Finite Group Databasebeta

← #30600433

Subgroups — #30600433

Order 15616.

648 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 100 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×#16840707C211yesmaximal, characteristic, complemented
2#21461711C211yesmaximal, characteristic, complemented
3C2×#16840614C212yesmaximal
4#21460692C212yesmaximal, complemented
5#21454152C211yesmaximal, characteristic
6C2×#14519834C2212yes—
7#16840707—24no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840628—21no—
13#16840614—24no—
14C2×#14519762—21no—
15C2×#14519752C2211yescharacteristic
16C2×C42×D61C2211yescharacteristic
17#16839664C411yescharacteristic, complemented
18#16839663C411yescharacteristic, complemented
19#16839658C411yescharacteristic, complemented
20#16839652—21no—
21#16839651—21no—
22#16839640—21no—
23#16839619C2211yescharacteristic
24C61×#389C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27#14519834—22no—
28#14519834—22no—
29(C2×#6923)⋊C4—24no—
30(C2×#6923)⋊C4—24no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4D414yes—
33C22×D244—21no—
34C22×D244C2×C411yescharacteristic
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61C2311yescharacteristic
38#14519762—44no—
39#14519752—22no—
40#14519752—22no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×#11758196—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49C2×C4×Dic61C2×C411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C42×D61—22no—
53C42×D61—22no—
54#14519684—21no—
55#14519684—21no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C42×C61C2×C411yescharacteristic
60#95×C61—21no—
61C23×D61—21no—
62C23×D61C2×D411yescharacteristic
63C22×#6923—24no—
64C22×#6923—42no—
65C22×#6923—42no—
66C22×#6923—42no—
67C22×#6923—42no—
68C22×#6923—42no—
69C22×#6923—42no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22×C411yescharacteristic
75C22×Dic61—21no—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C2×D244—22no—
80C2×D244—41no—
81C2×D244—41no—
82C2×D244—41no—
83C2×D244—41no—
84C2×D244—42no—
85C2×D244—42no—
86C2×C4×D61—21no—
87C2×C4×D61—21no—
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—22no—
91C2×C4×D61—22no—
92C2×C4×D61—41no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61C2×D412yes—
97C2×C4×D61#4112yes—
98C2×#92420—42no—
99C2×#92420—42no—
100C2×#92420—42no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—42no—
103C4⋊C4×C61—41no—
104C42×C61—22no—
105#11758198—42no—
106#11758198—42no—
107#11758198—42no—
108#11758198—42no—
109#11758198—42no—
110#11758198—42no—
111C4⋊Dic61—41no—
112#11758196—42no—
113C4×Dic61—22no—
114C4×Dic61—22no—
115C4×Dic61—22no—
116C23×C61C2×C22⋊C411yescharacteristic
117C22×D61—21no—
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—41no—
122C22×D61—41no—
123C22×D61—41no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61#12011yescharacteristic
129C22×D61#12311yescharacteristic
130C22×D61#13512yes—
131C22×D61AGL(1,ℤ/8)12yes—
132C2×#6923—44no—
133C2×#6923—44no—
134C2×#6923—44no—
135C2×#6923—82no—
136C2×#6923—82no—
137C2×#6923—82no—
138C2×#6923—82no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C4×D412yes—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61#12112yes—
166D244—82no—
167D244—82no—
168D244—82no—
169C4×D61—42no—
170C4×D61—42no—
171C4×D61—42no—
172C4×D61—42no—
173C4×D61—42no—
174C4×D61—42no—
175#92420—82no—
176#9157—611nomaximal, Sylow
177C22×C61#38611yescharacteristic
178C22×C61#41011yescharacteristic
179C22×C61#41011yescharacteristic
180C22×C61#41611yescharacteristic
181C22×C61#41811yescharacteristic
182C22×C61#42111yescharacteristic
183C22×C61#42111yescharacteristic
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—22no—
189C2×D61—22no—
190C2×D61—22no—
191C2×D61—22no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61#44912yes—
200C2×D61#48212yes—
201#6923—84no—
202C4×C61—22no—
203C4×C61—41no—
204C4×C61—41no—
205C4×C61—81no—
206C4×C61—81no—
207Dic61—22no—
208Dic61—41no—
209Dic61—41no—
210Dic61—81no—
211#2854—611no—
212C2×#383—611no—
213#1817—611no—
214C2×#327—612no—
215#1294—612no—
216C2×C61#151411yescharacteristic
217C2×C61#151411yescharacteristic
218C2×C61#152111yescharacteristic
219C2×C61#152111yescharacteristic
220C2×C61#187511yescharacteristic
221C2×C61#189711yescharacteristic
