Finite Group Databasebeta

← #30600416

Subgroups — #30600416

Order 15616.

800 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊#6923C211yesmaximal, characteristic, complemented
2#21461711C211yesmaximal, characteristic, complemented
3C2×#16840613C212yesmaximal
4#21460692C212yesmaximal, complemented
5#21454149C211yesmaximal, characteristic
6C22×C4⋊#6923—21no—
7C22×C4×#6923—21no—
8C2×#14519834C2212yes—
9C4×C4⋊#6923—24no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840628—21no—
13#16840613—24no—
14C2×#14519759—21no—
15C2×#14519748C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839665C411yescharacteristic, complemented
18#16839661C411yescharacteristic, complemented
19#16839660C411yescharacteristic, complemented
20#16839656—21no—
21#16839655—21no—
22#16839638—21no—
23#16839619C2211yescharacteristic
24C61×#390C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C2×C4⋊#6923—24no—
28C2×C4⋊#6923—24no—
29C2×C4⋊#6923—24no—
30C2×C4×#6923—24no—
31C2×C4×#6923—42no—
32C2×C4×#6923—42no—
33#14519834—22no—
34#14519834—22no—
35(C2×#6923)⋊C4—22no—
36(C2×#6923)⋊C4D414yes—
37C22×C22⋊D61—21no—
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C22×C4×D61C2311yescharacteristic
41#14519759—42no—
42#14519759—42no—
43#14519748—22no—
44#14519748—22no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61C2×C411yescharacteristic
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×#11758198—21no—
53C2×C4×Dic61C2×C411yescharacteristic
54C2×C4×Dic61D411yescharacteristic
55C2×C4×Dic61D411yescharacteristic
56C42×D61—22no—
57C42×D61—22no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61C22×D4×C61C2×C411yescharacteristic
62C2×C4⋊C4×C61C2×C411yescharacteristic
63C2×C22⋊C4×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C42×C61C2×C411yescharacteristic
66C24×C61—21no—
67C23×D61C2×D411yescharacteristic
68C22×#6923—24no—
69C22×#6923—24no—
70C22×#6923—24no—
71C22×#6923—42no—
72C22×#6923—42no—
73C22×#6923—42no—
74C22×#6923—42no—
75C2×D4×C61—22no—
76C2×D4×C61—41no—
77C2×D4×C61—41no—
78C22×C4×C61—21no—
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C2×C22⋊D61—41no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C2×C22⋊D61—41no—
86C2×C22⋊D61—41no—
87C2×C22⋊D61—41no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61C22⋊C411yescharacteristic
91C22×Dic61C22⋊C411yescharacteristic
92C22×Dic61C2×D411yescharacteristic
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—21no—
98C2×C4×D61—21no—
99C2×C4×D61—22no—
100C2×C4×D61—22no—
101C2×C4×D61—41no—
102C2×C4×D61—41no—
103C2×C4×D61D812yes—
104C2×C4×D61SD1612yes—
105C2×C4×D61C2×D412yes—
106C2×C4×D61#4112yes—
107C2×#92420—42no—
108C2×#92420—42no—
109C2×#92420—42no—
110C4⋊#6923—216no—
111C4×#6923—48no—
112C4⋊C4×C61—41no—
113C22⋊C4×C61—42no—
114C22⋊C4×C61—42no—
115C42×C61—22no—
116C22⋊Dic61—42no—
117C22⋊Dic61—42no—
118C22⋊Dic61—42no—
119#11758198—41no—
120#11758198—41no—
121C4⋊Dic61—22no—
122C4⋊Dic61—42no—
123C4⋊Dic61—41no—
124#11758196—42no—
125C4×Dic61—22no—
126C4×Dic61—22no—
127C4×Dic61—22no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61C2×C22⋊C411yescharacteristic
136C22×D61—21no—
137C22×D61—21no—
138C22×D61—21no—
139C22×D61—21no—
140C22×D61#12011yescharacteristic
141C22×D61#12311yescharacteristic
142C22×D61C2×D811yescharacteristic
