Finite Group Databasebeta

← #30610330

Subgroups — #30610330

Order 15616.

922 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462719C212yesmaximal, complemented
2#21461101C214yesmaximal, complemented
3#21461093C212yesmaximal, complemented
4#21461078C211yesmaximal, characteristic
5#21461072C212yesmaximal, complemented
6#21461058C211yesmaximal, characteristic
7#21461052C212yesmaximal, complemented
8C2×#16840451C211yesmaximal, characteristic
9#16840814—24no—
10#16840812—22no—
11#16840811—22no—
12#16840783C2212yes—
13#16840782C2214yes—
14#16840779C2212yes—
15#16840778C2214yes—
16#16840608C2214yes—
17#16840608C2214yes—
18#16840606C2211yescharacteristic
19#16840606C2212yes—
20#16840606C2211yescharacteristic
21#16840605C2212yes—
22#16840605C2212yes—
23#16840451—22no—
24#16840451—22no—
25#16840451—24no—
26C2×#14519770C2212yes—
27C2×#14519769C412yes—
28C2×#14519768C2211yescharacteristic
29C2×#14519759C412yes—
30C2×#14519757C411yescharacteristic
31C2×#14519756C411yescharacteristic
32C2×#14519742C2211yescharacteristic
33C2×C4×C22⋊D61C2211yescharacteristic
34C2×#14519734C2212yes—
35C2×C4⋊C4×D61C411yescharacteristic
36C2×#117×C61C411yescharacteristic
37#14519844—22no—
38#14519844—24no—
39C4⋊#92420—24no—
40C4⋊#92420—22no—
41C22⋊#92420C2312yes—
42C22⋊#92420C2314yes—
43C22⋊#92420C2312yes—
44#14519832—24no—
45#14519832—22no—
46C4×#92420—24no—
47C4×#92420—22no—
48C22×Dic122C2×C411yescharacteristic
49C22×C22⋊D61C2×C411yescharacteristic
50C22×C4×D61C2×C412yes—
51C22×C4×D61C2×C411yescharacteristic
52#14519770—24no—
53#14519770—24no—
54#14519769—24no—
55#14519769—24no—
56#14519768—22no—
57#14519768—22no—
58#14519759—24no—
59#14519759—24no—
60#14519757—22no—
61#14519757—22no—
62#14519756—22no—
63#14519756—22no—
64#14519742D412yes—
65#14519742D412yes—
66#14519742D414yes—
67C4×C22⋊D61—22no—
68C4×C22⋊D61—22no—
69#14519734—24no—
70#14519734—24no—
71C2×C4⋊Dic61C2×C411yescharacteristic
72C2×C4⋊Dic61C2×C412yes—
73C2×C4⋊Dic61C2×C412yes—
74C2×#11758196C2×C412yes—
75C2×#11758196C2×C412yes—
76C2×#11758196C2×C411yescharacteristic
77C2×#11758196C2×C412yes—
78C2×#11758196C2312yes—
79C4⋊C4×D61—22no—
80C4⋊C4×D61—21no—
81C4⋊C4×D61—21no—
82C2×C22⋊Dic61C2×C412yes—
83C2×C22⋊Dic61C2311yescharacteristic
84C2×#11758198C2×C412yes—
85C2×#11758198C2311yescharacteristic
86C2×C4×Dic61C2312yes—
87C2×C4×Dic61C2311yescharacteristic
88C23×C4×C61C2×C411yescharacteristic
89#117×C61—22no—
90#117×C61—21no—
91#117×C61—21no—
92C2×C4⋊C4×C61C2×C411yescharacteristic
93C2×C4⋊C4×C61C2×C411yescharacteristic
94C2×C22⋊C4×C61C2×C412yes—
95C2×C42×C61C2×C412yes—
96C24×C61C22×C411yescharacteristic
97C23×D61C22×C411yescharacteristic
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—42no—
101C22×C4×C61C22×C411yescharacteristic
102C22×C4×C61C22×C412yes—
103C22×C4×C61C22×C411yescharacteristic
104C22×C4×C61C22×C412yes—
105C2×C22⋊D61—22no—
106C2×C22⋊D61—22no—
107C2×C22⋊D61—41no—
108C22×Dic61C22×C412yes—
109C22×Dic61C22×C411yescharacteristic
110C22×Dic61C22×C411yescharacteristic
111C22×Dic61C22×C412yes—
112C22×Dic61C2411yescharacteristic
113C2×C4×D61—24no—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×Dic122—42no—
121C2×Dic122C22⋊C412yes—
122C2×Dic122C22⋊C412yes—
123C2×#92420—22no—
124C2×#92420—24no—
125C2×#92420—22no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61C22⋊C412yes—
129C4⋊C4×C61C22⋊C411yescharacteristic
130C4⋊C4×C61C22⋊C411yescharacteristic
