Finite Group Databasebeta

← #30600408

Subgroups — #30600408

Order 15616.

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

normal subgroups only

This group has 738 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
2#21461711C211yesmaximal, characteristic
3C2×#16840615C212yesmaximal
4#21460692C212yesmaximal
5#21454157C211yesmaximal, 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#16840615—24no—
14C2×#14519757C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16839668C411yescharacteristic, complemented
17#16839667C411yescharacteristic, complemented
18#16839658C411yescharacteristic, complemented
19#16839649—21no—
20#16839648—21no—
21#16839633—21no—
22#16839624—21no—
23#16839619C2211yescharacteristic
24C61×#389C411yescharacteristic, 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×Dic122C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C22×C4×D61C2311yescharacteristic
41#14519757—22no—
42#14519757—22no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×C4×Dic61C2×C411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61D411yescharacteristic
51C42×D61—22no—
52C42×D61—22no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C42×C61C2×C411yescharacteristic
63#95×C61—21no—
64C23×D61C2×D411yescharacteristic
65C22×#6923—24no—
66C22×#6923—24no—
67C22×#6923—24no—
68C22×#6923—42no—
69C22×#6923—42no—
70C22×#6923—42no—
71C22×#6923—42no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61—21no—
88C2×C4×D61—22no—
89C2×C4×D61—22no—
90C2×C4×D61—41no—
91C2×C4×D61—41no—
92C2×C4×D61SD1612yes—
93C2×C4×D61Q1612yes—
94C2×C4×D61C2×D412yes—
95C2×C4×D61#4112yes—
96C2×Dic122—22no—
97C2×Dic122—41no—
98C2×Dic122—41no—
99C2×#92420—42no—
100C2×#92420—42no—
101C2×#92420—42no—
102C4⋊#6923—216no—
103C4×#6923—48no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—42no—
107C42×C61—22no—
108#11758198—41no—
109#11758198—41no—
110#11758198—41no—
111#11758198—41no—
112C4⋊Dic61—42no—
113C4⋊Dic61—41no—
114#11758196—42no—
115C4×Dic61—22no—
116C4×Dic61—22no—
117C4×Dic61—22no—
118C23×C61C2×C22⋊C411yescharacteristic
119C22×D61—21no—
120C22×D61—21no—
121C22×D61—21no—
122C22×D61—21no—
123C22×D61#12011yescharacteristic
124C22×D61#12311yescharacteristic
125C22×D61C2×SD1611yescharacteristic
126C22×D61C2×Q1611yescharacteristic
127C22×D61AGL(1,ℤ/8)11yescharacteristic
128C22×D61#13711yescharacteristic
129C2×#6923—28no—
130C2×#6923—44no—
131C2×#6923—44no—
132C2×#6923—44no—
133C2×#6923—44no—
134C2×#6923—44no—
135C2×#6923—44no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61#10312yes—
149C2×C4×C61#10412yes—
150C2×C4×C61C4×D412yes—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61#10312yes—
164C2×Dic61#10412yes—
165C2×Dic61#12112yes—
166C4×D61—24no—
167C4×D61—24no—
168C4×D61—22no—
169C4×D61—22no—
170C4×D61—22no—
171C4×D61—22no—
172C4×D61—42no—
173C4×D61—42no—
174Dic122—82no—
175#92420—82no—
176#9144—611nomaximal, Sylow
177C22×C61#38611yescharacteristic
178C22×C61#41011yescharacteristic
179C22×C61#41011yescharacteristic
180C22×C61C2×#10311yescharacteristic
181C22×C61#41711yescharacteristic
182C22×C61C2×#10411yescharacteristic
183C22×C61C2×#10411yescharacteristic
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—21no—
189C2×D61—21no—
190C2×D61—21no—
191C2×D61—21no—
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—21no—
196C2×D61#45011yescharacteristic
197C2×D61#45111yescharacteristic
198C2×D61#48311yescharacteristic
199C2×D61#48411yescharacteristic
200#6923—48no—
201C4×C61—22no—
202C4×C61—22no—
203C4×C61—22no—
204C4×C61—81no—
205C4×C61—81no—
206Dic61—22no—
207Dic61—22no—
208Dic61—22no—
209Dic61—81no—
210Dic61—81no—
211#2855—611no—
212C2×C4×C4⋊C4—611no—
213#1817—611no—
214C2×#328—612no—
215#1294—612no—
216C2×C61#150311yescharacteristic
217C2×C61#150311yescharacteristic
218C2×C61#150911yescharacteristic
219C2×C61#150911yescharacteristic
220C2×C61#187211yescharacteristic
221C2×C61#189611yescharacteristic
222C2×C61#189911yescharacteristic
223D61—21no—
224D61—21no—
225D61—21no—
226D61—21no—
227C2×C22⋊Q8—611no—
228C22×C4⋊C4—1221no—
229C22×C42—611no—
230C22×C42—1221no—
231C2×C22⋊C8—612no—
232#391—611no—
