Finite Group Databasebeta

← #29571692

Subgroups — #29571692

Order 15616.

844 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16840458C211yesmaximal, characteristic
2#21445907C212yesmaximal, complemented
3#21444386C212yesmaximal, complemented
4#21444137C211yesmaximal, characteristic, complemented
5C61×#1799C211yesmaximal, characteristic, complemented
6#16840458—24no—
7#16840458—24no—
8C2×#14519772—21no—
9C2×#14519769C411yescharacteristic, complemented
10C2×#14519759—21no—
11C2×#14519751C411yescharacteristic, complemented
12C2×#14519751C2211yescharacteristic
13C2×#14519749—21no—
14C2×C4×D244C411yescharacteristic, complemented
15C2×#14519737—21no—
16C2×#14519736C411yescharacteristic, complemented
17C22×C4⋊Dic61C2211yescharacteristic
18C2×C22⋊#92418C2212yes—
19#16839524—24no—
20#16839495—24no—
21#16839488—21no—
22#16839486—21no—
23C2×C4×D4×C61C2211yescharacteristic
24C2×#99×C61—21no—
25C61×#380C2212yes—
26C61×#342—21no—
27C22×D244C2×C411yescharacteristic
28C22×C22⋊D61—21no—
29C22×C22⋊D61—21no—
30C23×Dic61—21no—
31C22×C4×D61—21no—
32C22×#92418—22no—
33#14519772—28no—
34#14519769—22no—
35#14519769—22no—
36#14519759—28no—
37#14519751—22no—
38#14519751—22no—
39#14519751—22no—
40#14519751D414yes—
41#14519749—28no—
42C4×D244—22no—
43C4×D244—22no—
44#14519737—28no—
45#14519736—22no—
46#14519736—22no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61C2×C411yescharacteristic
50C2×C4⋊Dic61C2×C411yescharacteristic
51C2×C4⋊Dic61C2×C411yescharacteristic
52C2×C4⋊Dic61D414yes—
53C2×#11758196—21no—
54C2×#11758196—21no—
55C2×C22⋊Dic61—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×C4×Dic61—21no—
60C22⋊#92418—22no—
61C22⋊#92418—22no—
62C22×D4×C61D411yescharacteristic
63C23×C4×C61C2×C411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65C4×D4×C61—22no—
66C4×D4×C61—22no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C22⋊C4×C61—22no—
70C2×C22⋊C4×C61D411yescharacteristic
71C2×C22⋊C4×C61D411yescharacteristic
72C2×C42×C61C2×C411yescharacteristic
73#99×C61—42no—
74#95×C61—22no—
75C24×C61C22⋊C411yescharacteristic
76C24×C61C2×D411yescharacteristic
77C23×D61—21no—
78C2×D4×C61—22no—
79C2×D4×C61—22no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—42no—
85C22×C4×C61—42no—
86C22×C4×C61—42no—
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22×C411yescharacteristic
91C22×C4×C61C2×D412yes—
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—24no—
95C2×C22⋊D61—24no—
96C2×C22⋊D61—42no—
97C2×C22⋊D61—41no—
98C2×C22⋊D61—41no—
99C2×C22⋊D61—82no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—24no—
104C22×Dic61—82no—
105C2×D244—21no—
106C2×D244—42no—
107C2×D244C22⋊C412yes—
108C2×C4×D61—24no—
109C2×C4×D61—82no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—44no—
112C22⋊C4×C61—22no—
113C22⋊C4×C61—22no—
114C22⋊C4×C61—42no—
115C22⋊C4×C61—42no—
116C42×C61—21no—
117C42×C61—21no—
118C22⋊Dic61—24no—
119#11758198—24no—
120#11758198—24no—
121#11758198—24no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—24no—
126C4⋊Dic61C22⋊C412yes—
127C4⋊Dic61C22⋊C412yes—
128C4⋊Dic61C22⋊C412yes—
129#11758196—24no—
130#11758196—24no—
131C4×Dic61—24no—
132C2×#92418—44no—
