Finite Group Databasebeta

← #29568747

Subgroups — #29568747

Order 15616.

546 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 87 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#21447053C211yesmaximal, characteristic, complemented
2#21447053C211yesmaximal, characteristic, complemented
3#21446889C211yesmaximal, characteristic
4#21443642C211yesmaximal, characteristic, complemented
5#21443642C211yesmaximal, characteristic, complemented
6#21443640C211yesmaximal, characteristic, complemented
7C61×#2727C211yesmaximal, characteristic, complemented
8C22×#11758199—21no—
9C2×#14519774C2211yescharacteristic
10C2×#14519697—21no—
11C2×#14519697—21no—
12C2×#14519697—21no—
13C2×#14519697—21no—
14C2×#14519697—21no—
15#16839517C2211yescharacteristic
16#16839517C2211yescharacteristic
17#16839517C2211yescharacteristic
18C2×#126×C61—21no—
19C61×#399—21no—
20C61×#396—21no—
21C61×#394—21no—
22C61×#390C2211yescharacteristic
23C61×#389—21no—
24C61×#389C2211yescharacteristic
25C61×#389C2211yescharacteristic
26C61×#388—21no—
27C22×D244—21no—
28C22×D244—21no—
29C22×D244—21no—
30C2×#11758199—42no—
31C2×#11758199—42no—
32C2×#11758199—42no—
33C22×#92418—21no—
34C22×#92418—21no—
35C22×#92418—21no—
36#14519774—42no—
37#14519697—42no—
38#14519697—42no—
39#14519697—42no—
40#14519697—42no—
41#14519697—42no—
42C22×D4×C61D411yescharacteristic
43#126×C61—42no—
44#126×C61—42no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C4⋊C4×C61D411yescharacteristic
51C2×C4⋊C4×C61D411yescharacteristic
52C2×C4⋊C4×C61D411yescharacteristic
53C2×C4⋊C4×C61D411yescharacteristic
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61—21no—
56C2×C22⋊C4×C61—21no—
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61—21no—
60C2×C42×C61—21no—
61C2×C42×C61C2311yescharacteristic
62#95×C61—21no—
63#95×C61—21no—
64#95×C61—21no—
65C24×C61—21no—
66C23×D61—41no—
67C2×D4×C61—21no—
68C2×D4×C61—21no—
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C2×D244—22no—
80C2×D244—22no—
81C2×D244—22no—
82C2×D244—41no—
83C2×D244—41no—
84C2×D244—41no—
85C2×D244—41no—
86C2×D244—41no—
87C2×D244—41no—
88C2×D244—41no—
89C2×D244—41no—
90C2×D244—41no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—41no—
94C2×D244—41no—
95C2×D244—41no—
96C2×D244—41no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—42no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—81no—
109C4⋊C4×C61D812yescomplemented
110C22⋊C4×C61—42no—
111C22⋊C4×C61—42no—
112C22⋊C4×C61—41no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—81no—
117C22⋊C4×C61—81no—
118C42×C61—41no—
119C42×C61—42no—
120#11758199—48no—
121C2×#92418—22no—
122C2×#92418—41no—
123C2×#92418—41no—
124C2×#92418—41no—
125C2×#92418—41no—
126C2×#92418—41no—
127C2×#92418—41no—
128C2×#92418—41no—
129C2×#92418—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61#12011yescharacteristic
138C22×D61—42no—
139C22×D61—81no—
140C22×D61—81no—
141C22×D61—81no—
142C22×D61—81no—
143C22×D61—81no—
144C22×D61—81no—
145D4×C61—42no—
146D4×C61—81no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61C2×D811yescharacteristic
169C2×C4×C61#13511yescharacteristic
170C2×C4×C61#13511yescharacteristic
171C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
174D244—42no—
175D244—42no—
176D244—42no—
177D244—42no—
178D244—42no—
179D244—42no—
180D244—81no—
181D244—81no—
182D244—81no—
183D244—81no—
184D244—81no—
185D244—81no—
186#92418—42no—
187#92418—81no—
188#92418—81no—
189#7740—611nomaximal, Sylow
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61#44711yescharacteristic
198C22×C61#44911yescharacteristic
199C22×C61#44911yescharacteristic
200C22×C61#45311yescharacteristic
201C22×C61#45411yescharacteristic
202C22×C61#45411yescharacteristic
203C22×C61B(4,𝔽2)11yescharacteristic
204C2×D61—82no—
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—161no—
208C2×D61—161no—
209C2×D61—161no—
210C2×D61—161no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—42no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61—81no—
221C4×C61—81no—
222#2727D6111yescharacteristic, complemented
223#2041—611no—
224#2041—611no—
225#2011—611no—
226#1271—611no—
227#1271—611no—
228#1270—611no—
229C2×C61—161no—
230C2×C61#159711yescharacteristic