233#391—611no—
234#389—611no—
235#389#692311yescharacteristic, complemented
236#387—1221no—
237#387—1221no—
238#385—1221no—
239#381—1221no—
240#380—611no—
241C4×C4⋊C4—1224no—
242C2×#95—612no—
243#336—1221no—
244#336—1221no—
245#328—1224no—
246C61#914411yescharacteristic, complemented, Sylow
247C22×Q8—611no—
248C23×C4—611no—
249C23×C4—611no—
250C23×C4—611no—
251C23×C4—1222no—
252C22×C8—1222no—
253C22⋊Q8—1222no—
254C22⋊Q8—1222no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1224no—
258C2×C4⋊C4—1224no—
259C2×C4⋊C4—1224no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4C2×#692311yescharacteristic
263C2×C4⋊C4C2×#692311yescharacteristic
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C42—611no—
267C2×C42—611no—
268C2×C42—611no—
269C2×C42—1222no—
270C2×C42—1222no—
271C2×C42—1224no—
272C2×C42—2442no—
273C2×C42—2442no—
274C2×C42C2×#692311yescharacteristic
275C22⋊C8—1222no—
276C22⋊C8—1222no—
277#95—21no—
278#95—614no—
279#95—1221no—
280#95—1221no—
281#95—1221no—
282#95—1221no—
283#95—1222no—
284#95—1221no—
285#95—1221no—
286C24—611no—
287C2×Q8—1222no—
288C2×Q8—2441no—
289C2×Q8—2441no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—611no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—612no—
296C22×C4—612no—
297C22×C4—612no—
298C22×C4—612no—
299C22×C4—1222no—
300C22×C4—1221no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1222no—
307C22×C4—1224no—
308C22×C4—1224no—
309C22×C4—1224no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—2441no—
313C22×C4—2441no—
314C22×C4—2442no—
315C22×C4—2442no—
316C22×C4—2442no—
317C22×C4—2442no—
318C22×C4C22⋊#692311yescharacteristic
319C22×C4C22⋊#692311yescharacteristic
320C22×C4C22×#692311yescharacteristic
321C2×C8—2442no—
322C2×C8—2442no—
323C2×C8—2442no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—12216no—
328C4⋊C4—2441no—
329C4⋊C4—2442no—
330C4⋊C4—2442no—
331C22⋊C4—2441no—
332C22⋊C4—2441no—
333C22⋊C4—2441no—
334C22⋊C4—2441no—
335C42—22no—
336C42—1222no—
337C42—1222no—
338C42—1222no—
339C42—2448no—
340C23—611no—
341C23—611no—
342C23—611no—
343C23—611no—
344C23—611no—
345C23—611no—
346C23—1221no—
347C23—1221no—
348C23—1221no—
349C23—1221no—
350C23C2×C22⋊#692311yescharacteristic
351Q8—4882no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—612no—
365C2×C4—612no—
366C2×C4—612no—
367C2×C4—1222no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—1222no—
371C2×C4—1224no—
372C2×C4—1224no—
373C2×C4—1221no—
374C2×C4—1221no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1228no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2444no—
389C2×C4—2444no—
390C2×C4—2444no—
391C2×C4—2444no—
392C2×C4—2444no—
393C2×C4—2444no—
394C2×C4#1451984012yes—
395C2×C4#1451984112yes—
396C2×C4D4×#692312yes—
397C8—4882no—
398C22—611no—
399C22—611no—
400C22—611no—
401C22—611no—
402C22—1221no—
403C22—1221no—
404C22—1221no—
405C22—1221no—
406C22—1221no—
407C22—1221no—
408C22—1221no—
409C22—1221no—
410C22—1221no—
411C22—1221no—
412C22—1221no—
413C22—1221no—
414C22#1684071011yescharacteristic
415C22#1684074111yescharacteristic
416C22#1684074311yescharacteristic
417C22C2×#1451984011yescharacteristic
418C22#1684075311yescharacteristic
419C22C2×#1451984111yescharacteristic
420C22D61⋊#10411yescharacteristic
421C4—22no—
422C4—22no—
423C4—22no—
424C4—81no—
425C4—81no—
426C4—1222no—
427C4—1222no—
428C4—1222no—
429C4—4881no—
430C4—4881no—
431C4—2448no—
432C2—1221no—
433C2—1221no—
434C2—1221no—
435C2—1221no—
436C2#2146120111yescharacteristic
437C2#2146120211yescharacteristic
438C2#2146122111yescharacteristic
439C2#2146122211yescharacteristic
440C2#2146180411yescharacteristic
441C2#2146183711yescharacteristic
442C2#2146184311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#389C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#389#6923
hypocenterC61#9144
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
16. #16839668C4semidirect
17. #16839667C4semidirect
18. #16839658C4semidirect
24. C61×#389C4semidirect
235. #389#6923semidirect
246. C61#9144semidirect

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×#168406152
#214606922
#214541572
#914461