Finite Group Databasebeta

← #30600166

Subgroups — #30600166

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840781C211yesmaximal, characteristic
2#21461674C211yesmaximal, characteristic
3C2×Q8⋊#92420C212yesmaximal, complemented
4#21460655C212yesmaximal, complemented
5#21448357C211yesmaximal, characteristic
6#16840781—24no—
7C2×C4⋊#92420—21no—
8C2×C4⋊#92420C2212yes—
9C2×#14519834—21no—
10C2×#14519834—21no—
11C2×C4×#92420C2212yes—
12#16840628—21no—
13Q8⋊#92420—24no—
14C2×#14519753C411yescharacteristic
15C2×Q8×Dic61C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839660C411yescharacteristic
18#16839649—21no—
19#16839636—21no—
20#16839619C411yescharacteristic
21C4⋊C4×Dic61C2211yescharacteristic
22#16839613—21no—
23#16839609—21no—
24C61×#391C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27C4⋊#92420—24no—
28C4⋊#92420—22no—
29C4⋊#92420—22no—
30#14519834—24no—
31#14519834—42no—
32C4×#92420—22no—
33C4×#92420D414yes—
34C22×C4×D61C2×C411yescharacteristic
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37#14519753—22no—
38#14519753—22no—
39Q8×Dic61—22no—
40Q8×Dic61—22no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×#11758196—21no—
45C2×#11758196—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61C2311yescharacteristic
53C42×D61—22no—
54C42×D61—22no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C22×Q8×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C42×C61C2×C411yescharacteristic
61#95×C61—21no—
62#95×C61—21no—
63C23×D61C22⋊C411yescharacteristic
64C2×Q8×C61—21no—
65C2×Q8×C61—22no—
66C2×Q8×C61—22no—
67C2×Q8×C61M1612yes—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C2×D411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—22no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—22no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×C4×D61SD1612yes—
89C2×C4×D61Q1612yes—
90C2×#92420—24no—
91C2×#92420—42no—
92C2×#92420—42no—
93C2×#92420—42no—
94C2×#92420—42no—
95C2×#92420—42no—
96C4⋊C4×C61—22no—
97C42×C61—22no—
98#11758198—24no—
99#11758198—42no—
100C4⋊Dic61—21no—
101C4⋊Dic61—22no—
102C4⋊Dic61—41no—
103C4⋊Dic61—41no—
104C4⋊Dic61M1612yes—
105#11758196—24no—
106#11758196—42no—
107C4×Dic61—21no—
108C4×Dic61—22no—
109C4×Dic61—21no—
110C4×Dic61—42no—
111C4×Dic61—42no—
112C4×Dic61C2×D412yes—
113C4×Dic61#4112yes—
114C23×C61C2×C22⋊C411yescharacteristic
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—21no—
121C22×D61#10312yes—
122Q8×C61—24no—
123Q8×C61—42no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—22no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61#10312yes—
136C2×C4×C61C4×D412yes—
137C2×C4×C61C2×M1611yescharacteristic
138C2×C4×C61C2×M1611yescharacteristic
139C2×C4×C61#13112yes—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—22no—
145C2×Dic61—21no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61#12011yescharacteristic
152C2×Dic61#12112yes—
153C2×Dic61#12311yescharacteristic
154C2×Dic61C2×SD1611yescharacteristic
155C2×Dic61C2×Q1611yescharacteristic
156C2×Dic61#13512yes—
157C4×D61—24no—
158C4×D61—24no—
159C4×D61—42no—
160C4×D61—42no—
161C4×D61—42no—
162C4×D61—42no—
163#92420—44no—
164#92420—82no—
165#9075—611nomaximal, Sylow
166C22×C61#38611yescharacteristic
167C22×C61#40711yescharacteristic
168C22×C61#40811yescharacteristic
169C22×C61C2×#10311yescharacteristic
170C22×C61#41611yescharacteristic
171C22×C61#42111yescharacteristic
172C22×C61#42111yescharacteristic
173C2×D61—22no—
174C2×D61—22no—
175C2×D61—22no—
176C2×D61—41no—
177C2×D61—41no—
178C2×D61—41no—
179C2×D61—41no—
180C4×C61—22no—
181C4×C61—42no—
182C4×C61—41no—
183C4×C61—81no—
184C4×C61#43512yes—
185C4×C61#43612yes—
186Dic61—21no—
187Dic61—21no—
188Dic61—41no—
189Dic61—42no—
190Dic61—81no—
191Dic61#46512yes—
192Dic61#46712yes—
193C2×#431—611no—
194C4×C22⋊Q8—611no—
195#1797—611no—
196C2×Q8⋊C8—612no—
197#1279—612no—
198C2×C61#148111yescharacteristic
199C2×C61#148211yescharacteristic
200C2×C61#148711yescharacteristic
201C2×C61#148811yescharacteristic