231C2×C61#159811yescharacteristic
232C2×C61#160011yescharacteristic
233C2×C61#162111yescharacteristic
234C2×C61#162511yescharacteristic
235C2×C61#162811yescharacteristic
236C2×C61#165711yescharacteristic
237D61—162no—
238C22×D8—1221no—
239C2×#127—611no—
240C2×#126—21no—
241C2×#102—1221no—
242C2×#102—1221no—
243C2×#102—1221no—
244C2×#102—1221no—
245C2×#102—1221no—
246#399—21no—
247#396—21no—
248#394—21no—
249#390C2×D6111yescharacteristic
250#389—21no—
251#389C2×D6111yescharacteristic
252#389C2×D6111yescharacteristic
253#388—21no—
254#336—611no—
255#336—611no—
256#336—611no—
257C61#774011yescharacteristic, complemented, Sylow
258C22×D4—1221no—
259C22×D4—1221no—
260C22×D4—1221no—
261C22×D4C22⋊D6111yescharacteristic
262C2×D8—2442no—
263C2×D8—2442no—
264C2×D8—2442no—
265C22×C8—1221no—
266C22×C8—1221no—
267C22×C8—1221no—
268#127—2442no—
269#126—42no—
270#126—42no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4C22⋊D6111yescharacteristic
276C2×C4⋊C4C22⋊D6111yescharacteristic
277C2×C4⋊C4C22⋊D6111yescharacteristic
278C2×C4⋊C4C22⋊D6111yescharacteristic
279C2×C4⋊C4C22⋊D6111yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C42—21no—
287C2×C42C22×D6111yescharacteristic
288#102—2442no—
289#102—2442no—
290#102—2442no—
291#102—2442no—
292#102—2442no—
293#95—21no—
294#95—21no—
295#95—21no—
296C24—21no—
297C24—2441no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—1222no—
303C2×D4—1222no—
304C2×D4—1222no—
305C2×D4—2441no—
306C2×D4—2441no—
307C2×D4—2441no—
308C2×D4—2441no—
309C2×D4—2441no—
310C2×D4—2441no—
311C2×D4—2441no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4—2441no—
315C2×D4—2441no—
316C2×D4—2441no—
317C2×D4—2441no—
318C2×D4—2441no—
319C2×D4—2441no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4C2×C22⋊D6111yescharacteristic
326C22×C4C2×C22⋊D6111yescharacteristic
327C22×C4C2×C22⋊D6111yescharacteristic
328D8—2448no—
329C2×C8—1222no—
330C2×C8—2441no—
331C2×C8—2441no—
332C2×C8—2441no—
333C2×C8—2441no—
334C2×C8—2441no—
335C2×C8—2441no—
336C2×C8—2441no—
337C2×C8—2441no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—81no—
350C4⋊C4#1175819912yescomplemented
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—81no—
358C22⋊C4—81no—
359C42—41no—
360C42—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—4881no—
369C23—4881no—
370C23—4881no—
371C23—4881no—
372C23—4881no—
373C23—4881no—
374C23—2442no—
375C23#1451974611yescharacteristic
376D4—42no—
377D4—81no—
378D4—4881no—
379D4—4881no—
380D4—4881no—
381D4—4881no—
382D4—4881no—
383D4—4881no—
384D4—2442no—
385D4—2442no—
386D4—2442no—
387D4—2442no—
388D4—2442no—
389D4—2442no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4C2×#1175819911yescharacteristic
412C2×C4#1451979711yescharacteristic
413C2×C4#1451979711yescharacteristic
414C2×C4#1451980211yescharacteristic
415C2×C4#1451980211yescharacteristic
416C2×C4#1451980311yescharacteristic
417C8—4881no—
418C8—4881no—
419C8—2442no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—9761no—
428C22—9761no—
429C22—9761no—
430C22—9761no—
431C22—4882no—
432C22—4882no—
433C22—4882no—
434C22#1683989611yescharacteristic
435C22#1683990711yescharacteristic
436C22#1683990711yescharacteristic
437C22#1683993011yescharacteristic
438C22#1683993511yescharacteristic
439C22#1683993511yescharacteristic
440C22#1683994911yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—81no—
452C2—161no—
453C2—9762no—
454C2#2144474811yescharacteristic
455C2#2144475411yescharacteristic
456C2#2144476511yescharacteristic
457C2#2144488411yescharacteristic
458C2#2144490811yescharacteristic
459C2#2144492911yescharacteristic
460C2#2144511711yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2727C2
socleC23×C61#120
prime radical#21446889C2
hypercenter#2727D61
hypocenterC61#7740
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
1. #21447053C2semidirect
2. #21447053C2semidirect
4. #21443642C2semidirect
5. #21443642C2semidirect
6. #21443640C2semidirect
7. C61×#2727C2semidirect
109. C4⋊C4×C61D8semidirect
222. #2727D61semidirect
257. C61#7740semidirect
350. C4⋊C4#11758199semidirect

Maximal subgroups

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

subgroupindex
#214470532
#214468892
#214436422
#214436402
C61×#27272
#774061