133C2×#92418—42no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61—21no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61C2×C22⋊C411yescharacteristic
142C23×C61#12012yes—
143C23×C61#13512yes—
144C23×C61AGL(1,ℤ/8)11yescharacteristic
145C23×C61#13711yescharacteristic
146C22×D61—22no—
147C22×D61—42no—
148C22×D61—42no—
149C22×D61—41no—
150C22×D61—41no—
151D4×C61—42no—
152D4×C61—42no—
153C2×C4×C61—24no—
154C2×C4×C61—22no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—82no—
167C2×C4×C61—82no—
168C2×C4×C61—82no—
169C2×C4×C61C2×C22⋊C411yescharacteristic
170C2×C4×C61C2×C22⋊C411yescharacteristic
171C2×C4×C61#12012yes—
172C22⋊D61—44no—
173C22⋊D61—44no—
174C22⋊D61—84no—
175C2×Dic61—22no—
176C2×Dic61—22no—
177C2×Dic61—22no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—84no—
183D244—42no—
184D244—42no—
185C4×D61—84no—
186#92418—82no—
187#8378—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—22no—
190C22×C61—21no—
191C22×C61—22no—
192C22×C61—42no—
193C22×C61—41no—
194C22×C61—41no—
195C22×C61—41no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61#37911yescharacteristic
201C22×C61#41611yescharacteristic
202C22×C61#41711yescharacteristic
203C22×C61#44912yes—
204C22×C61#45212yes—
205C2×D61—42no—
206C2×D61—42no—
207C2×D61—42no—
208C2×D61—82no—
209C2×D61—82no—
210C4×C61—22no—
211C4×C61—21no—
212C4×C61—41no—
213C4×C61—41no—
214C4×C61—41no—
215C4×C61—162no—
216C4×C61#41012yes—
217Dic61—42no—
218Dic61—42no—
219Dic61—42no—
220C2×#553—611no—
221#1824—612no—
222#1799D6111yescharacteristic, complemented
223#1519—612no—
224#1459—611no—
225C2×C61—22no—
226C2×C61—21no—
227C2×C61—21no—
228C2×C61—22no—
229C2×C61—81no—
230C2×C61#178211yescharacteristic
231C2×C61#178811yescharacteristic
232C2×C61#179511yescharacteristic
233D61—82no—
234#553—1224no—
235#553—1224no—
236C2×#126—1221no—
237C2×#125—611no—
238C2×#123—1221no—
239C2×#121—611no—
240C2×#121—611no—
241C2×#121—1221no—
242C2×C4×D4—611no—
243C2×C4×D4—1221no—
244C2×C4×D4C2×D6111yescharacteristic
245C2×#117—611no—
246C22×C4⋊C4—611no—
247C2×#99—21no—
248C2×C22⋊C8—612no—
249#380C2×D6112yes—
250#342—21no—
251#340—1224no—
252#327—1224no—
253#324—1221no—
254#323—1221no—
255C61#837811yescharacteristic, complemented, Sylow
256C22×D4—611no—
257C22×D4—1221no—
258C22×D4—1221no—
259C22×D4C22⋊D6111yescharacteristic
260C23×C4—1221no—
261C23×C4—1221no—
262C23×C4C4×D6111yescharacteristic
263C23×C4C22×D6111yescharacteristic
264C22×C8—1222no—
265#126—1228no—
266#125—1222no—
267#125—1222no—
268#123—1228no—
269#121—614no—
270#121—1222no—
271#121—1222no—
272#121—1228no—
273#121—1222no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—1222no—
277C4×D4—1222no—
278C4×D4—1228no—
279#117—1222no—
280#117—1222no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—611no—
283C2×C4⋊C4—611no—
284C2×C4⋊C4—611no—
285C2×C4⋊C4—614no—
286C2×C4⋊C4—1221no—
287C2×C4⋊C4—1221no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4—1222no—
290C2×C4⋊C4D24411yescharacteristic
291C2×C22⋊C4—22no—
292C2×C22⋊C4—1221no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4—1221no—
296C2×C22⋊C4D24411yescharacteristic
297C2×C22⋊C4C22⋊D6111yescharacteristic
