Finite Group Databasebeta

← #30598296

Subgroups — #30598296

Order 15616.

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

normal subgroups only

This group has 852 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
1#21461725C212yesmaximal, complemented
2C4×#14519835C211yesmaximal, characteristic, complemented
3#21461215C212yesmaximal, complemented
4#21461074C211yesmaximal, characteristic, complemented
5C2×#16840464C211yesmaximal, 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#16840464—24no—
15#16840464—24no—
16C2×#14519770—21no—
17C2×#14519768C411yescharacteristic, complemented
18C2×#14519761—21no—
19C2×#14519749—21no—
20C2×#14519748C2211yescharacteristic
21C2×D4×Dic61—21no—
22C2×D4×Dic61C411yescharacteristic, complemented
23C2×C4×C22⋊D61C2211yescharacteristic
24C2×#14519735C2211yescharacteristic
25C22×C4⋊Dic61C411yescharacteristic, complemented
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×C22⋊D61—21no—
35C22×C22⋊D61D411yescharacteristic
36C23×Dic61—21no—
37C23×Dic61—21no—
38C22×C4×D61C2311yescharacteristic
39#14519770—28no—
40#14519768—22no—
41#14519768—22no—
42#14519761—28no—
43#14519749—28no—
44#14519748—22no—
45#14519748D414yes—
46D4×Dic61—28no—
47D4×Dic61—22no—
48D4×Dic61—22no—
49C4×C22⋊D61—22no—
50C4×C22⋊D61—22no—
51#14519735—22no—
52#14519735D414yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61C2×C411yescharacteristic
58C2×C4⋊Dic61C2×C411yescharacteristic
59C2×#11758196—21no—
60C2×#11758196D411yescharacteristic
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61D411yescharacteristic
64C2×#11758198—21no—
65C2×#11758198D411yescharacteristic
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—42no—
83C2×D4×C61—42no—
84C2×D4×C61C22⋊C412yes—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—42no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22×C411yescharacteristic
91C2×C22⋊D61—22no—
92C2×C22⋊D61—22no—
93C2×C22⋊D61—24no—
94C2×C22⋊D61—41no—
95C2×C22⋊D61—82no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—24no—
99C22×Dic61—24no—
100C22×Dic61—82no—
101C22×Dic61—82no—
102C22×Dic61C22⋊C411yescharacteristic
103C22×Dic61C22⋊C411yescharacteristic
104C22×Dic61C2×D411yescharacteristic
105C2×C4×D61—42no—
106C2×C4×D61C2×D412yes—
107C2×C4×D61C2×D412yes—
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—22no—
119#11758198—24no—
120#11758198—22no—
121C4⋊Dic61—24no—
122C4⋊Dic61—24no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61C22⋊C412yes—
126C4⋊Dic61C22⋊C412yes—
127#11758196—24no—
128#11758196—22no—
129C4×Dic61—21no—
130C4×Dic61—21no—
131C4×Dic61—24no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—22no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—42no—
138C23×C61—42no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61#10412yes—
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—48no—
153D4×C61—42no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—21no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C22⋊D61—44no—
165C22⋊D61—84no—
166C2×Dic61—21no—
167C2×Dic61—21no—
168C2×Dic61—22no—
169C2×Dic61—22no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—41no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—41no—
176C2×Dic61—84no—
177C2×Dic61—84no—
178C2×Dic61#12012yes—
179C4×D61—24no—
180C4×D61—42no—
181#92420—82no—
182#8362—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—22no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—42no—
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—41no—
206C4×C61—42no—
207C4×C61#41012yes—
208Dic61—22no—
209Dic61—42no—
210Dic61—42no—
211Dic61—41no—
212Dic61—41no—
213C2×#553—611no—
214#1826—612no—
215C4×#99—611no—
216#1507—612no—
217#1465—611no—
218C2×C61—21no—
219C2×C61—21no—
220C2×C61—42no—
221C2×C61—82no—
222C2×C61#178211yescharacteristic
223C2×C61#178911yescharacteristic
224C2×C61#179511yescharacteristic
225D61—42no—
226#553—1224no—
227#553—1224no—
