Finite Group Databasebeta

← #29596780

Subgroups — #29596780

Order 15616.

608 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16839549C211yesmaximal, characteristic
2#21446228C211yesmaximal, characteristic
3#21446099C211yesmaximal, characteristic, complemented
4#21446028C211yesmaximal, characteristic, complemented
5#21443944C211yesmaximal, characteristic, complemented
6#21443937C211yesmaximal, characteristic, complemented
7C61×#2375C211yesmaximal, characteristic
8#16840237C2211yescharacteristic
9#16839802—21no—
10C2×#14519694C2211yescharacteristic
11C2×#14519692C2211yescharacteristic
12C2×#14519692C2211yescharacteristic
13#16839635—21no—
14#16839618—21no—
15#16839549C414yescomplemented
16#16839528—21no—
17#16839528—21no—
18#16839522—21no—
19C2×#120×C61C2211yescharacteristic
20C61×#390—21no—
21C61×#390C2211yescharacteristic
22C61×#388—21no—
23C61×#388—21no—
24C61×#388—21no—
25C61×#386—21no—
26C61×#386C2211yescharacteristic
27C61×#379—21no—
28C2×#11758194—21no—
29D4×Dic61—41no—
30D4×Dic61—41no—
31#14519736—21no—
32C2×#11758196—41no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—41no—
38C2×C22⋊Dic61—41no—
39#14519710—41no—
40#14519708—41no—
41#14519694—21no—
42#14519694C2×C412yes—
43#14519692—21no—
44#14519692—21no—
45#14519692—21no—
46#14519692C2×C412yes—
47#14519684—41no—
48C25×C61D411yescharacteristic
49C22×D4×C61—21no—
50C22×D4×C61C2311yescharacteristic
51C23×C4×C61D411yescharacteristic
52#120×C61—21no—
53#120×C61—21no—
54#120×C61C2×C412yes—
55#120×C61Q812yes—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61D411yescharacteristic
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61D411yescharacteristic
67C2×C22⋊C4×C61D411yescharacteristic
68C2×C42×C61—21no—
69C2×C42×C61D411yescharacteristic
70#95×C61—21no—
71#95×C61—21no—
72#95×C61—21no—
73C24×C61—21no—
74C24×C61—21no—
75C24×C61—41no—
76C24×C61—41no—
77C24×C61—81no—
78C24×C61—81no—
79C24×C61C4⋊C412yescomplemented
80C24×C61C2×D411yescharacteristic
81C24×C61C2×D411yescharacteristic
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—81no—
87C2×D4×C61—81no—
88C2×D4×C61C22×C411yescharacteristic
89C2×D4×C61C2×Q811yescharacteristic
90C2×D4×C61#4111yescharacteristic
91C2×D4×C61#4111yescharacteristic
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—41no—
98C22×C4×C61—81no—
99C22×C4×C61C2×D411yescharacteristic
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—41no—
105C22×Dic61—41no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—81no—
108C22⋊C4×C61—21no—
109C22⋊C4×C61—22no—
110C22⋊C4×C61—22no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—81no—
113C22⋊C4×C61—81no—
114C22⋊C4×C61—81no—
115C22⋊C4×C61—81no—
116C22⋊C4×C61—81no—
117C22⋊C4×C61—81no—
118C22⋊C4×C61—81no—
119C22⋊C4×C61C4⋊C412yescomplemented
120C42×C61—41no—
121C42×C61—81no—
122C22⋊Dic61—22no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125C22⋊Dic61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—41no—
128C22⋊Dic61—41no—
129C22⋊Dic61—82no—
130C22⋊Dic61—82no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133#11758196—82no—
134C4×Dic61—41no—
135#11758194—22no—
136#11758194—41no—
137C2×#92418—41no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—22no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—42no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61—42no—
153C23×C61—42no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—81no—
157C23×C61—82no—
158C23×C61—82no—
159C23×C61—84no—
160C23×C61—81no—
161C23×C61—81no—
162C23×C61C2×C4⋊C411yescharacteristic
163C23×C61C4×D411yescharacteristic
164C23×C61#12011yescharacteristic
165C23×C61C22⋊Q811yescharacteristic
166C23×C61#12311yescharacteristic
167D4×C61—41no—
168D4×C61—41no—
169D4×C61—41no—
170D4×C61—41no—
171D4×C61—82no—
172D4×C61—82no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—22no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—81no—
192C2×C4×C61C4×D411yescharacteristic
193C2×C4×C61C22⋊Q811yescharacteristic
194C2×Dic61—42no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—81no—
198C2×Dic61—81no—
199C2×Dic61—81no—
200C2×Dic61—81no—
201C2×Dic61—81no—
202C2×Dic61—81no—
203C2×Dic61—81no—
204C2×Dic61—81no—
205C2×Dic61—81no—
206#92418—42no—
207#12960—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—41no—
215C22×C61—41no—
216C22×C61—41no—
217C22×C61—41no—
218C22×C61—42no—
219C22×C61—42no—
220C22×C61—41no—
221C22×C61—41no—
222C22×C61—41no—
223C22×C61—41no—
224C22×C61—44no—
225C22×C61—42no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—82no—
231C22×C61—82no—
232C22×C61—82no—
233C22×C61—81no—
234C22×C61—81no—
235C22×C61#38511yescharacteristic
236C22×C61B(4,𝔽2)11yescharacteristic
237C22×C61#45811yescharacteristic
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—42no—
241C4×C61—81no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61—81no—
245Dic61—82no—
246Dic61—81no—
247Dic61—81no—
248Dic61—161no—
249#2375D6111yescharacteristic
250C2×AΣL(1,𝔽16)—611no—
