Finite Group Databasebeta

← #30600324

Subgroups — #30600324

Order 15616.

694 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 112 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×#16840780C211yesmaximal, characteristic, complemented
2C2×#16840613C212yesmaximal
3#21460995C211yesmaximal, characteristic, complemented
4#21460692C212yesmaximal, complemented
5#21454149C211yesmaximal, characteristic
6#16840780—24no—
7C2×#14519844—21no—
8C2×#14519834C2212yes—
9C2×C4×#92420—21no—
10C2×(C2×#6923)⋊C4—21no—
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4C2212yes—
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—
27#14519844—24no—
28#14519834—22no—
29#14519834D414yes—
30C4×#92420—24no—
31(C2×#6923)⋊C4—24no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—22no—
34(C2×#6923)⋊C4—42no—
35C22×C22⋊D61—21no—
36C22×C4×D61D411yescharacteristic
37C22×C4×D61D411yescharacteristic
38C22×C4×D61C2311yescharacteristic
39#14519759—42no—
40#14519759—42no—
41#14519748—22no—
42#14519748—22no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50C2×#11758198—21no—
51C2×C4×Dic61C2×C411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C42×D61—22no—
55C42×D61—22no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59C22×D4×C61C2×C411yescharacteristic
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C42×C61C2×C411yescharacteristic
64C24×C61—21no—
65C23×D61C2×D411yescharacteristic
66C22×#6923—24no—
67C22×#6923—42no—
68C22×#6923—42no—
69C22×#6923—42no—
70C22×#6923—42no—
71C22×#6923—42no—
72C22×#6923—42no—
73C2×D4×C61—22no—
74C2×D4×C61—41no—
75C2×D4×C61—41no—
76C22×C4×C61—21no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C2×C22⋊D61—42no—
81C2×C22⋊D61—41no—
82C2×C22⋊D61—41no—
83C2×C22⋊D61—42no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61C22⋊C411yescharacteristic
87C22×Dic61C22⋊C411yescharacteristic
88C22×Dic61C2×D411yescharacteristic
89C2×C4×D61—22no—
90C2×C4×D61—22no—
91C2×C4×D61—22no—
92C2×C4×D61—22no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61D812yes—
97C2×C4×D61SD1612yes—
98C2×C4×D61C2×D412yes—
99C2×C4×D61#4112yes—
100C2×#92420—24no—
101C2×#92420—42no—
102C2×#92420—42no—
103C4⋊C4×C61—41no—
104C22⋊C4×C61—42no—
105C22⋊C4×C61—42no—
106C42×C61—22no—
107C22⋊Dic61—42no—
108C22⋊Dic61—42no—
109C22⋊Dic61—41no—
110C22⋊Dic61—41no—
111#11758198—41no—
112#11758198—41no—
113C4⋊Dic61—22no—
114C4⋊Dic61—42no—
115C4⋊Dic61—41no—
116#11758196—42no—
117C4×Dic61—22no—
118C4×Dic61—21no—
119C4×Dic61—21no—
120C4×Dic61—21no—
121C4×Dic61—21no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61C2×C22⋊C411yescharacteristic
130C22×D61—21no—
131C22×D61—21no—
132C22×D61—21no—
133C22×D61—21no—
134C22×D61#12112yes—
135C22×D61C2×D811yescharacteristic
136C22×D61C2×SD1611yescharacteristic
137C22×D61#13512yes—
138C2×#6923—44no—
139C2×#6923—44no—
140C2×#6923—44no—
141C2×#6923—82no—
142C2×#6923—82no—
143C2×#6923—82no—
144C2×#6923—82no—
145D4×C61—82no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61#10212yes—
157C2×C4×C61C4×D412yes—
158C22⋊D61—82no—
159C22⋊D61—82no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61#10212yes—
173C2×Dic61#10412yes—
174C2×Dic61#12011yescharacteristic
175C2×Dic61#12311yescharacteristic
176C4×D61—24no—
177C4×D61—24no—
178C4×D61—42no—
179C4×D61—42no—
180C4×D61—44no—
181#92420—44no—
182#9114—611nomaximal, Sylow
183C22×C61—81no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61#38611yescharacteristic
191C22×C61#41011yescharacteristic
192C22×C61#41011yescharacteristic
193C22×C61C2×#10211yescharacteristic
194C22×C61#41611yescharacteristic
195C22×C61C2×#10411yescharacteristic
196C22×C61#42111yescharacteristic
197C2×D61—22no—
198C2×D61—22no—
199C2×D61—22no—
200C2×D61—21no—
201C2×D61—21no—
202C2×D61—21no—
203C2×D61—21no—
204C2×D61—22no—
205C2×D61#46512yes—
206C2×D61#46612yes—
207#6923—84no—
208C4×C61—22no—
209C4×C61—22no—
210C4×C61—41no—
211C4×C61—81no—
212Dic61—22no—
213Dic61—22no—
214Dic61—21no—
215Dic61—21no—
216Dic61—81no—
217Dic61—81no—
218C2×#431—611no—
219#2854—611no—
220C2×#327—612no—
221#1434—611no—
222#1294—612no—
223C2×C61—161no—
224C2×C61#150011yescharacteristic
225C2×C61#150211yescharacteristic
226C2×C61#151711yescharacteristic
227C2×C61#152111yescharacteristic
228C2×C61#187111yescharacteristic
229C2×C61#189311yescharacteristic