202C2×C61#187011yescharacteristic
203C2×C61#189211yescharacteristic
204C2×C61#189711yescharacteristic
205D61—42no—
206C2×C22⋊Q8—611no—
207C2×C4×Q8—611no—
208C22×C42—611no—
209#431—1224no—
210C2×C4⋊C8—612no—
211C2×C4⋊C8—1221no—
212C2×C22⋊C8—1221no—
213C2×C22⋊C8—1221no—
214C2×C4×C8—612no—
215#391#692311yescharacteristic
216#389—611no—
217#387—1221no—
218#385—1221no—
219#380—611no—
220C4×C4⋊C4—611no—
221C4×C4⋊C4—1221no—
222C4×C22⋊C4—1221no—
223#336—1221no—
224Q8⋊C8—1224no—
225C61#907511yescharacteristic, complemented, Sylow
226C22×Q8C2×#692311yescharacteristic
227C23×C4—611no—
228C23×C4—611no—
229C23×C4—611no—
230C22×C8—1222no—
231C22×C8—1222no—
232C22⋊Q8—1222no—
233C22⋊Q8—1222no—
234C4×Q8—1222no—
235C4×Q8—1222no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—611no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4C2×#692311yescharacteristic
242C2×C22⋊C4—1221no—
243C2×C22⋊C4—1221no—
244C2×C42—611no—
245C2×C42—611no—
246C2×C42—611no—
247C2×C42—1222no—
248C2×C42—1222no—
249C2×C42—1221no—
250C2×C42—1221no—
251C2×C42C2×#692311yescharacteristic
252C4⋊C8—1222no—
253C4⋊C8—1222no—
254C4⋊C8—1224no—
255C22⋊C8—1224no—
256C22⋊C8—2442no—
257C4×C8—614no—
258C4×C8—1222no—
259#95—21no—
260#95—21no—
261#95—1221no—
262#95—1221no—
263#95—1221no—
264C24—611no—
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8#1175821312yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—612no—
274C22×C4—612no—
275C22×C4—611no—
276C22×C4—1222no—
277C22×C4—1222no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1222no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—1222no—
285C22×C4—1221no—
286C22×C4—2441no—
287C22×C4—2441no—
288C22×C4C22⋊#692311yescharacteristic
289C22×C4C22⋊#692311yescharacteristic
290C22×C4C22×#692311yescharacteristic
291C2×C8—1224no—
292C2×C8—2442no—
293C2×C8—2442no—
294C2×C8—2442no—
295C2×C8—2442no—
296C2×C8—2442no—
297C4⋊C4—22no—
298C4⋊C4—612no—
299C4⋊C4—1224no—
300C4⋊C4—1221no—
301C4⋊C4—1222no—
302C4⋊C4—2442no—
303C4⋊C4—2441no—
304C4⋊C4—2441no—
305C22⋊C4—1224no—
306C22⋊C4—2442no—
307C42—22no—
308C42—612no—
309C42—612no—
310C42—1221no—
311C42—1221no—
312C42—1222no—
313C42—2442no—
314C42—2442no—
315C23—612no—
316C23—1221no—
317C23—1221no—
318C23—1221no—
319C23—1221no—
320C23—1221no—
321C23—1221no—
322C23C2×C22⋊#692311yescharacteristic
323Q8—24no—
324Q8—42no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—612no—
337C2×C4—611no—
338C2×C4—611no—
339C2×C4—612no—
340C2×C4—611no—
341C2×C4—611no—
342C2×C4—1224no—
343C2×C4—1224no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—1221no—
347C2×C4—1222no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—2442no—
351C2×C4—2442no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4#1451984012yes—
360C2×C4D4×#692312yes—
361C2×C4D61⋊M1611yescharacteristic
362C2×C4C2×#1175821311yescharacteristic
363C2×C4#1451986412yes—
364C8—4882no—
365C8—2444no—
366C22—1222no—
367C22—1222no—
368C22—1222no—
369C22—2441no—
370C22—2441no—
371C22—2441no—
372C22—2441no—
373C22#1684071011yescharacteristic
374C22#1684073311yescharacteristic
375C22#1684073611yescharacteristic
376C22C2×#1451984011yescharacteristic
377C22#1684075211yescharacteristic
378C22#1684075911yescharacteristic
379C22#1684076111yescharacteristic
380C4—22no—
381C4—42no—
382C4—41no—
383C4—81no—
384C4—612no—
385C4—612no—
386C4—1221no—
387C4—1221no—
388C4—4881no—
389C4—2441no—
390C4—2442no—
391C4#1684079112yes—
392C4#1684079512yes—
393C2—2442no—
394C2#2146112811yescharacteristic
395C2#2146113111yescharacteristic
396C2#2146114811yescharacteristic
397C2#2146115311yescharacteristic
398C2#2146179711yescharacteristic
399C2#2146183211yescharacteristic
400C2#2146183811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#9075
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
3. C2×Q8⋊#92420C2semidirect
4. #21460655C2semidirect
225. C61#9075semidirect

Maximal subgroups

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

subgroupindex
C2×#168407812
#214616742
C2×Q8⋊#924202
#214606552
#214483572
#907561