Finite Group Databasebeta

← #30598172

Subgroups — #30598172

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461674C211yesmaximal, characteristic, complemented
2C4×#14519834C211yesmaximal, characteristic, complemented
3C2×Q8⋊#92420C212yesmaximal, complemented
4C2×D4⋊#92420C212yesmaximal, complemented
5C2×#16840291C211yesmaximal, characteristic
6C2×C4⋊#92420C2212yes—
7C2×#14519834—21no—
8C2×#14519834—21no—
9C2×C4×#92420C2212yes—
10#16840628—21no—
11#16840628—21no—
12Q8⋊#92420—24no—
13D4⋊#92420—24no—
14#16840291—24no—
15#16840291—24no—
16C2×#14519770—21no—
17C2×#14519769C411yescharacteristic
18C2×#14519759—21no—
19C2×#14519753C411yescharacteristic
20C2×#14519748C411yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×D4×Dic61C2211yescharacteristic
23C2×#14519737—21no—
24C2×#14519733—21no—
25C2×C42×D61C2211yescharacteristic
26C2×#124×C61C411yescharacteristic
27C22×#92420—22no—
28C22×#92420—22no—
29C4⋊#92420—22no—
30C4⋊#92420—22no—
31#14519834—24no—
32#14519834—42no—
33C4×#92420—22no—
34C4×#92420—22no—
35C22×C22⋊D61—21no—
36C23×Dic61—21no—
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40#14519770—28no—
41#14519769—22no—
42#14519769—22no—
43#14519759—28no—
44#14519753—22no—
45#14519753—22no—
46#14519748—22no—
47#14519748—22no—
48Q8×Dic61—22no—
49Q8×Dic61D414yes—
50D4×Dic61—22no—
51D4×Dic61D414yes—
52#14519737—28no—
53#14519733—28no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—21no—
62C2×#11758198—21no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61D411yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67C42×D61—22no—
68C42×D61—22no—
69C22×Q8×C61C2×C411yescharacteristic
70C22×D4×C61C2×C411yescharacteristic
71#124×C61—22no—
72#124×C61—22no—
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C42×C61C2×C411yescharacteristic
76C24×C61—21no—
77C23×D61C22⋊C411yescharacteristic
78C2×Q8×C61—21no—
79C2×Q8×C61—22no—
80C2×Q8×C61—22no—
81C2×Q8×C61C22⋊C412yes—
82C2×D4×C61—21no—
83C2×D4×C61—22no—
84C2×D4×C61—22no—
85C2×D4×C61C22⋊C412yes—
86C22×C4×C61—21no—
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22×C411yescharacteristic
90C2×C22⋊D61—24no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—24no—
96C22×Dic61—42no—
97C22×Dic61—41no—
98C22×Dic61—41no—
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C2×D411yescharacteristic
101C22×Dic61C2×D411yescharacteristic
102C2×C4×D61—22no—
103C2×C4×D61—21no—
104C2×C4×D61—21no—
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—41no—
113C2×#92420—24no—
114C2×#92420—42no—
115C2×#92420—42no—
116C2×#92420—42no—
117C2×#92420—42no—
118C2×#92420—42no—
119C22⋊C4×C61—24no—
120C22⋊C4×C61—24no—
121C42×C61—21no—
122C42×C61—21no—
123C22⋊Dic61—24no—
124C22⋊Dic61—24no—
125#11758198—24no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—24no—
129C4⋊Dic61C22⋊C412yes—
130C4⋊Dic61C22⋊C412yes—
131#11758196—24no—
132#11758196—24no—
133#11758196—24no—
134C4×Dic61—22no—
135C4×Dic61—22no—
136C4×Dic61—24no—
137C4×Dic61C2×D412yes—
138C4×Dic61C2×D412yes—
139C23×C61—22no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61C2×C22⋊C411yescharacteristic
147C22×D61—21no—
148C22×D61—21no—