230C2×C61#189911yescharacteristic
231D61—22no—
232D61—22no—
233C2×#123—1221no—
234C2×#121—611no—
235C22×C42—611no—
236#431—1224no—
237C2×C4⋊C8—1221no—
238C2×C22⋊C8—612no—
239C2×C4×C8—1221no—
240#390—611no—
241#390#692311yescharacteristic, complemented
242#389—611no—
243#389—611no—
244#388—1221no—
245#388—1221no—
246#386—1221no—
247#380—611no—
248C2×#95—612no—
249C2×#95—1221no—
250C2×#95—1221no—
251#327—1224no—
252C61#911411yescharacteristic, complemented, Sylow
253C22×D4—1221no—
254C22×D4C2×#692311yescharacteristic
255C23×C4—611no—
256C23×C4—611no—
257C23×C4—611no—
258C23×C4—1222no—
259C22×C8—1222no—
260#123—2442no—
261#123—2442no—
262#121—1222no—
263#121—1222no—
264C2×C4⋊C4—611no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C4⋊C4C2×#692311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4—1221no—
273C2×C22⋊C4—1221no—
274C2×C22⋊C4—1221no—
275C2×C42—611no—
276C2×C42—611no—
277C2×C42—611no—
278C2×C42—1222no—
279C2×C42—1222no—
280C2×C42C2×#692311yescharacteristic
281C4⋊C8—1224no—
282C22⋊C8—614no—
283C22⋊C8—1222no—
284C4×C8—1224no—
285#95—1221no—
286#95—1221no—
287#95—1222no—
288#95—1222no—
289#95—1224no—
290#95—1221no—
291#95—2442no—
292C24—21no—
293C24—611no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—2442no—
298C2×D4—2441no—
299C2×D4—2441no—
300C2×D4—2442no—
301C22×C4—21no—
302C22×C4—611no—
303C22×C4—611no—
304C22×C4—611no—
305C22×C4—612no—
306C22×C4—612no—
307C22×C4—612no—
308C22×C4—612no—
309C22×C4—1222no—
310C22×C4—1222no—
311C22×C4—1222no—
312C22×C4—1222no—
313C22×C4—1224no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—2441no—
317C22×C4—2441no—
318C22×C4—2441no—
319C22×C4—2442no—
320C22×C4—2442no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4—2442no—
324C22×C4—2442no—
325C22×C4C22⋊#692311yescharacteristic
326C22×C4C22⋊#692311yescharacteristic
327C22×C4C22×#692311yescharacteristic
328C2×C8—1224no—
329C2×C8—2442no—
330C2×C8—2442no—
331C4⋊C4—41no—
332C4⋊C4—1222no—
333C4⋊C4—2441no—
334C4⋊C4—2442no—
335C4⋊C4—2442no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—2441no—
339C22⋊C4—2441no—
340C22⋊C4—2441no—
341C22⋊C4—2441no—
342C22⋊C4—2442no—
343C22⋊C4—2442no—
344C42—22no—
345C42—1221no—
346C42—1221no—
347C42—1221no—
348C42—1221no—
349C42—1222no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—612no—
358C23—611no—
359C23—611no—
360C23—612no—
361C23—1221no—
362C23—1221no—
363C23—1221no—
364C23—1221no—
365C23C2×C22⋊#692311yescharacteristic
366D4—82no—
367D4—4882no—
368D4—4882no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—611no—
380C2×C4—611no—
381C2×C4—612no—
382C2×C4—612no—
383C2×C4—1224no—
384C2×C4—1224no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—1221no—
391C2×C4—4882no—
392C2×C4—4882no—
393C2×C4—4882no—
394C2×C4—4882no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2444no—
398C2×C4—2444no—
399C2×C4—2444no—
400C2×C4—2444no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4#1451983912yes—
408C2×C4D4×#692312yes—
409C8—2444no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—612no—
418C22—612no—
419C22—1221no—
420C22—1221no—
421C22—1222no—
422C22—1222no—
423C22—1222no—
424C22—1222no—
425C22—1221no—
426C22—1221no—
427C22#1684071011yescharacteristic
428C22#1684074111yescharacteristic
429C22#1684074211yescharacteristic
430C22C2×#1451983911yescharacteristic
431C22#1684075211yescharacteristic
432C22D61⋊#10411yescharacteristic
433C22#1684076011yescharacteristic
434C4—22no—
435C4—22no—
436C4—41no—
437C4—81no—
438C4—1222no—
439C4—1222no—
440C4—1221no—
441C4—1221no—
442C4—4884no—
443C4—4881no—
444C4—4881no—
445C2—161no—
446C2—1222no—
447C2—1222no—
448C2#2146119111yescharacteristic
449C2#2146119811yescharacteristic
450C2#2146124811yescharacteristic
451C2#2146126111yescharacteristic
452C2#2146180311yescharacteristic
453C2#2146183311yescharacteristic
454C2#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#9114
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×#16840780C2semidirect
3. #21460995C2semidirect
4. #21460692C2semidirect
17. #16839665C4semidirect
18. #16839661C4semidirect
19. #16839660C4semidirect
24. C61×#390C4semidirect
241. #390#6923semidirect
252. C61#9114semidirect

Maximal subgroups

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

subgroupindex
C2×#168407802
C2×#168406132
#214609952
#214606922
#214541492
#911461