Finite Group Databasebeta

← #30605259

Subgroups — #30605259

Order 15616.

698 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 70 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#21461714C211yesmaximal, characteristic, complemented
2C2×#16840606C211yesmaximal, characteristic, complemented
3#21460814C212yesmaximal, complemented
4#21460797C212yesmaximal, complemented
5#21449134C211yesmaximal, characteristic
6C2×C22⋊#92420—21no—
7#16840703—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840606—21no—
11#16840606C2212yes—
12C2×#14519745—21no—
13C22×#11758198C2211yescharacteristic
14#16839673C411yescharacteristic, complemented
15#16839673C2211yescharacteristic
16#16839670C411yescharacteristic, complemented
17#16839670C2211yescharacteristic
18#16839642—21no—
19#16839641—21no—
20#16839641—21no—
21#16839625C411yescharacteristic, complemented
22C22×#41×C61C411yescharacteristic, complemented
23C22×#92420—41no—
24C2×C22⋊#6923—22no—
25C2×C22⋊#6923—41no—
26C2×C22⋊#6923—41no—
27C22⋊#92420—22no—
28C22⋊#92420—41no—
29C24×D61D411yescharacteristic
30C22×C22⋊D61—21no—
31C22×C22⋊D61—21no—
32C23×Dic61D411yescharacteristic
33#14519745—44no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×#11758198—21no—
38C2×#11758198—21no—
39C2×#11758198—41no—
40C2×#11758198—41no—
41C2×#11758198C2×C411yescharacteristic
42C2×#11758198D411yescharacteristic
43C2×#11758198D411yescharacteristic
44C2×#11758198C2311yescharacteristic
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684C2×C411yescharacteristic
49#14519684C2×C411yescharacteristic
50C2×#41×C61—41no—
51C2×#41×C61—41no—
52C2×#41×C61—44no—
53C22×Q8×C61C2×C411yescharacteristic
54C22×D4×C61—21no—
55C22×D4×C61C2×C411yescharacteristic
56C23×C4×C61—21no—
57C23×C4×C61C2×C411yescharacteristic
58C24×C61—21no—
59C24×C61C22⋊C411yescharacteristic
60C23×D61—21no—
61C23×D61—21no—
62C23×D61—22no—
63C23×D61—41no—
64C23×D61—42no—
65C23×D61—81no—
66C23×D61C22⋊C411yescharacteristic
67C23×D61C2×D411yescharacteristic
68C22×#6923—42no—
69#41×C61—44no—
70#41×C61—44no—
71#41×C61—44no—
72#41×C61—44no—
73C2×Q8×C61—42no—
74C2×Q8×C61—41no—
75C2×D4×C61—24no—
76C2×D4×C61—41no—
77C2×D4×C61—42no—
78C2×D4×C61—41no—
79C2×D4×C61—41no—
80C2×D4×C61—42no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—42no—
85C22×C4×C61—42no—
86C22×C4×C61—42no—
87C22×C4×C61—42no—
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22×C411yescharacteristic
90C2×C22⋊D61—42no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C2×C22⋊D61—42no—
94C2×C22⋊D61—81no—
95C2×C22⋊D61—81no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—81no—
101C22×Dic61C22⋊C411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C22⋊#6923—42no—
104C22⋊#6923—42no—
105C22⋊#6923—82no—
106C22⋊#6923—82no—
107C2×#92420—42no—
108C2×#92420—82no—
109C22⋊Dic61—42no—
110C22⋊Dic61—81no—
111C22⋊Dic61—81no—
112#11758198—41no—
113#11758198—41no—
114#11758198—41no—
115#11758198—41no—
116#11758198—42no—
117#11758198—42no—
118C23×C61—22no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—22no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61#9911yescharacteristic
130C23×C61#9911yescharacteristic
131C23×C61C2×C22⋊C411yescharacteristic
132C22×D61—21no—
133C22×D61—21no—
134C22×D61—21no—
135C22×D61—21no—
136C22×D61—21no—
137C22×D61—42no—
138C22×D61—42no—
139C22×D61—41no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61—42no—
144C22×D61—42no—
145C22×D61—81no—
146C22×D61—81no—
147C22×D61—84no—
148C22×D61—84no—
149C22×D61#13512yes—
150C2×#6923—82no—
151C2×#6923—82no—
152C2×#6923—82no—
153Q8×C61—42no—
154Q8×C61—42no—
155D4×C61—44no—
156D4×C61—42no—
157D4×C61—42no—
158D4×C61—44no—
159C2×C4×C61—22no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—48no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C22⋊D61—84no—
170C22⋊D61—84no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—42no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—81no—
183#92420—82no—
184#11839—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61—42no—
197C22×C61—81no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61AΣL(1,𝔽16)12yes—
202C22×C61C2×#9911yescharacteristic
203C22×C61#41611yescharacteristic
204C22×C61#42111yescharacteristic
205C2×D61—22no—
206C2×D61—41no—
207C2×D61—41no—
208C2×D61—42no—
209C2×D61—42no—
210C2×D61—41no—
211C2×D61—41no—
212C2×D61—41no—
213C2×D61—81no—
214C2×D61—84no—
215C2×D61—84no—
216C2×D61—81no—
217#6923—162no—
218C4×C61—41no—
219C4×C61—42no—