228C2×#126—1221no—
229C2×#125—611no—
230C2×#123—1221no—
231C2×#121—611no—
232C2×#121—1221no—
233C2×#121#692311yescharacteristic, complemented
234C2×C4×D4—611no—
235C2×C4×D4—611no—
236C2×C4×D4—1221no—
237C2×#117—611no—
238C22×C4⋊C4—611no—
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#836211yescharacteristic, complemented, Sylow
248C22×D4—21no—
249C22×D4—611no—
250C22×D4—1221no—
251C22×D4C2×#692311yescharacteristic
252C23×C4—611no—
253C23×C4—1221no—
254C23×C4—1221no—
255C23×C4C2×#692311yescharacteristic
256C2×M16—1222no—
257#126—1228no—
258#125—1222no—
259#125—1222no—
260#123—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—1228no—
270C4×D4—1222no—
271#117—614no—
272#117—1222no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—611no—
275C2×C4⋊C4—611no—
276C2×C4⋊C4—1222no—
277C2×C4⋊C4—1222no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4—1221no—
280C2×C4⋊C4—1221no—
281C2×C4⋊C4C2×#692311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—611no—
284C2×C22⋊C4—611no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4—1222no—
289C2×C42—611no—
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×D4—24no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—1224no—
305C2×D4—1222no—
306C2×D4—1222no—
307C2×D4—4882no—
308C2×D4—2441no—
309C2×D4C22⋊#692312yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—42no—
314C22×C4—611no—
315C22×C4—612no—
316C22×C4—612no—
317C22×C4—611no—
318C22×C4—611no—
319C22×C4—1224no—
320C22×C4—1224no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—4882no—
324C22×C4—4882no—
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—1224no—
337C4⋊C4—1224no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C4⋊C4C22⋊#692312yes—
341C22⋊C4—24no—
342C22⋊C4—1222no—
343C22⋊C4—1224no—
344C22⋊C4—1222no—
345C22⋊C4—1224no—
346C22⋊C4—1224no—
347C22⋊C4—1224no—
348C22⋊C4—2442no—
349C42—1224no—
350C42—1221no—
351C42—1221no—
352C42—2444no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—612no—
363C23—1221no—
364C23—1221no—
365C23—2441no—
366C23—2441no—
367C23#1451984112yes—
368C23C2×C22⋊#692311yescharacteristic
369D4—42no—
370D4—48no—
371D4—42no—
372D4—4884no—
373D4—2444no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—612no—
383C2×C4—1224no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1222no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1222no—
390C2×C4—4882no—
391C2×C4—4882no—
392C2×C4—4884no—
393C2×C4—4884no—
394C2×C4—2442no—
395C2×C4—2441no—
396C2×C4—2444no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2441no—
400C2×C4C2×C22⋊#692311yescharacteristic
401C2×C4C2×C22⋊#692311yescharacteristic
402C8—4882no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—42no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—82no—
414C22—82no—
415C22—1222no—
416C22—1222no—
417C22—1222no—
418C22—4881no—
419C22—4881no—
420C22#1684070311yescharacteristic
421C22C2×#1451984111yescharacteristic
422C22#1684076011yescharacteristic
423C4—21no—
424C4—41no—
425C4—42no—
426C4—1222no—
427C4—4884no—
428C4—2441no—
429C4—2441no—
430C4—2442no—
431C4—2442no—
432C4#1684074212yes—
433C2—21no—
434C2—21no—
435C2—42no—
436C2—82no—
437C2—2442no—
438C2#2146164911yescharacteristic
439C2#2146166011yescharacteristic
440C2#2146167011yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#121×C61C4
socleC22×C61#379
prime radicalC2×#16840464C2
cosocleC22×C4×D61C23
hypercenterC2×#121#6923
hypocenterC61#8362
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. #21461725C2semidirect
2. C4×#14519835C2semidirect
3. #21461215C2semidirect
4. #21461074C2semidirect
17. C2×#14519768C4semidirect
22. C2×D4×Dic61C4semidirect
25. C22×C4⋊Dic61C4semidirect
26. C2×#121×C61C4semidirect
233. C2×#121#6923semidirect
247. C61#8362semidirect

Maximal subgroups

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

subgroupindex
#214617252
C4×#145198352
#214612152
#214610742
C2×#168404642
#836261