Finite Group Databasebeta

← #29580588

Subgroups — #29580588

Order 15616.

772 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 117 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×#16840075C211yesmaximal, characteristic, complemented
2#21451923C211yesmaximal, characteristic
3C2×#16839497C211yesmaximal, characteristic, complemented
4C2×#16839496C211yesmaximal, characteristic, complemented
5#21444087C211yesmaximal, characteristic, complemented
6#21443661C211yesmaximal, characteristic, complemented
7C2×C61×#480C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519749—22no—
10#16840075—22no—
11#16840075—22no—
12C2×#14519710—21no—
13C2×#14519697—21no—
14C2×C22⋊#92418C2211yescharacteristic
15#16839698—21no—
16#16839697—21no—
17#16839671—21no—
18#16839670—21no—
19C2×#14519684—22no—
20C2×#14519684C2211yescharacteristic
21C2×#14519684C2211yescharacteristic
22#16839497—22no—
23#16839496—22no—
24C2×#121×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#480—22no—
27C61×#480—22no—
28C2×#107×C61—21no—
29C2×#102×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C22×C22⋊D61—22no—
34C23×Dic61—21no—
35C23×Dic61D412yes—
36C22×#92418—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519749—44no—
42#14519749—44no—
43C2×C4⋊Dic61D411yescharacteristic
44C2×#11758196—22no—
45C2×C22⋊Dic61—22no—
46C2×#11758198—22no—
47C2×#11758198—21no—
48#14519710—24no—
49#14519697—24no—
50C22⋊#92418—21no—
51C22⋊#92418D412yescomplemented
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—44no—
57C22×D4×C61—21no—
58C22×D4×C61D411yescharacteristic
59C23×C4×C61—21no—
60C23×C4×C61C2311yescharacteristic
61C22×C8×C61—21no—
62#121×C61—21no—
63#121×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C4⋊C4×C61D412yescomplemented
73C2×C22⋊C4×C61—21no—
74#107×C61—24no—
75#102×C61—24no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61D412yescomplemented
78C24×C61—21no—
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—42no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C2×D4×C61—42no—
87C2×D4×C61—42no—
88C22×C4×C61—21no—
89C22×C4×C61—22no—
90C22×C4×C61—42no—
91C22×C4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61—42no—
94C22×C4×C61—42no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C2×C22⋊D61—44no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—41no—
104C2×C22⋊D61—42no—
105C2×C22⋊D61—42no—
106C2×C22⋊D61—82no—
107C2×C22⋊D61—82no—
108C22×Dic61—22no—
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—21no—
112C22×Dic61—42no—
113C22×Dic61—42no—
114C22×Dic61—41no—
115C22×Dic61—42no—
116C22×Dic61—41no—
117C22×Dic61—42no—
118C22×Dic61D814yes—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—42no—
122C2×C8×C61—22no—
123C2×C8×C61—42no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—41no—
129C4⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C22⋊Dic61—44no—
133#11758198—44no—
134#11758198—41no—
135#11758198—41no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138#11758196—44no—
139C2×#92418—22no—
140C2×#92418—42no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—22no—
144C23×C61—41no—
145C23×C61—42no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—42no—
149C23×C61#12011yescharacteristic
150C23×C61#12111yescharacteristic
151C23×C61#12711yescharacteristic
152C23×C61C2×D812yes—
153C23×C61AGL(1,ℤ/8)12yes—
154C22×D61—41no—
155C22×D61—42no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—42no—
159D4×C61—42no—
160D4×C61—81no—
161D4×C61—82no—
162D4×C61—81no—
163D4×C61—81no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—44no—
174C2×C4×C61—82no—
175C2×C4×C61—81no—
176C2×C4×C61—81no—
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C22⋊D61—88no—
182C22⋊D61—82no—
183C22⋊D61—81no—
184C22⋊D61—81no—
185C2×Dic61—24no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—44no—
189C2×Dic61—44no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193C2×Dic61—82no—
194C2×Dic61—82no—
195C2×Dic61—81no—
196C2×Dic61—81no—
197D244—42no—
198D244—81no—
199C8×C61—42no—
200#92418—42no—
201#9970—611nomaximal, Sylow
202C22×C61—22no—
203C22×C61—22no—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—82no—
211C22×C61—82no—
212C22×C61—81no—
213C22×C61—81no—
214C22×C61—81no—
215C22×C61#39211yescharacteristic
216C22×C61#44712yes—
217C22×C61#45511yescharacteristic
218C22×C61#45611yescharacteristic
219C22×C61#45812yes—
220C22×C61#45912yes—
221C22×C61#49311yescharacteristic
222C22×C61#49611yescharacteristic
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—81no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—82no—
233C4×C61—81no—
234Dic61—44no—
235Dic61—82no—
236Dic61—81no—
237C2×#480—611no—
238C2×#480D6111yescharacteristic, complemented
239#2656—611no—
240C2×#327—611no—
241C2×#327—611no—
242#1445—611no—
243#1280—611no—
244C2×C61—21no—
245C2×C61—22no—
246C2×C61—21no—