251#1889—611no—
252#1867—611no—
253#1856—611no—
254#1393—611no—
255#1390—611no—
256C2×C61—21no—
257C2×C61—21no—
258C2×C61—41no—
259C2×C61—41no—
260C2×C61—41no—
261C2×C61—41no—
262C2×C61—82no—
263C2×C61#186711yescharacteristic
264C2×C61#219911yescharacteristic
265C2×C61#220011yescharacteristic
266C2×#120C2×D6111yescharacteristic
267#518—611no—
268#428—1221no—
269C2×#100—611no—
270C2×#99—611no—
271C2×#99—611no—
272#390—21no—
273#390C2×D6111yescharacteristic
274#388—21no—
275#388—21no—
276#388—21no—
277#386—21no—
278#386C2×D6111yescharacteristic
279#385—1221no—
280#379—21no—
281#379—1221no—
282AΣL(1,𝔽16)—614no—
283#342—1221no—
284#342—1221no—
285#339—1221no—
286C61#1296011yescharacteristic, complemented, Sylow
287C25C22⋊D6111yescharacteristic
288C22×D4—21no—
289C22×D4C22×D6111yescharacteristic
290C23×C4C22⋊D6111yescharacteristic
291C2×M16—1221no—
292#120—21no—
293#120—21no—
294#120Dic12212yes—
295#120C4×D6112yes—
296C4×D4—2441no—
297C4×D4—2441no—
298#117—1221no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—2441no—
301C2×C4⋊C4C22⋊D6111yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—1221no—
311C2×C22⋊C4—1221no—
312C2×C22⋊C4—1221no—
313C2×C22⋊C4—1221no—
314C2×C22⋊C4—2441no—
315C2×C22⋊C4—2441no—
316C2×C22⋊C4C22⋊D6111yescharacteristic
317C2×C22⋊C4C22⋊D6111yescharacteristic
318C2×C42—21no—
319C2×C42C22⋊D6111yescharacteristic
320#107—2441no—
321#106—2441no—
322#100—612no—
323#100—1221no—
324#99—612no—
325#99—1221no—
326#99—1221no—
327#99—1221no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—2441no—
332C24—21no—
333C24—21no—
334C24—41no—
335C24—41no—
336C24—81no—
337C24—81no—
338C24#1175819612yescomplemented
339C24C2×C22⋊D6111yescharacteristic
340C24C2×C22⋊D6111yescharacteristic
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D4C2×Dic12211yescharacteristic
348C2×D4C2×C4×D6111yescharacteristic
349C2×D4#1175822211yescharacteristic
350C2×D4#1175822211yescharacteristic
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—81no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—1221no—
362C22×C4—2441no—
363C22×C4—2441no—
364C22×C4C2×C22⋊D6111yescharacteristic
365M16—1222no—
366M16—2441no—
367C2×C8—2441no—
368C4⋊C4—41no—
369C4⋊C4—81no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—4882no—
373C22⋊C4—21no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—41no—
377C22⋊C4—81no—
378C22⋊C4—81no—
379C22⋊C4—81no—
380C22⋊C4—81no—
381C22⋊C4—81no—
382C22⋊C4—81no—
383C22⋊C4—81no—
384C22⋊C4—1222no—
385C22⋊C4—4882no—
386C22⋊C4—4882no—
387C22⋊C4—2441no—
388C22⋊C4—2441no—
389C22⋊C4—2441no—
390C22⋊C4—2441no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C22⋊C4#1175819612yescomplemented
394C42—41no—
395C42—81no—
396C42—2441no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23—41no—
414C23—41no—
415C23—81no—
416C23—82no—
417C23—82no—
418C23—81no—
419C23—81no—
420C23—84no—
421C23C2×#1175819611yescharacteristic
422C23C4×C22⋊D6111yescharacteristic
423C23#1451974611yescharacteristic
424C23#1451975611yescharacteristic
425C23#1451976011yescharacteristic
426D4—41no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—82no—
431D4—82no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—81no—
451C2×C4—4881no—
452C2×C4—4881no—
453C2×C4—4881no—
454C2×C4—4881no—
455C2×C4—4881no—
456C2×C4—4881no—
457C2×C4—4881no—
458C2×C4—4881no—
459C2×C4—4881no—
460C2×C4—2442no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4C4×C22⋊D6111yescharacteristic
464C2×C4#1451975611yescharacteristic
465C8—2442no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—42no—
477C22—42no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—44no—
483C22—42no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—81no—
492C22—81no—
493C22#1683963511yescharacteristic
494C22#1683994911yescharacteristic
495C22#1683995211yescharacteristic
496C4—21no—
497C4—21no—
498C4—42no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—9761no—
504C4—4882no—
505C4—4881no—
506C4—4881no—
507C2—21no—
508C2—21no—
509C2—41no—
510C2—41no—
511C2—41no—
512C2—41no—
513C2—82no—
514C2#2144610011yescharacteristic
515C2#2144782011yescharacteristic
516C2#2144782611yescharacteristic
subgroupG/H
centerC22#16839949
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#2375C2
socleC22×C61B(4,𝔽2)
prime radicalC2×#120×C61C22
cosocleC22×D4×C61C23
hypercenter#2375D61
hypocenterC61#12960
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2375

Complements

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

NH ≅ G/Ntype
3. #21446099C2semidirect
4. #21446028C2semidirect
5. #21443944C2semidirect
6. #21443937C2semidirect
15. #16839549C4semidirect
79. C24×C61C4⋊C4semidirect
119. C22⋊C4×C61C4⋊C4semidirect
286. C61#12960semidirect
338. C24#11758196semidirect
393. C22⋊C4#11758196semidirect

Maximal subgroups

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

subgroupindex
C2×#168395492
#214462282
#214460992
#214460282
#214439442
#214439372
C61×#23752
#1296061