143C22×D61C2×SD1611yescharacteristic
144C22×D61AGL(1,ℤ/8)11yescharacteristic
145C22×D61#13711yescharacteristic
146C2×#6923—28no—
147C2×#6923—44no—
148C2×#6923—44no—
149C2×#6923—44no—
150C2×#6923—44no—
151C2×#6923—44no—
152C2×#6923—44no—
153D4×C61—82no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61#10212yes—
164C2×C4×C61#10412yes—
165C2×C4×C61C4×D412yes—
166C22⋊D61—81no—
167C22⋊D61—81no—
168C22⋊D61—81no—
169C22⋊D61—81no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61#10212yes—
183C2×Dic61#10412yes—
184C2×Dic61#12112yes—
185C4×D61—24no—
186C4×D61—24no—
187C4×D61—22no—
188C4×D61—22no—
189C4×D61—22no—
190C4×D61—22no—
191C4×D61—42no—
192C4×D61—42no—
193#92420—82no—
194#9148—611nomaximal, Sylow
195C22×C61—81no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61—81no—
202C22×C61#38611yescharacteristic
203C22×C61#41011yescharacteristic
204C22×C61#41011yescharacteristic
205C22×C61C2×#10211yescharacteristic
206C22×C61#41711yescharacteristic
207C22×C61C2×#10411yescharacteristic
208C22×C61C2×#10411yescharacteristic
209C2×D61—21no—
210C2×D61—21no—
211C2×D61—21no—
212C2×D61—21no—
213C2×D61—21no—
214C2×D61—21no—
215C2×D61—21no—
216C2×D61—21no—
217C2×D61—21no—
218C2×D61—21no—
219C2×D61—21no—
220C2×D61—21no—
221C2×D61#44711yescharacteristic
222C2×D61#44811yescharacteristic
223C2×D61#48011yescharacteristic
224C2×D61#48111yescharacteristic
225#6923—48no—
226C4×C61—22no—
227C4×C61—22no—
228C4×C61—22no—
229C4×C61—81no—
230Dic61—22no—
231Dic61—22no—
232Dic61—22no—
233Dic61—81no—
234Dic61—81no—
235#2854—611no—
236C2×C4×C4⋊C4—611no—
237#1817—611no—
238C2×#327—612no—
239#1294—612no—
240C2×C61—161no—
241C2×C61#150211yescharacteristic
242C2×C61#150211yescharacteristic
243C2×C61#150911yescharacteristic
244C2×C61#150911yescharacteristic
245C2×C61#187411yescharacteristic
246C2×C61#189611yescharacteristic
247C2×C61#189911yescharacteristic
248D61—21no—
249D61—21no—
250D61—21no—
251D61—21no—
252C2×#123—1221no—
253C2×#121—611no—
254C22×C4⋊C4—1221no—
255C22×C42—611no—
256C22×C42—1221no—
257C2×C22⋊C8—612no—
258#390—611no—
259#390#692311yescharacteristic, complemented
260#389—611no—
261#389—611no—
262#388—1221no—
263#388—1221no—
264#386—1221no—
265#380—611no—
266C4×C4⋊C4—1224no—
267C2×#95—612no—
268#336—1221no—
269#336—1221no—
270#327—1224no—
271C61#914811yescharacteristic, complemented, Sylow
272C22×D4—1221no—
273C22×D4C2×#692311yescharacteristic
274C23×C4—611no—
275C23×C4—611no—
276C23×C4—611no—
277C23×C4—1222no—
278C22×C8—1222no—
279#123—2442no—
280#123—2442no—
281#121—1222no—
282#121—1222no—
283C2×C4⋊C4—611no—
284C2×C4⋊C4—611no—
285C2×C4⋊C4—1224no—
286C2×C4⋊C4—1224no—
287C2×C4⋊C4—1224no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4C2×#692311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4—1221no—
296C2×C22⋊C4—1221no—
297C2×C42—611no—
298C2×C42—611no—
299C2×C42—611no—
300C2×C42—1222no—
301C2×C42—1222no—
302C2×C42—1224no—
303C2×C42—2442no—
304C2×C42—2442no—
305C2×C42C2×#692311yescharacteristic
306C22⋊C8—1222no—
307C22⋊C8—1222no—
308#95—614no—
309#95—1221no—
310#95—1221no—
311#95—1222no—
312#95—1221no—
313C24—21no—
314C24—611no—
315C2×D4—22no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—2441no—
319C2×D4—2441no—
320C2×D4—2441no—
321C2×D4—2441no—
322C2×D4—2441no—
323C2×D4—2441no—
