Finite Group Databasebeta

← #29712846

Subgroups — #29712846

Order 15616.

1450 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1450 conjugacy classes of subgroups — too many to draw. Only its 187 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#21459454C211yesmaximal, characteristic
2#21455750C212yesmaximal, complemented
3#21447029C214yesmaximal, complemented
4#21446267C212yesmaximal, complemented
5#21446028C212yesmaximal, complemented
6#21445772C212yesmaximal, complemented
7#21445740C211yesmaximal, characteristic, complemented
8C61×#2309C211yesmaximal, characteristic, complemented
9C2×#14519822C2211yescharacteristic
10#16840406—24no—
11D4×C22⋊D61—24no—
12#16840329C2212yes—
13#16840321—22no—
14#16840273C2212yes—
15C2×#14519746—22no—
16#16840237C2211yescharacteristic
17#16840103—24no—
18#16840086—24no—
19#16839802C2214yescomplemented
20C2×#14519705C2214yescomplemented
21#16839761C2214yescomplemented
22C2×#14519694C2212yes—
23C2×#14519694C2212yescomplemented
24#16839726C2212yescomplemented
25#16839522C2214yescomplemented
26C23×D4×C61C2211yescharacteristic
27C2×#117×C61C2211yescharacteristic, complemented
28C2×#117×C61C2211yescharacteristic, complemented
29C61×#390—22no—
30C61×#390C2212yes—
31C61×#390C2212yescomplemented
32C61×#379—22no—
33#14519822—24no—
34#14519822—42no—
35C2×#11758222C2312yes—
36C2×#11758224—22no—
37C22×C22⋊D61—22no—
38C22×C22⋊D61—24no—
39C2×D4×D61—22no—
40C2×#11758194C2314yes—
41#14519773—22no—
42#14519764—22no—
43#14519760—22no—
44#14519758—24no—
45#14519756D414yes—
46#14519751D414yes—
47#14519749—28no—
48#14519748—24no—
49#14519746—28no—
50#14519746—22no—
51#14519746—24no—
52#14519745—24no—
53D4×Dic61—24no—
54C4×C22⋊D61—24no—
55#14519736C2312yes—
56C2×C22⋊Dic61—24no—
57C2×C22⋊Dic61—22no—
58#14519710—24no—
59#14519708—24no—
60#14519705—44no—
61#14519700—24no—
62#14519697—24no—
63#14519694—24no—
64#14519694—42no—
65C22⋊#92418—24no—
66C25×C61D412yes—
67C22×D4×C61—24no—
68C22×D4×C61—24no—
69C22×D4×C61—42no—
70C22×D4×C61D412yes—
71C22×D4×C61C2311yescharacteristic
72C22×D4×C61C2311yescharacteristic
73C23×C4×C61C2311yescharacteristic
74#117×C61—24no—
75#117×C61—42no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61C2312yes—
78C2×C22⋊C4×C61—21no—
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—24no—
81C2×C22⋊C4×C61—24no—
82C2×C42×C61—21no—
83C2×C42×C61C2312yes—
84C24×C61—24no—
85C24×C61—24no—
86C24×C61—42no—
87C24×C61—84no—
88C24×C61C2×D412yes—
89C24×C61C2×D411yescharacteristic
90C24×C61C2×D412yes—
91C23×D61—22no—
92C2×D4×C61—24no—
93C2×D4×C61—24no—
94C2×D4×C61—22no—
95C2×D4×C61—22no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61—22no—
99C2×D4×C61—416no—
100C2×D4×C61—42no—
101C2×D4×C61#4114yes—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D414yes—
108C22×C4×C61C2411yescharacteristic
109C2×C22⋊D61—24no—
110C2×C22⋊D61—24no—
111C2×C22⋊D61—22no—
112C2×C22⋊D61—28no—
113C2×C22⋊D61—44no—
114C2×C22⋊D61—44no—
115C2×C22⋊D61—88no—
116C22×Dic61—22no—
117C22×Dic61—24no—
118#11758224—48no—
119D4×D61—48no—
120#11758222—24no—
121#11758222—24no—
122C2×D244C2×D412yes—
123C2×C4×D61—22no—
124C2×Dic122C2×D412yes—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—42no—
127C22⋊C4×C61—42no—
128C22⋊C4×C61—42no—
129C22⋊C4×C61—84no—
130C22⋊C4×C61—84no—
131C42×C61—24no—
132C42×C61—42no—
133C22⋊Dic61—24no—
