Finite Group Databasebeta

← #30606500

Subgroups — #30606500

Order 15616.

844 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 102 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×#16840662C212yesmaximal
2#21461569C211yesmaximal, characteristic, complemented
3#21461566C211yesmaximal, characteristic, complemented
4#21460904C212yesmaximal, complemented
5#21448318C211yesmaximal, characteristic
6C2×#14519837C2212yes—
7C2×#14519835C2212yes—
8C4×#11758217—21no—
9C4×C22⋊#6923—21no—
10#16840662C418yescomplemented
11#16840642—21no—
12#16840640—21no—
13C24×Dic61C411yescharacteristic, complemented
14C2×#14519746C2211yescharacteristic
15C2×#14519737—21no—
16C2×#14519737C2211yescharacteristic
17#16839639—21no—
18#16839639C2211yescharacteristic
19#16839634—21no—
20#16839634C411yescharacteristic, complemented
21C22⋊C4×Dic61—21no—
22C22⋊C4×Dic61C411yescharacteristic, complemented
23C61×#379C411yescharacteristic, complemented
24C2×#11758217—22no—
25C2×C22⋊#6923—22no—
26C2×C4×#6923—41no—
27#14519837—22no—
28#14519837C2×C414yescomplemented
29#14519835—22no—
30#14519835C2×C414yescomplemented
31#14519832—41no—
32C4×#92420—41no—
33(C2×#6923)⋊C4—41no—
34C22×C22⋊D61—21no—
35C22×C22⋊D61C2311yescharacteristic
36C23×Dic61—21no—
37C23×Dic61—21no—
38C23×Dic61—44no—
39C23×Dic61—41no—
40C23×Dic61—41no—
41C23×Dic61C2×C411yescharacteristic
42C23×Dic61C2×C411yescharacteristic
43C22×C4×D61D411yescharacteristic
44#14519746—22no—
45#14519746C2×C414yes—
46#14519737—41no—
47#14519737—41no—
48#14519737—44no—
49C2×#11758196—21no—
50C2×#11758196D411yescharacteristic
51C2×C22⋊Dic61—21no—
52C2×C22⋊Dic61C2×C411yescharacteristic
53C2×#11758198—21no—
54C2×#11758198D411yescharacteristic
55C2×C4×Dic61—21no—
56C2×C4×Dic61D411yescharacteristic
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60C25×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61C2×C411yescharacteristic
64C2×C22⋊C4×C61C2×C411yescharacteristic
65C24×C61—21no—
66C24×C61—21no—
67C24×C61—42no—
68C24×C61—41no—
69C24×C61—42no—
70C24×C61—41no—
71C24×C61C4212yescomplemented
72C24×C61C22×C411yescharacteristic
73C23×D61C2×D411yescharacteristic
74C22×#6923—42no—
75C22×C4×C61—21no—
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C22⋊C411yescharacteristic
78C2×C22⋊D61—24no—
79C2×C22⋊D61—41no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61—42no—
82C2×C22⋊D61—42no—
83C2×C22⋊D61C22×C412yes—
84C2×C22⋊D61#4112yes—
85C22×Dic61—24no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—42no—
91C22×Dic61—42no—
92C22×Dic61—42no—
93C22×Dic61—42no—
94C22×Dic61—42no—
95C22×Dic61—42no—
96C22×Dic61—44no—
97C22×Dic61—44no—
98C22×Dic61—44no—
99C22×Dic61—41no—
100C22×Dic61—41no—
101C22×Dic61—41no—
102C22×Dic61—44no—
103C22×Dic61C22⋊C411yescharacteristic
104C22×Dic61C22⋊C411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61—42no—
109C22⋊#6923—24no—
110C22⋊#6923—42no—
111#11758217—24no—
112#11758217—42no—
113C2×#92420—42no—
114C4×#6923—82no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—42no—
117C22⋊C4×C61—42no—
118C22⋊C4×C61—41no—
119C22⋊Dic61—21no—
120C22⋊Dic61—24no—
121C22⋊Dic61C4212yescomplemented
122#11758198—41no—
123#11758198—42no—
124#11758196—41no—
125#11758196—42no—
126C4×Dic61—21no—
127C4×Dic61—21no—
128C4×Dic61—42no—
129C23×C61—22no—
130C23×C61—22no—
131C23×C61—22no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—44no—
137C23×C61—42no—
138C23×C61—41no—
