Finite Group Databasebeta

← #30598288

Subgroups — #30598288

Order 15616.

846 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 98 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
1#21461726C212yesmaximal, complemented
2C4×#14519835C211yesmaximal, characteristic, complemented
3#21461215C212yesmaximal, complemented
4#21461074C211yesmaximal, characteristic, complemented
5C2×#16840476C211yesmaximal, characteristic
6C2×#14519835—21no—
7#16840733C2212yes—
8C4×C22⋊#6923C2212yes—
9#16840640—21no—
10#16840632—24no—
11#16840613—24no—
12#16840608—21no—
13#16840605—21no—
14#16840476—24no—
15#16840476—24no—
16C2×#14519776C411yescharacteristic, complemented
17C2×#14519764—21no—
18C2×#14519763—21no—
19C2×#14519749—21no—
20C2×#14519748C2211yescharacteristic
21C2×D4×Dic61C411yescharacteristic, complemented
22C2×C4×C22⋊D61C2211yescharacteristic
23C2×#14519736C411yescharacteristic, complemented
24C2×#14519735C2211yescharacteristic
25C2×#14519733—21no—
26C2×#121×C61C411yescharacteristic, complemented
27C2×#11758213—22no—
28C2×C22⋊#6923—22no—
29C2×C4×#6923—22no—
30#14519835—24no—
31#14519834—22no—
32C22⋊#92420—22no—
33(C2×#6923)⋊C4—22no—
34C22×Dic122—21no—
35C22×C22⋊D61—21no—
36C22×C22⋊D61D411yescharacteristic
37C23×Dic61—21no—
38C22×C4×D61C2311yescharacteristic
39#14519776—22no—
40#14519776—22no—
41#14519764—28no—
42#14519763—28no—
43#14519749—28no—
44#14519748—22no—
45#14519748D414yes—
46D4×Dic61—22no—
47D4×Dic61—22no—
48C4×C22⋊D61—22no—
49C4×C22⋊D61—22no—
50#14519736—22no—
51#14519736—22no—
52#14519735—22no—
53#14519735D414yes—
54#14519733—28no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196—21no—
58C2×#11758196D411yescharacteristic
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61D411yescharacteristic
63C2×#11758198—21no—
64C2×#11758198D411yescharacteristic
65C2×C4×Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61C2×C411yescharacteristic
68C22×D4×C61—21no—
69C22×D4×C61C2×C411yescharacteristic
70C23×C4×C61C2×C411yescharacteristic
71#121×C61—22no—
72#121×C61—22no—
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C22⋊C4×C61—21no—
75C24×C61—21no—
76C24×C61C22⋊C411yescharacteristic
77C23×D61C2×D411yescharacteristic
78C22×#6923—42no—
79C2×D4×C61—24no—
80C2×D4×C61—21no—
81C2×D4×C61—42no—
82C2×D4×C61—82no—
83C2×D4×C61C22⋊C412yes—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—42no—
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22×C411yescharacteristic
90C2×C22⋊D61—22no—
91C2×C22⋊D61—24no—
92C2×C22⋊D61—22no—
93C2×C22⋊D61—41no—
94C2×C22⋊D61—82no—
95C22×Dic61—24no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—42no—
99C22×Dic61—42no—
100C22×Dic61C22⋊C411yescharacteristic
101C22×Dic61C22⋊C411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C2×C4×D61—42no—
104C2×C4×D61C2×D412yes—
105C2×C4×D61C2×D412yes—
106C2×Dic122—24no—
107C2×Dic122—82no—
108C22⋊#6923—24no—
109C22⋊#6923—42no—
110C2×#92420—42no—
111C4×#6923—44no—
112#11758213—44no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61C22⋊C412yes—
115C22⋊C4×C61—24no—
116C22⋊Dic61—24no—
117C22⋊Dic61—24no—
118C22⋊Dic61—24no—
119C22⋊Dic61—22no—
120#11758198—24no—