220C4×C61—41no—
221Dic61—41no—
222Dic61—41no—
223Dic61—81no—
224Dic61—81no—
225#2404—611no—
226#1819—611no—
227C2×#323—611no—
228#1351—612no—
229#1343—612no—
230C2×C61—21no—
231C2×C61—21no—
232C2×C61—22no—
233C2×C61—41no—
234C2×C61—82no—
235C2×C61#151611yescharacteristic
236C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
237C2×C61#212711yescharacteristic
238D61—42no—
239D61—81no—
240C22×#41#692311yescharacteristic, complemented
241C2×#120—1221no—
242C22×C22⋊C4—611no—
243C2×C22⋊C8—1221no—
244#393—611no—
245#393—611no—
246#392—611no—
247#392—611no—
248#386—1221no—
249#386—1221no—
250#386—1221no—
251#381—611no—
252#379—1221no—
253#342—612no—
254#342—1221no—
255#323—612no—
256#323—1221no—
257C61#1183911yescharacteristic, complemented, Sylow
258C25—611no—
259C2×#41—41no—
260C2×#41—41no—
261C2×#41—44no—
262C22×Q8C2×#692311yescharacteristic
263C22×D4—21no—
264C22×D4—1221no—
265C22×D4—1221no—
266C22×D4C2×#692311yescharacteristic
267C23×C4—21no—
268C23×C4—611no—
269C23×C4C2×#692311yescharacteristic
270C22×C8—2441no—
271#120—2444no—
272C2×C22⋊C4—611no—
273C2×C22⋊C4—611no—
274C2×C22⋊C4—611no—
275C2×C22⋊C4—611no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1221no—
281C2×C22⋊C4—1222no—
282C2×C22⋊C4—2441no—
283C2×C22⋊C4—2441no—
284C2×C22⋊C4—2441no—
285C2×C22⋊C4—2441no—
286C22⋊C8—1222no—
287C22⋊C8—2441no—
288#95—611no—
289#95—611no—
290#95—1221no—
291#95—1221no—
292#95—1221no—
293C24—21no—
294C24—611no—
295C24—611no—
296C24—1222no—
297C24—1221no—
298C24—1221no—
299C24—4881no—
300C24—2441no—
301C24—2442no—
302C24C22⋊#692311yescharacteristic
303#41—44no—
304#41—44no—
305#41—44no—
306#41—44no—
307C2×Q8—42no—
308C2×Q8—41no—
309C2×D4—24no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—4881no—
316C2×D4—4881no—
317C2×D4—2442no—
318C2×D4—2442no—
319C2×D4—2442no—
320C2×D4—2442no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—611no—
329C22×C4—611no—
330C22×C4—1221no—
331C22×C4—1221no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—4881no—
335C22×C4—2442no—
336C22×C4C22⋊#692311yescharacteristic
337C22×C4C22×#692311yescharacteristic
338C2×C8—4882no—
339C2×C8—2442no—
340C22⋊C4—4882no—
341C22⋊C4—4882no—
342C22⋊C4—4881no—
343C22⋊C4—4881no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C22⋊C4—2442no—
349C22⋊C4—2442no—
350C22⋊C4—2442no—
351C22⋊C4—2442no—
352C22⋊C4—2442no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—612no—
365C23—1221no—
366C23—1221no—
367C23—1221no—
368C23—1221no—
369C23—1221no—
370C23—4881no—
371C23—4881no—
372C23—4884no—
373C23—4884no—
374C23—2442no—
375C23—2442no—
376C23—2442no—
377C23—2441no—
378C23—2441no—
379C23—2441no—
380C23—2441no—
381C23—2442no—
382C23#1451983511yescharacteristic
383C23#1451983511yescharacteristic
384C23C2×C22⋊#692311yescharacteristic
385Q8—42no—
386Q8—42no—
387D4—44no—
388D4—42no—
389D4—42no—
390D4—44no—
391D4—4884no—
392D4—4884no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—48no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—4882no—
406C2×C4—4882no—
407C2×C4—4882no—
408C2×C4—4881no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4—2442no—
416C2×C4—2441no—
417C2×C4—2441no—
418C8—4882no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—1222no—
436C22—4881no—
437C22—4884no—
438C22—4881no—
439C22—4884no—
440C22—2442no—
441C22—2442no—
442C22—2441no—
443C22—2441no—
444C22—2441no—
445C22—2441no—
446C22—2441no—
447C22#1684066212yes—
448C22C2×#1451983511yescharacteristic
449C22#1684075211yescharacteristic
450C22#1684075911yescharacteristic
451C4—41no—
452C4—42no—
453C4—41no—
454C4—9762no—
455C4—4881no—
456C4—4881no—
457C4—2441no—
458C4—2441no—
459C2—21no—
460C2—21no—
461C2—22no—
462C2—41no—
463C2—82no—
464C2—4881no—
465C2—2442no—
466C2#2146124511yescharacteristic
467C2C2×#1684066211yescharacteristic
468C2#2146209411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×#41×C61C4
socleC22×C61C2×#99
prime radical#21449134C2
cosocleC2×#11758198C23
hypercenterC22×#41#6923
hypocenterC61#11839
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #21461714C2semidirect
2. C2×#16840606C2semidirect
3. #21460814C2semidirect
4. #21460797C2semidirect
14. #16839673C4semidirect
16. #16839670C4semidirect
21. #16839625C4semidirect
22. C22×#41×C61C4semidirect
240. C22×#41#6923semidirect
257. C61#11839semidirect

Maximal subgroups

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

subgroupindex
#214617142
C2×#168406062
#214608142
#214607972
#214491342
#1183961