134C22⋊Dic61—28no—
135C22⋊Dic61—24no—
136C22⋊Dic61—22no—
137#11758198—24no—
138C4⋊Dic61C2×D414yes—
139#11758196—24no—
140C4×Dic61—22no—
141#11758194—24no—
142#11758194—24no—
143C2×#92418—24no—
144C23×C61—22no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—22no—
148C23×C61—22no—
149C23×C61—22no—
150C23×C61—22no—
151C23×C61—42no—
152C23×C61—48no—
153C23×C61—44no—
154C23×C61—44no—
155C23×C61—48no—
156C23×C61—84no—
157C23×C61—88no—
158C23×C61—88no—
159C23×C61#12012yes—
160C23×C61#12314yes—
161C23×C61C22×D411yescharacteristic
162C22×D61—22no—
163C22×D61—44no—
164C22×D61—84no—
165D4×C61—28no—
166D4×C61—48no—
167D4×C61—44no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—28no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—44no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61#12012yes—
177C2×C4×C61C22×D412yes—
178C2×C4×C61C2×#4112yes—
179C2×C4×C61H(5,𝔽2)12yes—
180C22⋊D61—48no—
181C22⋊D61—48no—
182C22⋊D61—42no—
183C22⋊D61—42no—
184C22⋊D61—88no—
185C2×Dic61—24no—
186C2×Dic61—22no—
187C2×Dic61—44no—
188C2×Dic61—84no—
189D244—24no—
190C4×D61—44no—
191Dic122—24no—
192#92418—44no—
193#24022—611nomaximal, Sylow
194C22×C61—22no—
195C22×C61—24no—
196C22×C61—22no—
197C22×C61—22no—
198C22×C61—22no—
199C22×C61—44no—
200C22×C61—42no—
201C22×C61—42no—
202C22×C61—44no—
203C22×C61—44no—
204C22×C61—42no—
205C22×C61—44no—
206C22×C61—82no—
207C22×C61—82no—
208C22×C61—816no—
209C22×C61C2×#12011yescharacteristic
210C22×C61C2×#12311yescharacteristic
211C22×C61#53411yescharacteristic
212C2×D61—42no—
213C2×D61—44no—
214C2×D61—164no—
215C4×C61—22no—
216C4×C61—82no—
217C4×C61—82no—
218C4×C61#54614yes—
219Dic61—42no—
220Dic61—44no—
221#3484—611no—
222#3093—612no—
223#2309D6111yescharacteristic, complemented
224#2037—614no—
225#1897—612no—
226#1856—612no—
227#1792—612no—
228#1783—611no—
229C2×C61—21no—
230C2×C61—21no—
231C2×C61—44no—
232C2×C61—42no—
233C2×C61—82no—
234C2×C61#240511yescharacteristic
235C2×C61#301912yes—
236D61—82no—
237C2×H(5,𝔽2)—611no—
238C23×D4C2×D6111yescharacteristic
239#546—1224no—
240D42—1224no—
241#535—612no—
242#534—1222no—
243#525—612no—
244C2×#120—1222no—
245#518—611no—
246C2×#117C2×D6111yescharacteristic, complemented
247C2×#117C2×D6111yescharacteristic, complemented
248#485—1224no—
249#482—1224no—
250#428—614no—
251C2×#104—614no—
252#417—614no—
253C2×#100—612no—
254C2×#100—612no—
255#407—612no—
256#390—22no—
257#390C2×D6112yes—
258#390C2×D6112yescomplemented
259#379—22no—
260#339—614no—
261C61#2402211yescharacteristic, complemented, Sylow
262C25C22⋊D6112yes—
263H(5,𝔽2)—1224no—
264H(5,𝔽2)—2442no—
265C2×#41—612no—
266C2×#41—1222no—
267C22×D4—24no—
268C22×D4—24no—
269C22×D4—42no—
270C22×D4—1222no—
271C22×D4—1224no—
272C22×D4—1222no—
273C22×D4C22⋊D6112yes—
274C22×D4C22×D6111yescharacteristic
275C22×D4C22×D6111yescharacteristic
276C23×C4C22×D6111yescharacteristic
277C2×M16—614no—
278#127—1222no—
279#124—1222no—
280#123—1224no—
281#123—1222no—
282C22⋊Q8—614no—
283#121—614no—
284#121—1224no—
285#121—1228no—
286#120—1224no—
287#120—1228no—
288#120—1222no—
289#120—1224no—
290C4×D4—1224no—
291C4×D4—1224no—
292#117—24no—
293#117—42no—
294#117—612no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4C22×D6112yes—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—1224no—
302C2×C22⋊C4—1222no—
303C2×C42—21no—
304C2×C42C22×D6112yes—
305#107—1224no—