247C2×C61—161no—
248C2×C61#160712yes—
249C2×C61#161012yes—
250C2×C61#200711yescharacteristic
251C2×C61#201111yescharacteristic
252C2×C61#202211yescharacteristic
253D61—161no—
254C2×#121—611no—
255C2×#121—1222no—
256C2×#121C2×D6111yescharacteristic
257C22×C4⋊C4C2×D6111yescharacteristic
258#480—22no—
259#480—22no—
260#480—1222no—
261#480—1222no—
262C2×#107—21no—
263C2×#107—1221no—
264C2×#102—21no—
265C2×#102—1221no—
266C2×C22⋊C8—611no—
267C2×C22⋊C8C2×D6111yescharacteristic
268#399—1221no—
269#399—1221no—
270#392—1221no—
271#392—1221no—
272C2×#95—611no—
273C2×#95—611no—
274C2×#95—1222no—
275#327—1222no—
276#327—1222no—
277C61#997011yescharacteristic, complemented, Sylow
278C22×D4—21no—
279C22×D4—611no—
280C22×D4—1222no—
281C22×D4—1221no—
282C22×D4C22⋊D6111yescharacteristic
283C23×C4—21no—
284C23×C4—612no—
285C23×C4—1221no—
286C23×C4C22×D6111yescharacteristic
287C22×C8—21no—
288C22×C8—1221no—
289#121—21no—
290#121—21no—
291#121—21no—
292#121—21no—
293#121—1221no—
294#121—1221no—
295#121—1221no—
296#121—1221no—
297#121—2444no—
298#121—2444no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—1222no—
306C2×C4⋊C4C22⋊D6111yescharacteristic
307C2×C4⋊C4C22⋊D6112yescomplemented
308C2×C22⋊C4—21no—
309C2×C22⋊C4—1222no—
310C2×C22⋊C4—1222no—
311C2×C22⋊C4—1221no—
312#107—24no—
313#107—1224no—
314#102—24no—
315#102—1224no—
316C22⋊C8—21no—
317C22⋊C8—612no—
318C22⋊C8—1221no—
319C22⋊C8C22⋊D6112yescomplemented
320#95—1221no—
321#95—1221no—
322#95—1221no—
323#95—1221no—
324#95—2444no—
325C24—21no—
326C24—1221no—
327C24D4×D6111yescharacteristic
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—1221no—
336C2×D4—1221no—
337C2×D4—4882no—
338C2×D4—4882no—
339C2×D4—2442no—
340C2×D4—2444no—
341C2×D4—2441no—
342C2×D4—2441no—
343C2×D4—2442no—
344C2×D4—2442no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—614no—
353C22×C4—1222no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1221no—
357C22×C4—2441no—
358C22×C4—2442no—
359C22×C4—2441no—
360C22×C4—2442no—
361C22×C4—2442no—
362C22×C4—2442no—
363C22×C4D4×D6111yescharacteristic
364C22×C4D4×D6111yescharacteristic
365C22×C4#1175822411yescharacteristic
366C22×C4C2×C22⋊D6111yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368C22×C4C2×C22⋊D6111yescharacteristic
369C2×C8—22no—
370C2×C8—42no—
371C2×C8—1222no—
372C2×C8—2442no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—2444no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—2444no—
385C22⋊C4—2444no—
386C22⋊C4—2441no—
387C22⋊C4—2441no—
388C23—21no—
389C23—21no—
390C23—22no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23—41no—
396C23—2441no—
397C23—2441no—
398C23—2442no—
399C23—2442no—
400C23—2441no—
401C23#1451974511yescharacteristic
402C23#1451974911yescharacteristic
403C23#1451977311yescharacteristic
404C23D8×D6112yes—
405C23#1451980112yes—
406D4—42no—
407D4—81no—
408D4—82no—
409D4—81no—
410D4—81no—
411D4—4881no—
412D4—4882no—
413D4—4881no—
414D4—4881no—
415D4—4888no—
416D4—2442no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—1224no—
431C2×C4—4881no—
432C2×C4—4882no—
433C2×C4—4881no—
434C2×C4—4882no—
435C2×C4—2442no—
436C2×C4—2444no—
437C2×C4—2442no—
438C2×C4—2441no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2444no—
442C2×C4#1451974511yescharacteristic
443C2×C4#1451974611yescharacteristic
444C2×C4#1451974811yescharacteristic
445C2×C4#1451974911yescharacteristic
446C8—42no—
447C8—2442no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—22no—
456C22—81no—
457C22—82no—
458C22—81no—
459C22—82no—
460C22—81no—
461C22—4881no—
462C22—4882no—
463C22—4881no—
464C22—4882no—
465C22—4881no—
466C22#1683967011yescharacteristic
467C22#1683989412yes—
468C22#1683993811yescharacteristic
469C22#1683994311yescharacteristic
470C22#1683995112yes—
471C22#1683995512yes—
472C22#1684013711yescharacteristic
473C22#1684015011yescharacteristic
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—82no—
479C4—81no—
480C4—4882no—
481C4—4881no—
482C4—2444no—
483C2—21no—
484C2—22no—
485C2—21no—
486C2—161no—
487C2—9761no—
488C2#2144480312yes—
489C2#2144482012yes—
490C2#2144686411yescharacteristic
491C2#2144688811yescharacteristic
492C2#2144694711yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#480C2
socleC23×C61#120
prime radical#21451923C2
hypercenterC2×#480D61
hypocenterC61#9970
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
1. C2×#16840075C2semidirect
3. C2×#16839497C2semidirect
4. C2×#16839496C2semidirect
5. #21444087C2semidirect
6. #21443661C2semidirect
7. C2×C61×#480C2semidirect
51. C22⋊#92418D4semidirect
72. C2×C4⋊C4×C61D4semidirect
77. C22⋊C8×C61D4semidirect
238. C2×#480D61semidirect
277. C61#9970semidirect
307. C2×C4⋊C4C22⋊D61semidirect
319. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400752
#214519232
C2×#168394972
C2×#168394962
#214440872
#214436612
C2×C61×#4802
#997061