Finite Group Databasebeta

← #30600398

Subgroups — #30600398

Order 15616.

612 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 120 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×#16840767C211yesmaximal, characteristic
2#21461676C211yesmaximal, characteristic
3C2×#16840621C212yesmaximal, complemented
4#21460697C212yesmaximal, complemented
5#21454170C211yesmaximal, characteristic
6C22×#11758213—21no—
7#16840767—24no—
8C2×#14519844—21no—
9C2×#14519844C2212yes—
10C2×#14519844C2212yes—
11C2×#14519834—21no—
12#16840628—21no—
13#16840621—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519755—21no—
16C2×C42×D61C2211yescharacteristic
17#16839666—21no—
18#16839660C411yescharacteristic
19#16839657—21no—
20#16839631C2211yescharacteristic
21#16839621—21no—
22#16839619C411yescharacteristic
23#16839619C411yescharacteristic
24C61×#389C411yescharacteristic
25C2×#11758213—24no—
26C2×#11758213—42no—
27C2×#11758213—42no—
28C22×#92420—22no—
29C22×#92420—22no—
30#14519844—24no—
31#14519844—22no—
32#14519844—22no—
33#14519844—22no—
34#14519844D414yes—
35#14519834—42no—
36C22×Dic122—21no—
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40#14519776—22no—
41#14519776—22no—
42#14519755—42no—
43#14519755—42no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×C4⋊Dic61C2×C411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196—21no—
50C2×#11758198—21no—
51C2×#11758198—21no—
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61C2311yescharacteristic
55C42×D61—22no—
56C42×D61—22no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C4⋊C4×C61C2×C411yescharacteristic
63C2×C42×C61C2×C411yescharacteristic
64#95×C61—21no—
65C23×D61C22⋊C411yescharacteristic
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—22no—
81C2×C4×D61—22no—
82C2×C4×D61—22no—
83C2×C4×D61—21no—
84C2×C4×D61—22no—
85C2×C4×D61—21no—
86C2×C4×D61—41no—
87C2×C4×D61D812yes—
88C2×C4×D61SD1612yes—
89C2×Dic122—41no—
90C2×Dic122—41no—
91C2×Dic122—41no—
92C2×Dic122—41no—
93C2×Dic122—41no—
94C2×Dic122—41no—
95C2×#92420—24no—
96C2×#92420—42no—
97C2×#92420—42no—
98C2×#92420—42no—
99C2×#92420—42no—
100C2×#92420—42no—
101#11758213—48no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—42no—
105C42×C61—22no—
106#11758198—42no—
107#11758198—42no—
108C4⋊Dic61—22no—
109C4⋊Dic61—41no—
110#11758196—42no—
111#11758196—42no—
112#11758196—41no—
113#11758196—41no—
114#11758196—41no—
115#11758196—41no—
116C4×Dic61—21no—
117C4×Dic61—22no—
118C4×Dic61—21no—
119C4×Dic61C2×D412yes—
120C4×Dic61#4112yes—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—21no—
128C22×D61#10212yes—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61#10212yes—
143C2×C4×C61C4×D412yes—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61#10112yes—
153C2×Dic61#12011yescharacteristic
154C2×Dic61#12112yes—
155C2×Dic61#12311yescharacteristic
156C2×Dic61C2×D811yescharacteristic
157C2×Dic61C2×SD1611yescharacteristic
158C2×Dic61AGL(1,ℤ/8)11yescharacteristic
159C2×Dic61AGL(1,ℤ/8)11yescharacteristic
160C2×Dic61#13711yescharacteristic
161C2×Dic61#13712yes—
162C2×Dic61#13711yescharacteristic
163C4×D61—22no—
164C4×D61—22no—
165C4×D61—22no—
166C4×D61—22no—
167C4×D61—44no—
168C4×D61—42no—
169C4×D61—42no—
170Dic122—81no—
171Dic122—81no—
172Dic122—81no—
173Dic122—81no—
174#92420—44no—
175#92420—82no—
176#9139—611nomaximal, Sylow
177C22×C61#38611yescharacteristic
178C22×C61C2×#10111yescharacteristic
179C22×C61#41311yescharacteristic
180C22×C61C2×#10211yescharacteristic
181C22×C61#41711yescharacteristic
182C22×C61#41711yescharacteristic
183C22×C61#41911yescharacteristic
184C2×D61—22no—
185C2×D61—22no—
186C2×D61—22no—
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—41no—
190C2×D61—41no—
191C4×C61—22no—
192C4×C61—22no—
193C4×C61—41no—
194C4×C61—81no—
195C4×C61—81no—
196Dic61—22no—
197Dic61—81no—
198Dic61—81no—
199Dic61#44711yescharacteristic
200Dic61#44811yescharacteristic
201Dic61#46812yes—
202Dic61#47012yes—
203Dic61#48011yescharacteristic
204Dic61#48111yescharacteristic
205C2×#423—611no—
206#2857—611no—
207#1797—611no—
208C2×#332—612no—
209#1296—612no—
210C2×C61#154611yescharacteristic