324C22×C4—21no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—611no—
328C22×C4—612no—
329C22×C4—612no—
330C22×C4—612no—
331C22×C4—612no—
332C22×C4—1222no—
333C22×C4—1221no—
334C22×C4—1221no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—1221no—
338C22×C4—1221no—
339C22×C4—1222no—
340C22×C4—1224no—
341C22×C4—1224no—
342C22×C4—1224no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—2441no—
346C22×C4—2441no—
347C22×C4—2442no—
348C22×C4—2442no—
349C22×C4—2442no—
350C22×C4—2442no—
351C22×C4C22⋊#692311yescharacteristic
352C22×C4C22⋊#692311yescharacteristic
353C22×C4C22×#692311yescharacteristic
354C2×C8—2442no—
355C2×C8—2442no—
356C2×C8—2442no—
357C4⋊C4—41no—
358C4⋊C4—12216no—
359C4⋊C4—1222no—
360C4⋊C4—2441no—
361C4⋊C4—2442no—
362C4⋊C4—2442no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—2441no—
366C22⋊C4—2441no—
367C22⋊C4—2442no—
368C22⋊C4—2442no—
369C22⋊C4—2442no—
370C42—22no—
371C42—1222no—
372C42—1222no—
373C42—1222no—
374C42—2448no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—611no—
383C23—611no—
384C23—611no—
385C23—611no—
386C23—611no—
387C23—611no—
388C23—1221no—
389C23—1221no—
390C23—1221no—
391C23—1221no—
392C23C2×C22⋊#692311yescharacteristic
393D4—82no—
394D4—4881no—
395D4—4881no—
396D4—4881no—
397D4—4881no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—612no—
408C2×C4—612no—
409C2×C4—612no—
410C2×C4—1222no—
411C2×C4—1222no—
412C2×C4—1222no—
413C2×C4—1222no—
414C2×C4—1224no—
415C2×C4—1224no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—1221no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—1228no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2444no—
432C2×C4—2444no—
433C2×C4—2444no—
434C2×C4—2444no—
435C2×C4—2444no—
436C2×C4—2444no—
437C2×C4#1451983912yes—
438C2×C4#1451984112yes—
439C2×C4D4×#692312yes—
440C8—4882no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—611no—
449C22—611no—
450C22—611no—
451C22—611no—
452C22—1221no—
453C22—1221no—
454C22—1221no—
455C22—1221no—
456C22—1221no—
457C22—1221no—
458C22—1221no—
459C22—1221no—
460C22—1221no—
461C22—1221no—
462C22—1221no—
463C22—1221no—
464C22#1684071011yescharacteristic
465C22#1684074111yescharacteristic
466C22#1684074211yescharacteristic
467C22C2×#1451983911yescharacteristic
468C22#1684075311yescharacteristic
469C22C2×#1451984111yescharacteristic
470C22D61⋊#10411yescharacteristic
471C4—22no—
472C4—22no—
473C4—22no—
474C4—81no—
475C4—1222no—
476C4—1222no—
477C4—1222no—
478C4—4881no—
479C4—4881no—
480C4—2448no—
481C2—161no—
482C2—1221no—
483C2—1221no—
484C2—1221no—
485C2—1221no—
486C2#2146119811yescharacteristic
487C2#2146119911yescharacteristic
488C2#2146122011yescharacteristic
489C2#2146122111yescharacteristic
490C2#2146180611yescharacteristic
491C2#2146183711yescharacteristic
492C2#2146184211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519748C22
cosocleC22×C4×D61C23
hypercenter#390#6923
hypocenterC61#9148
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×C4×C4⋊#6923C2semidirect
2. #21461711C2semidirect
4. #21460692C2semidirect
17. #16839665C4semidirect
18. #16839661C4semidirect
19. #16839660C4semidirect
24. C61×#390C4semidirect
259. #390#6923semidirect
271. C61#9148semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊#69232
#214617112
C2×#168406132
#214606922
#214541492
#914861