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C42×C61—22no—
134C42×C61—22no—
135C22⋊Dic61—24no—
136C22⋊Dic61—22no—
137#11758198—24no—
138#11758198—22no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—24no—
142C4⋊Dic61—24no—
143#11758196—22no—
144#11758196—22no—
145#11758196—24no—
146#11758196—22no—
147#11758196—24no—
148#11758196C22⋊C414yes—
149#11758196C22⋊C414yes—
150C4×Dic61C2×D412yes—
151C4×Dic61C2×D414yes—
152C4×Dic61C2×D414yes—
153C4×Dic61C2×D412yes—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61C23×C411yescharacteristic
160C22×D61—21no—
161C22×D61—21no—
162C22×D61—41no—
163C22×D61—41no—
164C22×D61#13112yes—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—21no—
169C2×C4×C61—22no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61C2×C22⋊C412yes—
175C2×C4×C61C2×C22⋊C412yes—
176C2×C4×C61C2×C22⋊C411yescharacteristic
177C2×C4×C61C2×C22⋊C411yescharacteristic
178C22⋊D61—44no—
179C2×Dic61—21no—
180C2×Dic61—21no—
181C2×Dic61—22no—
182C2×Dic61—21no—
183C2×Dic61—22no—
184C2×Dic61—22no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61C2×C22⋊C412yes—
189C2×Dic61C2×C22⋊C414yes—
190C2×Dic61C22×D412yes—
191C4×D61—44no—
192C4×D61—44no—
193C4×D61—42no—
194C4×D61—42no—
195Dic122—24no—
196Dic122—42no—
197#92420—42no—
198#92420—44no—
199#92420—42no—
200#17423—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—41no—
204C22×C61—41no—
205C22×C61—41no—
206C22×C61—81no—
207C22×C61—81no—
208C22×C61C22×C22⋊C411yescharacteristic
209C22×C61C2×#13111yescharacteristic
210C22×C61#56811yescharacteristic
211C2×D61—22no—
212C2×D61—42no—
213C2×D61—41no—
214C2×D61—41no—
215C2×D61—42no—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—21no—
219C4×C61—42no—
220C4×C61—41no—
221C4×C61#41312yes—
222Dic61—22no—
223Dic61—22no—
224Dic61—24no—
225Dic61—42no—
226Dic61—41no—
227C2×#552—611no—
228#2984—612no—
229#1473—614no—
230#1470—612no—
231#1466—611no—
232#1464—612no—
233#1460—611no—
234#1458—612no—
235C2×C61—21no—
236C2×C61—21no—
237C2×C61—81no—
238C2×C61#288011yescharacteristic
239C2×C61#288411yescharacteristic
240C2×C61C2×#41311yescharacteristic
241D61—42no—
242#552—1222no—
243#552—1224no—
244#552—1222no—
245C2×#126—612no—
246C2×#125—611no—
247C2×#125—612no—
248C2×#123—612no—
249C2×C22⋊Q8—611no—
250C2×C22⋊Q8—611no—
251C2×C4×Q8—611no—
252C2×C4×D4—611no—
253C2×#117—612no—
254C2×#117#692311yescharacteristic
255C22×C4⋊C4—611no—
256#446—1222no—
257#446—1224no—
258C8×Q8—1222no—
259#432—614no—
260#432—612no—
261#431—614no—
262#431—612no—
263#324—614no—
264#324—614no—
265#323—612no—
266#323—612no—
267#323—611no—
268#323—612no—
269#323—611no—
270C61#1742311yescharacteristic, complemented, Sylow
271C22×Q8—611no—
272C22×D4—611no—
273C23×C4—612no—
274C23×C4—611no—
275C23×C4C2×#692311yescharacteristic
276#126—1224no—
277#126—1224no—
278#125—1224no—
279#125—1224no—
280#125—1222no—
281#125—1222no—
282#123—1224no—
283#123—1224no—
284C22⋊Q8—1222no—
285C22⋊Q8—1222no—
286C22⋊Q8—1222no—
287C22⋊Q8—1222no—
288C4×Q8—612no—
289C4×Q8—614no—
290C4×Q8—612no—
291C4×D4—1222no—
292C4×D4—1222no—
293#117—22no—
294#117—21no—
295#117—21no—
296#117—1224no—
297#117—1224no—
298C2×C4⋊C4—612no—
299C2×C4⋊C4—611no—
300C2×C4⋊C4—612no—
301C2×C4⋊C4—612no—
302C2×C4⋊C4—612no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—612no—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4—1221no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4C2×#692311yescharacteristic