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—21no—
153C22×D61#10412yes—
154Q8×C61—24no—
155Q8×C61—42no—
156D4×C61—24no—
157D4×C61—42no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—22no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61C2×C22⋊C411yescharacteristic
168C2×C4×C61C2×C22⋊C411yescharacteristic
169C2×C4×C61#13112yes—
170C2×C4×C61#13112yes—
171C22⋊D61—48no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—22no—
176C2×Dic61—22no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—44no—
184C2×Dic61—44no—
185C2×Dic61#12012yes—
186C2×Dic61#12012yes—
187C2×Dic61#13512yes—
188C2×Dic61AGL(1,ℤ/8)11yescharacteristic
189C2×Dic61#13711yescharacteristic
190C4×D61—44no—
191C4×D61—42no—
192C4×D61—42no—
193C4×D61—42no—
194C4×D61—42no—
195#92420—44no—
196#92420—82no—
197#8315—611nomaximal, Sylow
198C22×C61—42no—
199C22×C61—42no—
200C22×C61—42no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61#37911yescharacteristic
206C22×C61#40811yescharacteristic
207C22×C61#40811yescharacteristic
208C22×C61#41611yescharacteristic
209C22×C61#41711yescharacteristic
210C22×C61C2×#10411yescharacteristic
211C22×C61#42111yescharacteristic
212C2×D61—22no—
213C2×D61—22no—
214C2×D61—22no—
215C2×D61—41no—
216C2×D61—41no—
217C2×D61—41no—
218C2×D61—41no—
219C4×C61—42no—
220C4×C61—41no—
221C4×C61—41no—
222C4×C61#40812yes—
223C4×C61#40812yes—
224Dic61—22no—
225Dic61—41no—
226Dic61—42no—
227Dic61—42no—
228Dic61#44912yes—
229Dic61#45212yes—
230C2×#529—611no—
231#1797—611no—
232C4×C22⋊C8—611no—
233C2×Q8⋊C8—612no—
234C2×D4⋊C8—612no—
235C2×C61—82no—
236C2×C61#148211yescharacteristic
237C2×C61#148311yescharacteristic
238C2×C61#149411yescharacteristic
239C2×C61#149511yescharacteristic
240C2×C61#178011yescharacteristic
241C2×C61#178811yescharacteristic
242C2×C61#178911yescharacteristic
243D61—42no—
244#529—1224no—
245#529—1224no—
246C2×#126—1221no—
247C2×#125—611no—
248C2×#124#692311yescharacteristic
249C2×#123—1221no—
250C2×C22⋊Q8—611no—
251C2×#121—611no—
252C2×C4×Q8—611no—
253C2×C4×D4—611no—
254C2×C4×D4—1221no—
255C2×#117—1221no—
256C22×C42—611no—
257C2×C4⋊C8—612no—
258C2×C22⋊C8—1221no—
259C2×C22⋊C8—1221no—
260C2×C4×C8—612no—
261#336—1221no—
262#336—1221no—
263Q8⋊C8—1224no—
264D4⋊C8—1224no—
265C61#831511yescharacteristic, complemented, Sylow
266C22×Q8C2×#692311yescharacteristic
267C22×D4—1221no—
268C22×D4C2×#692311yescharacteristic
269C23×C4—611no—
270C23×C4—611no—
271C23×C4—611no—
272C23×C4—1221no—
273C22×C8—1222no—
274C22×C8—1222no—
275#126—1228no—
276#125—1222no—
277#125—1222no—
278#124—22no—
279#124—22no—
280#123—1228no—
281C22⋊Q8—1222no—
282C22⋊Q8—1222no—
283#121—1222no—
284#121—1222no—
285C4×Q8—614no—
286C4×Q8—1222no—
287C4×D4—614no—
288C4×D4—1222no—
289C4×D4—1228no—
290#117—1228no—
291C2×C4⋊C4—611no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—1221no—
300C2×C22⋊C4—1221no—
301C2×C22⋊C4—1221no—
302C2×C42—611no—
303C2×C42—611no—
304C2×C42—611no—
305C2×C42—1222no—
306C2×C42—1222no—
307C2×C42—1221no—
308C2×C42C2×#692311yescharacteristic
309C4⋊C8—1222no—
310C4⋊C8—1222no—
311C22⋊C8—1224no—
312C22⋊C8—2442no—
313C4×C8—1222no—
314C4×C8—1222no—
315C24—21no—