306#106—1224no—
307#104—2444no—
308#103—1224no—
309#102—1224no—
310#100—1224no—
311#100—2442no—
312C22⋊C8—1224no—
313C24—24no—
314C24—24no—
315C24—42no—
316C24—84no—
317C24—1222no—
318C24C2×C22⋊D6112yes—
319C24C2×C22⋊D6111yescharacteristic
320C24C2×C22⋊D6112yes—
321#41—1224no—
322#41—1224no—
323#41—2448no—
324C2×Q8—612no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—416no—
333C2×D4—42no—
334C2×D4—612no—
335C2×D4—1228no—
336C2×D4—1224no—
337C2×D4—1222no—
338C2×D4—1224no—
339C2×D4—4888no—
340C2×D4—2448no—
341C2×D4—2444no—
342C2×D4—2444no—
343C2×D4#1175822214yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—1224no—
349C22×C4—1222no—
350C22×C4—1222no—
351C22×C4D4×D6114yes—
352C22×C4C2×C22⋊D6111yescharacteristic
353C22×C4C23×D6111yescharacteristic
354M16—1224no—
355M16—1224no—
356C2×C8—1224no—
357C4⋊C4—24no—
358C4⋊C4—42no—
359C4⋊C4—614no—
360C4⋊C4—1224no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—84no—
364C22⋊C4—84no—
365C22⋊C4—1224no—
366C22⋊C4—1228no—
367C22⋊C4—1222no—
368C22⋊C4—1224no—
369C22⋊C4—1224no—
370C42—24no—
371C42—42no—
372C42—1222no—
373C23—22no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—48no—
382C23—44no—
383C23—44no—
384C23—48no—
385C23—84no—
386C23—88no—
387C23—88no—
388C23—1222no—
389C23—4884no—
390C23—2444no—
391C23#1451974512yes—
392C23#1451976014yes—
393C23C22×C22⋊D6111yescharacteristic
394Q8—1224no—
395D4—28no—
396D4—48no—
397D4—44no—
398D4—1224no—
399D4—4888no—
400D4—2448no—
401D4—2448no—
402D4—2442no—
403D4—2442no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—28no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—1224no—
413C2×C4—1222no—
414C2×C4—4884no—
415C2×C4—2444no—
416C2×C4—2444no—
417C2×C4#1451974512yes—
418C2×C4C2×D4×D6112yes—
419C2×C4C2×#1175822212yes—
420C2×C4#1451982212yes—
421C8—2444no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—24no—
427C22—44no—
428C22—42no—
429C22—42no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22—42no—
434C22—82no—
435C22—82no—
436C22—816no—
437C22—9764no—
438C22—2444no—
439C22—2442no—
440C22C2×#1451974511yescharacteristic
441C22C2×#1451976011yescharacteristic
442C22#1684032111yescharacteristic
443C4—22no—
444C4—82no—
445C4—82no—
446C4—2444no—
447C4—2442no—
448C4#1684040214yes—
449C2—21no—
450C2—21no—
451C2—42no—
452C2—44no—
453C2—82no—
454C2—4882no—
455C2#2144914611yescharacteristic
456C2#2145517512yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2309C2
socleC22×C61C2×#120
prime radical#21459454C2
hypercenter#2309D61
hypocenterC61#24022
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. #21455750C2semidirect
3. #21447029C2semidirect
4. #21446267C2semidirect
5. #21446028C2semidirect
6. #21445772C2semidirect
7. #21445740C2semidirect
8. C61×#2309C2semidirect
19. #16839802C22semidirect
20. C2×#14519705C22semidirect
21. #16839761C22semidirect
23. C2×#14519694C22semidirect
24. #16839726C22semidirect
25. #16839522C22semidirect
27. C2×#117×C61C22semidirect
28. C2×#117×C61C22semidirect
31. C61×#390C22semidirect
223. #2309D61semidirect
246. C2×#117C2×D61semidirect
247. C2×#117C2×D61semidirect
258. #390C2×D61semidirect
261. C61#24022semidirect

Maximal subgroups

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

subgroupindex
#214594542
#214557502
#214470292
#214462672
#214460282
#214457722
#214457402
C61×#23092
#2402261