Finite Group Databasebeta

← #30604846

Subgroups — #30604846

Order 15616.

648 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C4×#14519835C211yesmaximal, characteristic, complemented
2#21461058C211yesmaximal, characteristic, complemented
3#21460814C212yesmaximal, complemented
4#21460797C212yesmaximal, complemented
5#21449032C211yesmaximal, characteristic
6#16840779—21no—
7C2×#14519835—21no—
8C4×C22⋊#6923—21no—
9#16840640C2212yes—
10#16840606C2212yes—
11#16840605—21no—
12C2×C4×C22⋊D61—21no—
13C2×C4×C22⋊D61C2211yescharacteristic
14#16839681C411yescharacteristic, complemented
15#16839676—21no—
16#16839676C411yescharacteristic, complemented
17#16839673C2211yescharacteristic
18#16839670—21no—
19#16839670C2211yescharacteristic
20C4×C22⋊Dic61—21no—
21C4×C22⋊Dic61C411yescharacteristic, complemented
22C22×#41×C61C411yescharacteristic, complemented
23C2×C22⋊#6923—22no—
24C2×C4×#6923—41no—
25#14519844—41no—
26#14519835—24no—
27C22⋊#92420—22no—
28C4×#92420—41no—
29(C2×#6923)⋊C4—41no—
30C22×C22⋊D61—21no—
31C22×C22⋊D61D411yescharacteristic
32C22×C4×D61D411yescharacteristic
33C4×C22⋊D61—41no—
34C4×C22⋊D61—41no—
35C4×C22⋊D61—44no—
36C2×#11758196—21no—
37C2×#11758196C2×C411yescharacteristic
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61D411yescharacteristic
43C2×#11758198—21no—
44C2×#11758198C2311yescharacteristic
45C2×C4×Dic61—21no—
46C2×C4×Dic61D411yescharacteristic
47#14519684—21no—
48#14519684—21no—
49#14519684C2×C411yescharacteristic
50#14519684C2×C411yescharacteristic
51C2×#41×C61—44no—
52C2×#41×C61—41no—
53C2×#41×C61—41no—
54C22×Q8×C61C2×C411yescharacteristic
55C22×D4×C61—21no—
56C22×D4×C61C2×C411yescharacteristic
57C23×C4×C61—21no—
58C23×C4×C61C2×C411yescharacteristic
59C24×C61—21no—
60C24×C61C22⋊C411yescharacteristic
61C23×D61C2×D411yescharacteristic
62C22×#6923—42no—
63#41×C61—44no—
64#41×C61—44no—
65#41×C61—44no—
66#41×C61—44no—
67C2×Q8×C61—42no—
68C2×Q8×C61—41no—
69C2×D4×C61—42no—
70C2×D4×C61—41no—
71C2×D4×C61—42no—
72C2×D4×C61—42no—
73C2×D4×C61—42no—
74C22×C4×C61—24no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—42no—
79C22×C4×C61—42no—
80C22×C4×C61—41no—
81C22×C4×C61—41no—
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22×C411yescharacteristic
84C2×C22⋊D61—22no—
85C2×C22⋊D61—22no—
86C2×C22⋊D61—41no—
87C2×C22⋊D61—42no—
88C2×C22⋊D61—42no—
89C2×C22⋊D61—42no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61C22⋊C411yescharacteristic
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C2×D411yescharacteristic
95C2×C4×D61—21no—
96C2×C4×D61—21no—
97C2×C4×D61—42no—
98C22⋊#6923—24no—
99C22⋊#6923—42no—
100C2×#92420—42no—
101C4×#6923—82no—
102C22⋊Dic61—41no—
103C22⋊Dic61—42no—
104C22⋊Dic61—42no—
105C22⋊Dic61—42no—
106C22⋊Dic61—42no—
107#11758198—41no—
108#11758198—42no—
109#11758196—41no—
110#11758196—42no—
111C4×Dic61—21no—
112C4×Dic61—21no—
113C4×Dic61—42no—
114C23×C61—22no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61C2×C22⋊C411yescharacteristic
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61#13512yes—
132C2×#6923—44no—
133C2×#6923—82no—
134C2×#6923—82no—
135Q8×C61—42no—
136Q8×C61—42no—
137D4×C61—42no—
138D4×C61—48no—
139D4×C61—42no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—21no—
143C2×C4×C61—44no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—44no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61#9911yescharacteristic
151C2×C4×C61#9911yescharacteristic
152C22⋊D61—44no—
153C22⋊D61—48no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—22no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61#10412yes—
167C4×D61—42no—
168C4×D61—81no—
169#92420—82no—
170#11725—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—42no—
174C22×C61—42no—
175C22×C61—41no—
176C22×C61—42no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61C2×#9911yescharacteristic
186C22×C61#41611yescharacteristic
187C22×C61C2×#10411yescharacteristic
188C2×D61—22no—
189C2×D61—41no—
190C2×D61—41no—
191C2×D61—42no—
192C2×D61—81no—
193#6923—84no—
194C4×C61—21no—
195C4×C61—42no—
196C4×C61—41no—
197C4×C61AΣL(1,𝔽16)12yes—
198Dic61—22no—