121#11758198—22no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61C22⋊C412yes—
125C4⋊Dic61C22⋊C412yes—
126#11758196—24no—
127#11758196—22no—
128C4×Dic61—24no—
129C4×Dic61—21no—
130C4×Dic61—21no—
131C4×Dic61—24no—
132C23×C61—21no—
133C23×C61—22no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—42no—
137C23×C61—42no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61C2×C22⋊C411yescharacteristic
143C22×D61—21no—
144C22×D61—21no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61#12012yes—
148C2×#6923—44no—
149C2×#6923—82no—
150C2×#6923—82no—
151D4×C61—42no—
152D4×C61—42no—
153D4×C61—84no—
154C2×C4×C61—22no—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—41no—
161C2×C4×C61C2×C22⋊C411yescharacteristic
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C22⋊D61—44no—
164C22⋊D61—84no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—21no—
170C2×Dic61—41no—
171C2×Dic61—48no—
172C2×Dic61—42no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61#10412yes—
177C2×Dic61#12012yes—
178C4×D61—24no—
179C4×D61—42no—
180Dic122—84no—
181#92420—82no—
182#8360—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—42no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61—42no—
189C22×C61—42no—
190C22×C61—41no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—82no—
194C22×C61—82no—
195C22×C61#37911yescharacteristic
196C22×C61C2×#10411yescharacteristic
197C22×C61#42111yescharacteristic
198C2×D61—22no—
199C2×D61—22no—
200C2×D61—22no—
201C2×D61—81no—
202C2×D61—81no—
203#6923—84no—
204C4×C61—21no—
205C4×C61—42no—
206C4×C61—41no—
207C4×C61#41012yes—
208Dic61—22no—
209Dic61—22no—
210Dic61—41no—
211Dic61—42no—
212Dic61—42no—
213C2×#555—611no—
214#1826—612no—
215C4×#99—611no—
216#1507—612no—
217#1465—611no—
218C2×C61—21no—
219C2×C61—21no—
220C2×C61—81no—
221C2×C61—82no—
222C2×C61#178211yescharacteristic
223C2×C61#178911yescharacteristic
224C2×C61#179511yescharacteristic
225D61—42no—
226#555—1224no—
227#555—1224no—
228C2×C4⋊Q8—611no—
229C2×#124—1221no—
230C2×#124—1221no—
231C2×#121—611no—
232C2×#121—1221no—
233C2×#121#692311yescharacteristic, complemented
234C2×C4×D4—611no—
235C2×C4×D4—611no—
236C2×#117—611no—
237C2×#117—611no—
238C2×#117—1221no—
239C2×#99—1221no—
240#407—612no—
241C4×C22⋊C4—612no—
242#342—1221no—
243#339—1224no—
244#327—1224no—
245#324—1221no—
246#323—1221no—
247C61#836011yescharacteristic, complemented, Sylow
248C22×Q8—1221no—
249C22×D4—21no—
250C22×D4—611no—
251C22×D4—1221no—
252C22×D4C2×#692311yescharacteristic
253C23×C4—611no—
254C23×C4—1221no—
255C23×C4C2×#692311yescharacteristic
256C2×M16—1222no—
257C4⋊Q8—1222no—
258C4⋊Q8—1222no—
259#124—1228no—
260#124—1228no—
261#121—22no—
262#121—22no—
263#121—614no—
264#121—1222no—
265#121—1228no—
266C4×D4—1222no—
267C4×D4—1222no—
268C4×D4—1222no—
269C4×D4—1222no—
270#117—614no—
271#117—1228no—
272#117—1222no—
273#117—1222no—
274#117—1222no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—611no—
277C2×C4⋊C4—611no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4C2×#692311yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—611no—
282C2×C22⋊C4—611no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1222no—