222C2×C61#189911yescharacteristic
223D61—22no—
224D61—22no—
225D61—161no—
226C2×#123—1221no—
227C2×#121—611no—
228C22×C42—611no—
229C2×C22⋊C8—612no—
230#390—611no—
231#390—611no—
232#389—611no—
233#389#692311yescharacteristic, complemented
234#388—1221no—
235#388—1221no—
236#386—1221no—
237#383—1224no—
238#380—611no—
239C2×#95—612no—
240C2×#95—1221no—
241C2×#95—1221no—
242#336—1221no—
243#336—1221no—
244#327—1224no—
245C61#915711yescharacteristic, complemented, Sylow
246C22×D4—611no—
247C22×D4—1221no—
248C23×C4—611no—
249C23×C4—611no—
250C23×C4—611no—
251C23×C4—1222no—
252C22×C8—1222no—
253#123—2444no—
254#121—1222no—
255#121—1222no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—611no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4C2×#692311yescharacteristic
260C2×C4⋊C4C2×#692311yescharacteristic
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C42—611no—
268C2×C42—611no—
269C2×C42—611no—
270C2×C42—1222no—
271C2×C42—1222no—
272C2×C42C2×#692311yescharacteristic
273C22⋊C8—1222no—
274C22⋊C8—1222no—
275#95—21no—
276#95—614no—
277#95—1221no—
278#95—1221no—
279#95—1222no—
280#95—1224no—
281#95—1224no—
282C24—611no—
283C24—1221no—
284C2×D4—1222no—
285C2×D4—2441no—
286C2×D4—2441no—
287C2×D4—2441no—
288C2×D4—2441no—
289C2×D4—2442no—
290C2×D4—2442no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—611no—
294C22×C4—612no—
295C22×C4—612no—
296C22×C4—611no—
297C22×C4—611no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1222no—
303C22×C4—1222no—
304C22×C4—1224no—
305C22×C4—1221no—
306C22×C4—2441no—
307C22×C4—2441no—
308C22×C4—2441no—
309C22×C4—2441no—
310C22×C4—2442no—
311C22×C4—2442no—
312C22×C4—2442no—
313C22×C4—2442no—
314C22×C4—2442no—
315C22×C4—2442no—
316C22×C4C22⋊#692311yescharacteristic
317C22×C4C22⋊#692311yescharacteristic
318C22×C4C22×#692311yescharacteristic
319C2×C8—2442no—
320C2×C8—2442no—
321C2×C8—2442no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—2441no—
326C4⋊C4—2442no—
327C22⋊C4—2442no—
328C22⋊C4—2442no—
329C22⋊C4—2442no—
330C22⋊C4—2442no—
331C22⋊C4—2442no—
332C22⋊C4—2442no—
333C42—22no—
334C42—1222no—
335C42—1222no—
336C42—1222no—
337C23—611no—
338C23—611no—
339C23—612no—
340C23—612no—
341C23—1221no—
342C23—1221no—
343C23—1221no—
344C23—1221no—
345C23—2441no—
346C23—2441no—
347C23—2441no—
348C23—2441no—
349C23—2441no—
350C23—2441no—
351C23—2441no—
352C23C2×C22⋊#692311yescharacteristic
353D4—4882no—
354D4—4882no—
355D4—4882no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—612no—
371C2×C4—1221no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—4882no—
380C2×C4—4882no—
381C2×C4—4882no—
382C2×C4—4882no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4—2442no—
389C2×C4—2444no—
390C2×C4—2444no—
391C2×C4—2444no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4D4×#692312yes—
396C8—4882no—
397C22—612no—
398C22—612no—
399C22—1222no—
400C22—1222no—
401C22—1221no—
402C22—1221no—
403C22—1221no—
404C22—1221no—
405C22—1222no—
406C22—1222no—
407C22—4881no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22—4881no—
413C22—4881no—
414C22#1684071011yescharacteristic
415C22#1684074111yescharacteristic
416C22#1684074311yescharacteristic
417C22#1684075211yescharacteristic
418C22#1684075411yescharacteristic
419C22#1684075911yescharacteristic
420C22#1684076111yescharacteristic
421C4—22no—
422C4—41no—
423C4—41no—
424C4—81no—
425C4—81no—
426C4—1222no—
427C4—4884no—
428C4—4881no—
429C4—2441no—
430C4—2441no—
431C2—1222no—
432C2—1222no—
433C2—9761no—
434C2#2146123811yescharacteristic
435C2#2146124011yescharacteristic
436C2#2146126211yescharacteristic
437C2#2146126311yescharacteristic
438C2#2146180711yescharacteristic
439C2#2146183811yescharacteristic
440C2#2146184311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519752C22
cosocleC22×C4×D61C23
hypercenter#389#6923
hypocenterC61#9157
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#16840707C2semidirect
2. #21461711C2semidirect
4. #21460692C2semidirect
17. #16839664C4semidirect
18. #16839663C4semidirect
19. #16839658C4semidirect
24. C61×#389C4semidirect
233. #389#6923semidirect
245. C61#9157semidirect

Maximal subgroups

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

subgroupindex
C2×#168407072
#214617112
C2×#168406142
#214606922
#214541522
#915761