310C2×C4⋊C4C2×#692311yescharacteristic
311C2×C22⋊C4—611no—
312C2×C22⋊C4—612no—
313C2×C22⋊C4—611no—
314C2×C22⋊C4—612no—
315C2×C22⋊C4C2×#692312yes—
316C2×C42—612no—
317C2×C42—611no—
318C2×C42C2×#692312yes—
319C4⋊C8—1224no—
320C4⋊C8—1222no—
321C4⋊C8—1222no—
322C4⋊C8—1224no—
323C22⋊C8—612no—
324C22⋊C8—614no—
325C22⋊C8—612no—
326#97—1224no—
327#97—1222no—
328C4×C8—1224no—
329C4×C8—1222no—
330C24—611no—
331C24C22×#692311yescharacteristic
332C2×Q8—612no—
333C2×Q8—612no—
334C2×Q8—2442no—
335C2×D4—1222no—
336C2×D4—1222no—
337C2×D4—2441no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—42no—
341C22×C4—611no—
342C22×C4—611no—
343C22×C4—612no—
344C22×C4—612no—
345C22×C4—611no—
346C22×C4—1224no—
347C22×C4—1222no—
348C22×C4—1222no—
349C22×C4—1221no—
350C22×C4—1221no—
351C22×C4—2442no—
352C22×C4—2442no—
353C22×C4C22×#692312yes—
354C22×C4C22×#692311yescharacteristic
355C22×C4C22×#692311yescharacteristic
356C22×C4C22×#692312yes—
357C2×C8—1222no—
358C2×C8—1224no—
359C2×C8—1222no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—614no—
363C4⋊C4—614no—
364C4⋊C4—1224no—
365C4⋊C4—1224no—
366C4⋊C4—1222no—
367C4⋊C4—1224no—
368C4⋊C4—1224no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4C22⋊#692312yes—
374C4⋊C4C22⋊#692311yescharacteristic
375C4⋊C4C22⋊#692311yescharacteristic
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—1222no—
379C22⋊C4—1224no—
380C22⋊C4—1222no—
381C22⋊C4—1224no—
382C42—22no—
383C42—22no—
384C42—614no—
385C42—612no—
386C42—614no—
387C42—612no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—41no—
392C23—41no—
393C23—612no—
394C23—1221no—
395C23—1221no—
396C23—2441no—
397C23—2441no—
398C23C23×#692311yescharacteristic
399Q8—1224no—
400Q8—2442no—
401D4—2444no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—614no—
412C2×C4—612no—
413C2×C4—612no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1222no—
417C2×C4—1221no—
418C2×C4—1222no—
419C2×C4—1222no—
420C2×C4—2444no—
421C2×C4—2444no—
422C2×C4—2442no—
423C2×C4—2441no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2441no—
427C2×C4C2×C22⋊#692312yes—
428C2×C4C2×C22⋊#692312yes—
429C2×C4C2×C22⋊#692311yescharacteristic
430C2×C4C2×C22⋊#692311yescharacteristic
431C8—2442no—
432C8—2444no—
433C8—2442no—
434C22—21no—
435C22—21no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—81no—
440C22—81no—
441C22—1222no—
442C22—2442no—
443C22—2442no—
444C22—2441no—
445C22—2441no—
446C22C22×C22⋊#692311yescharacteristic
447C22D61⋊#13111yescharacteristic
448C22#1684084111yescharacteristic
449C4—22no—
450C4—22no—
451C4—21no—
452C4—42no—
453C4—41no—
454C4—1222no—
455C4—1222no—
456C4—1224no—
457C4—2442no—
458C4—2441no—
459C4#1684074812yes—
460C2—21no—
461C2—21no—
462C2—81no—
463C2—2442no—
464C2#2146240911yescharacteristic
465C2#2146242111yescharacteristic
466C2C2×#1684074811yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#117×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×C22⋊D61C2×C4
cosocleC22×Dic61C24
hypercenterC2×#117#6923
hypocenterC61#17423
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #21462719C2semidirect
2. #21461101C2semidirect
3. #21461093C2semidirect
5. #21461072C2semidirect
7. #21461052C2semidirect
270. C61#17423semidirect

Maximal subgroups

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

subgroupindex
#214627192
#214611012
#214610932
#214610782
#214610722
#214610582
#214610522
C2×#168404512
#1742361