288C2×C42—611no—
289C2×C42—1221no—
290C2×C42—1221no—
291C2×C42—1222no—
292#99—1224no—
293C22⋊C8—1222no—
294C22⋊C8—1222no—
295#95—1222no—
296C24—21no—
297C24—611no—
298C24C22⋊#692311yescharacteristic
299C2×Q8—1224no—
300C2×Q8—4882no—
301C2×D4—24no—
302C2×D4—21no—
303C2×D4—42no—
304C2×D4—82no—
305C2×D4—1222no—
306C2×D4—1224no—
307C2×D4—1222no—
308C2×D4—4882no—
309C2×D4—2441no—
310C2×D4C22⋊#692312yes—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—42no—
315C22×C4—611no—
316C22×C4—611no—
317C22×C4—612no—
318C22×C4—612no—
319C22×C4—611no—
320C22×C4—1224no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—2442no—
324C22×C4—2442no—
325C22×C4—2442no—
326C22×C4—2442no—
327C22×C4C22⋊#692311yescharacteristic
328C22×C4C22×#692311yescharacteristic
329M16—2444no—
330C2×C8—2442no—
331C4⋊C4—21no—
332C4⋊C4—612no—
333C4⋊C4—612no—
334C4⋊C4—1222no—
335C4⋊C4—1224no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4C22⋊#692312yes—
339C22⋊C4—24no—
340C22⋊C4—1222no—
341C22⋊C4—1224no—
342C22⋊C4—1222no—
343C22⋊C4—1224no—
344C22⋊C4—1224no—
345C22⋊C4—1224no—
346C22⋊C4—1224no—
347C22⋊C4—2442no—
348C42—1224no—
349C42—1224no—
350C42—1221no—
351C42—1221no—
352C42—2444no—
353C23—22no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—612no—
364C23—1221no—
365C23—1221no—
366C23—2441no—
367C23—2441no—
368C23C2×C22⋊#692311yescharacteristic
369Q8—4884no—
370D4—42no—
371D4—42no—
372D4—84no—
373D4—4884no—
374D4—2444no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—612no—
383C2×C4—612no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1224no—
387C2×C4—1222no—
388C2×C4—1221no—
389C2×C4—1222no—
390C2×C4—4882no—
391C2×C4—4882no—
392C2×C4—2442no—
393C2×C4—2441no—
394C2×C4—2448no—
395C2×C4—2444no—
396C2×C4—2442no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4C2×C22⋊#692311yescharacteristic
401C2×C4C2×C22⋊#692311yescharacteristic
402C8—4882no—
403C22—21no—
404C22—21no—
405C22—41no—
406C22—42no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—81no—
412C22—81no—
413C22—82no—
414C22—82no—
415C22—1222no—
416C22—1222no—
417C22—1222no—
418C22—4881no—
419C22—4881no—
420C22#1684070311yescharacteristic
421C22D61⋊#10411yescharacteristic
422C22#1684075911yescharacteristic
423C4—21no—
424C4—41no—
425C4—42no—
426C4—1222no—
427C4—1222no—
428C4—4884no—
429C4—2442no—
430C4—2442no—
431C4—2441no—
432C4#1684074212yes—
433C2—21no—
434C2—21no—
435C2—82no—
436C2—81no—
437C2—2442no—
438C2#2146164911yescharacteristic
439C2#2146166111yescharacteristic
440C2#2146166911yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#121×C61C4
socleC22×C61#379
prime radicalC2×#16840476C2
cosocleC22×C4×D61C23
hypercenterC2×#121#6923
hypocenterC61#8360
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #21461726C2semidirect
2. C4×#14519835C2semidirect
3. #21461215C2semidirect
4. #21461074C2semidirect
16. C2×#14519776C4semidirect
21. C2×D4×Dic61C4semidirect
23. C2×#14519736C4semidirect
26. C2×#121×C61C4semidirect
233. C2×#121#6923semidirect
247. C61#8360semidirect

Maximal subgroups

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

subgroupindex
#214617262
C4×#145198352
#214612152
#214610742
C2×#168404762
#836061