199Dic61—42no—
200Dic61—41no—
201Dic61—41no—
202Dic61—81no—
203#2388—611no—
204C4×#99—611no—
205#1460—611no—
206#1351—612no—
207#1343—612no—
208C2×C61—21no—
209C2×C61—21no—
210C2×C61—81no—
211C2×C61—82no—
212C2×C61#150111yescharacteristic
213C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
214C2×C61#212611yescharacteristic
215D61—42no—
216C22×#41#692311yescharacteristic, complemented
217C2×C4×D4—611no—
218C2×C4×D4—1221no—
219#431—1221no—
220C2×#99—1221no—
221#395—611no—
222#394—611no—
223#394—1221no—
224#393—611no—
225#392—611no—
226#392—1221no—
227C4×C22⋊C4—611no—
228C4×C22⋊C4—1221no—
229C4×C22⋊C4—1221no—
230#342—612no—
231#323—612no—
232#323—1221no—
233C61#1172511yescharacteristic, complemented, Sylow
234C2×#41—44no—
235C2×#41—41no—
236C2×#41—41no—
237C22×Q8C2×#692311yescharacteristic
238C22×D4—21no—
239C22×D4—611no—
240C22×D4—1221no—
241C22×D4C2×#692311yescharacteristic
242C23×C4—21no—
243C23×C4—611no—
244C23×C4C2×#692311yescharacteristic
245C4×D4—2441no—
246C4×D4—2441no—
247C4×D4—2444no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1221no—
250C2×C22⋊C4—611no—
251C2×C22⋊C4—611no—
252C2×C22⋊C4—1222no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C42—611no—
259C2×C42—1221no—
260C2×C42—2441no—
261C4⋊C8—2441no—
262#99—1224no—
263C22⋊C8—1222no—
264C4×C8—2441no—
265#95—611no—
266#95—611no—
267#95—1221no—
268#95—1221no—
269#95—2441no—
270C24—21no—
271C24—611no—
272C24C22⋊#692311yescharacteristic
273#41—44no—
274#41—44no—
275#41—44no—
276#41—44no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—1222no—
285C2×D4—1222no—
286C2×D4—2441no—
287C2×D4—2442no—
288C2×D4—2442no—
289C2×D4—2442no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—611no—
301C22×C4—1221no—
302C22×C4—1221no—
303C22×C4—1221no—
304C22×C4—1221no—
305C22×C4—2442no—
306C22×C4—2442no—
307C22×C4C22⋊#692311yescharacteristic
308C22×C4C22×#692311yescharacteristic
309C2×C8—2442no—
310C4⋊C4—2441no—
311C4⋊C4—2442no—
312C22⋊C4—1224no—
313C22⋊C4—2441no—
314C22⋊C4—2442no—
315C22⋊C4—2442no—
316C22⋊C4—2441no—
317C22⋊C4—2442no—
318C22⋊C4—2442no—
319C22⋊C4—2442no—
320C22⋊C4—2442no—
321C42—1221no—
322C42—1221no—
323C42—4882no—
324C42—2442no—
325C23—22no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—612no—
338C23—1221no—
339C23—1221no—
340C23—2441no—
341C23—2441no—
342C23C2×C22⋊#692311yescharacteristic
343Q8—42no—
344Q8—42no—
345D4—42no—
346D4—48no—
347D4—42no—
348D4—2444no—
349D4—2448no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—612no—
361C2×C4—1221no—
362C2×C4—1221no—
363C2×C4—1222no—
364C2×C4—1221no—
365C2×C4—1221no—
366C2×C4—4881no—
367C2×C4—4882no—
368C2×C4—4882no—
369C2×C4—2441no—
370C2×C4—2441no—
371C2×C4—2444no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2442no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4#1451983511yescharacteristic
379C2×C4#1451983511yescharacteristic
380C8—4882no—
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—1222no—
396C22—4881no—
397C22—2442no—
398C22—2441no—
399C22—2441no—
400C22C2×#1451983511yescharacteristic
401C22#1684075211yescharacteristic
402C22D61⋊#10411yescharacteristic
403C4—21no—
404C4—42no—
405C4—41no—
406C4—1222no—
407C4—4881no—
408C4—4884no—
409C4—2442no—
410C4—2441no—
411C4—2441no—
412C4#1684066212yes—
413C2—21no—
414C2—21no—
415C2—82no—
416C2—81no—
417C2—2442no—
418C2#2146119411yescharacteristic
419C2C2×#1684066211yescharacteristic
420C2#2146209011yescharacteristic
subgroupG/H
centerC2×C4#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×#41×C61C4
socleC22×C61C2×#99
prime radical#21449032C2
cosocleC2×#11758198C23
hypercenterC22×#41#6923
hypocenterC61#11725
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C4×#14519835C2semidirect
2. #21461058C2semidirect
3. #21460814C2semidirect
4. #21460797C2semidirect
14. #16839681C4semidirect
16. #16839676C4semidirect
21. C4×C22⋊Dic61C4semidirect
22. C22×#41×C61C4semidirect
216. C22×#41#6923semidirect
233. C61#11725semidirect

Maximal subgroups

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

subgroupindex
C4×#145198352
#214610582
#214608142
#214607972
#214490322
#1172561