298C2×C42—1221no—
299C2×C42C4×D6111yescharacteristic
300#99—42no—
301C22⋊C8—1222no—
302C22⋊C8—1222no—
303#95—22no—
304C24—1221no—
305C24#1175819811yescharacteristic
306C24C2×C22⋊D6111yescharacteristic
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—21no—
311C2×D4—612no—
312C2×D4—1224no—
313C2×D4—1224no—
314C2×D4—1221no—
315C2×D4—4882no—
316C2×D4—2442no—
317C2×D4—2441no—
318C2×D4—2442no—
319C2×D4—2441no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—1224no—
326C22×C4—1224no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4—4882no—
331C22×C4—4882no—
332C22×C4#1175819811yescharacteristic
333C22×C4#1175819811yescharacteristic
334C22×C4#1175819811yescharacteristic
335C22×C4C2×C4×D6111yescharacteristic
336C22×C4C2×D24411yescharacteristic
337C22×C4D4×D6112yes—
338C22×C4D4×D6112yes—
339C2×C8—2444no—
340C2×C8—2442no—
341C4⋊C4—22no—
342C4⋊C4—44no—
343C4⋊C4—612no—
344C4⋊C4—612no—
345C4⋊C4—612no—
346C4⋊C4—1224no—
347C4⋊C4—1224no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—1221no—
351C4⋊C4—1224no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—1224no—
357C22⋊C4—1224no—
358C22⋊C4—1224no—
359C22⋊C4—1224no—
360C42—21no—
361C42—21no—
362C42—1224no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—1222no—
371C23—2442no—
372C23—2442no—
373C23—2441no—
374C23—2441no—
375C23C2×#1175819811yescharacteristic
376C23#1451974512yes—
377C23#1451979712yes—
378C23#1451980311yescharacteristic
379C23#1451980811yescharacteristic
380D4—42no—
381D4—42no—
382D4—4884no—
383D4—2442no—
384D4—2444no—
385D4—2444no—
386D4—2442no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—1222no—
404C2×C4—1222no—
405C2×C4—1222no—
406C2×C4—4884no—
407C2×C4—4884no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4C22⋊C4×D6111yescharacteristic
413C2×C4C22⋊C4×D6111yescharacteristic
414C2×C4#1451974712yes—
415C8—4882no—
416C22—21no—
417C22—22no—
418C22—21no—
419C22—22no—
420C22—41no—
421C22—42no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—81no—
428C22—4882no—
429C22—4882no—
430C22—2442no—
431C22—2442no—
432C22—2442no—
433C22#1683961711yescharacteristic
434C22#1683975611yescharacteristic
435C22#1683976111yescharacteristic
436C22#1683990512yes—
437C22#1683992212yes—
438C4—22no—
439C4—21no—
440C4—41no—
441C4—41no—
442C4—41no—
443C4—162no—
444C4—2442no—
445C4—2442no—
446C4—2442no—
447C4#1683973412yes—
448C2—22no—
449C2—21no—
450C2—21no—
451C2—22no—
452C2—81no—
453C2—4882no—
454C2#2144573511yescharacteristic
455C2#2144575711yescharacteristic
456C2#2144578711yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1799C2
socleC22×C61#379
prime radicalC2×#16840458C2
cosocleC23×C4×C61C23
hypercenter#1799D61
hypocenterC61#8378
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #21445907C2semidirect
3. #21444386C2semidirect
4. #21444137C2semidirect
5. C61×#1799C2semidirect
9. C2×#14519769C4semidirect
11. C2×#14519751C4semidirect
14. C2×C4×D244C4semidirect
16. C2×#14519736C4semidirect
222. #1799D61semidirect
255. C61#8378semidirect

Maximal subgroups

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

subgroupindex
C2×#168404582
#214459072
#214443862
#214441372
C61×#17992
#837861