no—
212C22×Q8—1221no—
213C23×C4—22no—
214C23×C4C22⋊D6111yescharacteristic
215C23×C4C22⋊D6111yescharacteristic
216C23×C4C22×D6111yescharacteristic
217C22×C8—1222no—
218C4⋊Q8—1224no—
219C4⋊Q8—1224no—
220#125—1228no—
221C22⋊Q8—618no—
222C22⋊Q8—1228no—
223C22⋊Q8—1224no—
224C4×Q8—1224no—
225C4×Q8—1224no—
226C4×Q8—1228no—
227#117—1228no—
228C2×C4⋊C4—612no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1222no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1222no—
234C2×C22⋊C4—1222no—
235C2×C42—24no—
236C2×C42—1222no—
237C2×C42C4×D6111yescharacteristic
238C2×C42C4×D6111yescharacteristic
239C2×C42D24411yescharacteristic
240C2×C42D24411yescharacteristic
241C22⋊C8—1224no—
242#95—24no—
243#95—24no—
244#95—42no—
245C24C2×C22⋊D6111yescharacteristic
246C2×Q8—614no—
247C2×Q8—1224no—
248C2×Q8—1222no—
249C2×Q8—2442no—
250C2×Q8—2442no—
251C2×Q8—2442no—
252C2×Q8—2442no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—1222no—
268C22×C4—1222no—
269C22×C4#1175819811yescharacteristic
270C22×C4#1175819811yescharacteristic
271C22×C4#1175819811yescharacteristic
272C22×C4#1175819811yescharacteristic
273C22×C4#1175820012yes—
274C22×C4#1175820212yes—
275C22×C4C2×C4×D6111yescharacteristic
276C22×C4C2×D24411yescharacteristic
277C22×C4D4×D6114yes—
278C2×C8—2442no—
279C2×C8—2442no—
280C2×C8—2442no—
281C4⋊C4—614no—
282C4⋊C4—1228no—
283C4⋊C4—1224no—
284C4⋊C4—1224no—
285C4⋊C4—1224no—
286C4⋊C4—1222no—
287C4⋊C4—1228no—
288C22⋊C4—1228no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—1228no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C2×#1175819811yescharacteristic
299C23#1451974512yes—
300C23C2×#1175820011yescharacteristic
301C23C2×#1175820211yescharacteristic
302C23#1451980712yes—
303Q8—2444no—
304Q8—2444no—
305Q8—2448no—
306C2×C4—21no—
307C2×C4—21no—
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—24no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—1224no—
331C2×C4—1224no—
332C2×C4—2442no—
333C2×C4—2442no—
334C2×C4—2442no—
335C2×C4—2442no—
336C2×C4#1451970012yes—
337C2×C4#1451970012yes—
338C2×C4#1451970512yes—
339C2×C4C22⋊C4×D6112yes—
340C2×C4#1451974712yes—
341C8—4882no—
342C22—22no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—22no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22#1683961711yescharacteristic
351C22#1683973511yescharacteristic
352C22#1683973511yescharacteristic
353C22C2×#1451970011yescharacteristic
354C22#1683977011yescharacteristic
355C22C2×#1451970511yescharacteristic
356C22#1683978011yescharacteristic
357C22#1683990012yes—
358C22#1683991712yes—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—41no—
365C4—84no—
366C4—2444no—
367C4—2444no—
368C2—22no—
369C2—22no—
370C2#2144431711yescharacteristic
371C2#2144433011yescharacteristic
372C2#2144436911yescharacteristic
373C2#2144438211yescharacteristic
374C2#2144573611yescharacteristic
375C2#2144575311yescharacteristic
376C2#2144576411yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1434C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1434D61
hypocenterC61#8333
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
6. C2×#14519777C4semidirect
7. C2×#14519777C4semidirect
12. C2×C4×Dic122C4semidirect
210. C61#8333semidirect

Maximal subgroups

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

subgroupindex
C2×#168403192
#214458872
C2×#168394982
C61×#14342
#833361