316C24—611no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8C22⋊#692312yes—
321C2×D4—21no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—1224no—
325C2×D4—2442no—
326C2×D4—2442no—
327C2×D4C22⋊#692312yes—
328C22×C4—21no—
329C22×C4—611no—
330C22×C4—611no—
331C22×C4—611no—
332C22×C4—1221no—
333C22×C4—1222no—
334C22×C4—1222no—
335C22×C4—1222no—
336C22×C4—1221no—
337C22×C4—1222no—
338C22×C4—1224no—
339C22×C4—1221no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1222no—
344C22×C4—1222no—
345C22×C4—2441no—
346C22×C4—2442no—
347C22×C4—2441no—
348C22×C4—2441no—
349C22×C4C22⋊#692311yescharacteristic
350C22×C4C22⋊#692311yescharacteristic
351C22×C4C22×#692311yescharacteristic
352C2×C8—1224no—
353C2×C8—2442no—
354C2×C8—2442no—
355C2×C8—2442no—
356C2×C8—2442no—
357C2×C8—2442no—
358C4⋊C4—612no—
359C4⋊C4—612no—
360C4⋊C4—1224no—
361C4⋊C4—1224no—
362C4⋊C4—1224no—
363C4⋊C4—1224no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—1224no—
369C22⋊C4—1224no—
370C22⋊C4—1224no—
371C42—21no—
372C42—21no—
373C42—612no—
374C42—612no—
375C42—1222no—
376C42—1222no—
377C42—1224no—
378C23—22no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—612no—
386C23—1221no—
387C23—1221no—
388C23—1221no—
389C23—1221no—
390C23—1221no—
391C23—1221no—
392C23C2×C22⋊#692311yescharacteristic
393Q8—24no—
394Q8—42no—
395D4—24no—
396D4—42no—
397D4—2448no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—612no—
408C2×C4—611no—
409C2×C4—611no—
410C2×C4—612no—
411C2×C4—612no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—1221no—
417C2×C4—1222no—
418C2×C4—1222no—
419C2×C4—2442no—
420C2×C4—2442no—
421C2×C4—2444no—
422C2×C4—2444no—
423C2×C4—2444no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4C2×C22⋊#692311yescharacteristic
431C2×C4C2×C22⋊#692311yescharacteristic
432C2×C4#1451986412yes—
433C2×C4#1451986412yes—
434C8—4882no—
435C8—2444no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—1222no—
444C22—1222no—
445C22—1222no—
446C22—2441no—
447C22—2441no—
448C22—2441no—
449C22—2441no—
450C22#1684070311yescharacteristic
451C22#1684073611yescharacteristic
452C22#1684073611yescharacteristic
453C22#1684075211yescharacteristic
454C22#1684075311yescharacteristic
455C22D61⋊#10411yescharacteristic
456C22#1684075911yescharacteristic
457C4—42no—
458C4—41no—
459C4—41no—
460C4—612no—
461C4—612no—
462C4—1222no—
463C4—2441no—
464C4—2442no—
465C4—2442no—
466C4#1684073512yes—
467C4#1684073512yes—
468C2—82no—
469C2—2442no—
470C2#2146113111yescharacteristic
471C2#2146113511yescharacteristic
472C2#2146117111yescharacteristic
473C2#2146117511yescharacteristic
474C2#2146164311yescharacteristic
475C2#2146165911yescharacteristic
476C2#2146166111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519748C4
cosocleC2×C4×Dic61C23
hypercenterC2×#124#6923
hypocenterC61#8315
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #21461674C2semidirect
2. C4×#14519834C2semidirect
3. C2×Q8⋊#92420C2semidirect
4. C2×D4⋊#92420C2semidirect
265. C61#8315semidirect

Maximal subgroups

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

subgroupindex
#214616742
C4×#145198342
C2×Q8⋊#924202
C2×D4⋊#924202
C2×#168402912
#831561