139C23×C61—44no—
140C23×C61—42no—
141C23×C61—41no—
142C23×C61—42no—
143C23×C61—48no—
144C23×C61—42no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—42no—
148C23×C61—42no—
149C23×C61C2×C4211yescharacteristic
150C23×C61C2×C22⋊C411yescharacteristic
151C23×C61#11711yescharacteristic
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—41no—
155C22×D61—41no—
156C22×D61C4×D412yes—
157C2×#6923—44no—
158C2×#6923—82no—
159C2×#6923—82no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C22⋊D61—42no—
168C22⋊D61—42no—
169C22⋊D61—48no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—22no—
173C2×Dic61—22no—
174C2×Dic61—22no—
175C2×Dic61—21no—
176C2×Dic61—44no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—48no—
182C2×Dic61—42no—
183C2×Dic61—41no—
184C2×Dic61—44no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61#9911yescharacteristic
189C2×Dic61#9911yescharacteristic
190C2×Dic61C4×D412yes—
191C4×D61—42no—
192C4×D61—81no—
193#92420—44no—
194C4×AΣL(1,𝔽16)—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—24no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—41no—
201C22×C61—41no—
202C22×C61—42no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61—44no—
207C22×C61—44no—
208C22×C61—44no—
209C22×C61—44no—
210C22×C61—42no—
211C22×C61—41no—
212C22×C61—42no—
213C22×C61—41no—
214C22×C61—42no—
215C22×C61C4×C22⋊C411yescharacteristic
216C22×C61C2×#9911yescharacteristic
217C22×C61#41011yescharacteristic
218C2×D61—22no—
219C2×D61—41no—
220C2×D61—41no—
221C2×D61—42no—
222C2×D61—81no—
223#6923—84no—
224C4×C61—41no—
225C4×C61—41no—
226C4×C61—81no—
227Dic61—21no—
228Dic61—22no—
229Dic61—42no—
230Dic61—42no—
231Dic61—41no—
232Dic61AΣL(1,𝔽16)12yescomplemented
233C4×#120—611no—
234C2×AΣL(1,𝔽16)—612no—
235C4×#100—611no—
236C4×#99—611no—
237#1390—612no—
238C2×C61—21no—
239C2×C61—21no—
240C2×C61—44no—
241C2×C61—41no—
242C2×C61—41no—
243C2×C61C4×#9911yescharacteristic
244C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
245C2×C61#212011yescharacteristic
246D61—42no—
247C24×C4—611no—
248C2×#120—611no—
249C2×C4×D4—611no—
250C2×C4×D4—1221no—
251C2×#100—612no—
252C2×#99—612no—
253C4×M16—1221no—
254#386—611no—
255#386—1221no—
256#385—611no—
257#385—1221no—
258#379#692311yescharacteristic, complemented
259C4×C22⋊C4—611no—
260C4×C22⋊C4—1221no—
261C4×C22⋊C4—1221no—
262AΣL(1,𝔽16)—618no—
263#343—1221no—
264#342—1221no—
265C61C4×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
266C25C2×#692311yescharacteristic
267C22×D4—611no—
268C22×D4—1221no—
269C23×C4—611no—
270C23×C4—611no—
271C23×C4—611no—
272C23×C4—1221no—
273C23×C4—1221no—
274C23×C4—2441no—
275C23×C4—2441no—
276C23×C4—2444no—
277C2×M16—1222no—
278#120—614no—
279#120—1222no—
280C4×D4—2441no—
281C4×D4—2441no—
282C4×D4—2444no—
283C2×C4⋊C4—611no—
284C2×C4⋊C4—1221no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—611no—
288C2×C22⋊C4—611no—
289C2×C22⋊C4—1221no—
290C2×C22⋊C4—1222no—
291C2×C22⋊C4—1221no—
292C2×C22⋊C4C2×#692311yescharacteristic
293C2×C22⋊C4C2×#692311yescharacteristic
294C2×C42—611no—
295C2×C42—1221no—
296C2×C42—2441no—
297#100—614no—
298#100—1222no—
299#99—614no—
300#99—1222no—
301#97—2441no—
302C4×C8—2441no—
303#95—1221no—
304#95—1221no—
305#95—1221no—
306#95—2441no—
307C24—21no—
308C24—21no—
309C24—42no—
310C24—41no—
311C24—42no—
312C24—41no—
313C24—611no—
314C24C4×#692312yescomplemented
315C24C22×#692311yescharacteristic
316C2×D4—612no—