211C2×C61#154711yescharacteristic
212C2×C61#155911yescharacteristic
213C2×C61#156111yescharacteristic
214C2×C61#187411yescharacteristic
215C2×C61#189411yescharacteristic
216C2×C61#190111yescharacteristic
217D61—42no—
218C22×M16—1221no—
219C2×C4⋊Q8—611no—
220C2×C22⋊Q8—1221no—
221C22×C42—611no—
222#423—1224no—
223C2×C4⋊C8—612no—
224C2×C4⋊C8—612no—
225C2×C4⋊C8—1221no—
226C2×C22⋊C8—1221no—
227#391—1221no—
228#389—611no—
229#389—1221no—
230#389#692311yescharacteristic
231#384—611no—
232#380—611no—
233#380—611no—
234#380—1221no—
235#336—1221no—
236#332—1224no—
237C61#913911yescharacteristic, complemented, Sylow
238C22×Q8—1221no—
239C23×C4—611no—
240C23×C4—611no—
241C23×C4—611no—
242C2×M16—1224no—
243C2×M16—2442no—
244C2×M16—2442no—
245C22×C8—1222no—
246C22×C8—1222no—
247C4⋊Q8—1222no—
248C4⋊Q8—1222no—
249C22⋊Q8—2442no—
250C22⋊Q8—2442no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—1221no—
255C2×C4⋊C4—1221no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4C2×#692311yescharacteristic
259C2×C4⋊C4C2×#692311yescharacteristic
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C42—611no—
263C2×C42—611no—
264C2×C42—611no—
265C2×C42—1222no—
266C2×C42—1222no—
267C2×C42C2×#692311yescharacteristic
268C4⋊C8—614no—
269C4⋊C8—1222no—
270C4⋊C8—1222no—
271C4⋊C8—1222no—
272C4⋊C8—1224no—
273C22⋊C8—2442no—
274#95—21no—
275#95—1221no—
276#95—1221no—
277#95—1221no—
278C24—611no—
279C2×Q8—2441no—
280C2×Q8—2441no—
281C2×Q8—2441no—
282C2×Q8—2441no—
283C2×Q8—2441no—
284C2×Q8—2441no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—611no—
290C22×C4—612no—
291C22×C4—612no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1222no—
298C22×C4—1222no—
299C22×C4—1221no—
300C22×C4—1222no—
301C22×C4—1222no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—2441no—
305C22×C4C22⋊#692311yescharacteristic
306C22×C4C22⋊#692311yescharacteristic
307C22×C4C22×#692311yescharacteristic
308M16—2448no—
309C2×C8—1224no—
310C2×C8—2442no—
311C2×C8—2442no—
312C2×C8—2442no—
313C2×C8—2442no—
314C2×C8—2442no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—1222no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C4⋊C4—2442no—
325C4⋊C4—2442no—
326C22⋊C4—2442no—
327C22⋊C4—2442no—
328C42—22no—
329C42—612no—
330C42—612no—
331C42—1221no—
332C42—1222no—
333C42—1221no—
334C23—612no—
335C23—1221no—
336C23—1221no—
337C23—1221no—
338C23—1221no—
339C23—1221no—
340C23—1221no—
341C23C2×C22⋊#692311yescharacteristic
342Q8—4881no—
343Q8—4881no—
344Q8—4881no—
345Q8—4881no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—611no—
360C2×C4—612no—
361C2×C4—611no—
362C2×C4—611no—
363C2×C4—611no—
364C2×C4—611no—
365C2×C4—611no—
366C2×C4—611no—
367C2×C4—611no—
368C2×C4—612no—
369C2×C4—612no—
370C2×C4—1222no—
371C2×C4—1222no—
372C2×C4—1222no—
373C2×C4—1222no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—2444no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4#1451983912yes—
386C2×C4D4×#692312yes—
387C8—4882no—
388C8—2444no—
389C22—1222no—
390C22—1222no—
391C22—1222no—
392C22—2441no—
393C22—2441no—
394C22—2441no—
395C22—2441no—
396C22#1684071011yescharacteristic
397C22D61⋊#10111yescharacteristic
398C22#1684074811yescharacteristic
399C22C2×#1451983911yescharacteristic
400C22#1684075311yescharacteristic
401C22#1684075311yescharacteristic
402C22#1684075511yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—81no—
407C4—81no—
408C4—611no—
409C4—611no—
410C4—611no—
411C4—611no—
412C4—612no—
413C4—612no—
414C4—1222no—
415C4—4881no—
416C4—4881no—
417C2—2442no—
418C2#2146132911yescharacteristic
419C2#2146133011yescharacteristic
420C2#2146135411yescharacteristic
421C2#2146135911yescharacteristic
422C2#2146180611yescharacteristic
423C2#2146183411yescharacteristic
424C2#2146184711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#389#6923
hypocenterC61#9139
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
3. C2×#16840621C2semidirect
4. #21460697C2semidirect
237. C61#9139semidirect

Maximal subgroups

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

subgroupindex
C2×#168407672
#214616762
C2×#168406212
#214606972
#214541702
#913961