317C2×D4—612no—
318C2×D4—1224no—
319C2×D4—2441no—
320C2×D4—2441no—
321C2×D4—2442no—
322C2×D4—2442no—
323C22×C4—21no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4—1221no—
331C22×C4—1224no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—2442no—
335C22×C4—2441no—
336C22×C4—2441no—
337C22×C4—2442no—
338C22×C4—2442no—
339C22×C4—2442no—
340C22×C4—2442no—
341C22×C4—2442no—
342C22×C4—2442no—
343C22×C4—2442no—
344C22×C4—2444no—
345C22×C4—2444no—
346C22×C4—2444no—
347C22×C4—2441no—
348C22×C4—2444no—
349C22×C4C22⋊#692311yescharacteristic
350C22×C4C22⋊#692311yescharacteristic
351M16—1224no—
352M16—2442no—
353C2×C8—2442no—
354C4⋊C4—2441no—
355C4⋊C4—2442no—
356C22⋊C4—41no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—41no—
360C22⋊C4—612no—
361C22⋊C4—1224no—
362C22⋊C4—1221no—
363C22⋊C4—1224no—
364C22⋊C4—2441no—
365C22⋊C4—2442no—
366C22⋊C4—2442no—
367C42—1221no—
368C42—1221no—
369C42—4882no—
370C42—2442no—
371C23—22no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—22no—
376C23—41no—
377C23—44no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—44no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—48no—
388C23—42no—
389C23—41no—
390C23—41no—
391C23—612no—
392C23—1221no—
393C23—1221no—
394C23—2441no—
395C23—2441no—
396C23C2×C4×#692311yescharacteristic
397C23C2×C22⋊#692311yescharacteristic
398C23#1451985611yescharacteristic
399D4—2442no—
400D4—2442no—
401D4—2448no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—611no—
410C2×C4—611no—
411C2×C4—612no—
412C2×C4—1222no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—1222no—
416C2×C4—1221no—
417C2×C4—1222no—
418C2×C4—4881no—
419C2×C4—4882no—
420C2×C4—4882no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2444no—
424C2×C4—2442no—
425C2×C4—2444no—
426C2×C4—2444no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2448no—
431C2×C4—2442no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4—2442no—
435C8—2444no—
436C22—21no—
437C22—21no—
438C22—24no—
439C22—21no—
440C22—21no—
441C22—41no—
442C22—41no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—42no—
452C22—41no—
453C22—42no—
454C22—41no—
455C22—42no—
456C22—1222no—
457C22—4881no—
458C22—2442no—
459C22—2441no—
460C22—2441no—
461C22C4×C22⋊#692311yescharacteristic
462C22D61⋊#9911yescharacteristic
463C22#1684074211yescharacteristic
464C4—41no—
465C4—41no—
466C4—81no—
467C4—612no—
468C4—1221no—
469C4—1222no—
470C4—4884no—
471C4—2442no—
472C4—2442no—
473C4—2441no—
474C2—21no—
475C2—21no—
476C2—44no—
477C2—41no—
478C2—41no—
479C2—2442no—
480C2#2146156611yescharacteristic
481C2D61⋊AΣL(1,𝔽16)11yescharacteristic
482C2#2146207111yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC61×#379C4
socleC22×C61C2×#99
prime radicalC2×#14519746C22
cosocleC22×C22⋊D61C23
hypercenter#379#6923
hypocenterC61C4×AΣL(1,𝔽16)
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #21461569C2semidirect
3. #21461566C2semidirect
4. #21460904C2semidirect
10. #16840662C4semidirect
13. C24×Dic61C4semidirect
20. #16839634C4semidirect
22. C22⋊C4×Dic61C4semidirect
23. C61×#379C4semidirect
28. #14519837C2×C4semidirect
30. #14519835C2×C4semidirect
71. C24×C61C42semidirect
121. C22⋊Dic61C42semidirect
232. Dic61AΣL(1,𝔽16)semidirect
258. #379#6923semidirect
265. C61C4×AΣL(1,𝔽16)semidirect
314. C24C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406622
#214615692
#214615662
#214609042
#214483182
C4×AΣL(1,𝔽16)61