Finite Group Databasebeta

← #30595842

Subgroups — #30595842

Order 15616.

640 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461709C211yesmaximal, characteristic, complemented
2#21461545C211yesmaximal, characteristic, complemented
3#21460636C212yesmaximal, complemented
4#21460635C212yesmaximal, complemented
5#21454162C211yesmaximal, characteristic
6C2×C22⋊#92420—21no—
7C2×C22⋊#92420—21no—
8C2×#14519832—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519773C411yescharacteristic
13C2×#14519751C411yescharacteristic
14C2×#14519749—21no—
15C2×#14519749—21no—
16C22×C4×Dic61C2211yescharacteristic
17#16839665C2211yescharacteristic
18#16839663C2211yescharacteristic
19#16839662—21no—
20#16839662—21no—
21#16839631C411yescharacteristic
22C2×#121×C61C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25C22⋊#92420—24no—
26C22⋊#92420—42no—
27#14519832—42no—
28C22×D244C2×C411yescharacteristic
29C22×C22⋊D61—21no—
30C22×C22⋊D61—21no—
31C22×C22⋊D61—21no—
32C23×Dic61D411yescharacteristic
33C23×Dic61D411yescharacteristic
34#14519773—42no—
35#14519751—42no—
36#14519749—44no—
37#14519749—44no—
38C2×C4⋊Dic61C2×C411yescharacteristic
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×C4×Dic61—42no—
47C2×C4×Dic61C2×C411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61C2311yescharacteristic
51C22×D4×C61—21no—
52C22×D4×C61C2×C411yescharacteristic
53C23×C4×C61C2×C411yescharacteristic
54#121×C61—42no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61—21no—
57C24×C61—21no—
58C24×C61C22⋊C411yescharacteristic
59C23×D61—21no—
60C2×D4×C61—22no—
61C2×D4×C61—41no—
62C2×D4×C61—41no—
63C2×D4×C61—42no—
64C2×D4×C61—42no—
65C2×D4×C61—42no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—21no—
69C22×C4×C61—41no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22×C411yescharacteristic
72C2×C22⋊D61—42no—
73C2×C22⋊D61—42no—
74C2×C22⋊D61—42no—
75C2×C22⋊D61—42no—
76C2×C22⋊D61—42no—
77C2×C22⋊D61—42no—
78C2×C22⋊D61—42no—
79C2×C22⋊D61—42no—
80C2×C22⋊D61—42no—
81C22×Dic61—22no—
82C22×Dic61—22no—
83C22×Dic61—22no—
84C22×Dic61—22no—
85C22×Dic61—21no—
86C22×Dic61—41no—
87C22×Dic61—41no—
88C22×Dic61C22⋊C411yescharacteristic
89C22×Dic61C22⋊C411yescharacteristic
90C22×Dic61C2×D411yescharacteristic
91C22×Dic61C2×D411yescharacteristic
92C2×D244—22no—
93C2×D244—41no—
94C2×D244—41no—
95C2×#92420—24no—
96C2×#92420—42no—
97C2×#92420—42no—
98C2×#92420—42no—
99C2×#92420—42no—
100C2×#92420—42no—
101C4⋊C4×C61—41no—
102C22⋊C4×C61—42no—
103C22⋊Dic61—42no—
104C22⋊Dic61—42no—
105#11758198—42no—
106#11758198—42no—
107#11758198—42no—
108C4⋊Dic61—41no—
109#11758196—42no—
110#11758196—42no—
111C4×Dic61—22no—
112C4×Dic61—22no—
113C4×Dic61—41no—
114C4×Dic61—41no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61#10012yes—
129C23×C61C2×C22⋊C411yescharacteristic
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137D4×C61—82no—
138D4×C61—84no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—44no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61#13112yes—
149C22⋊D61—84no—
150C22⋊D61—84no—
151C22⋊D61—84no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—21no—
157C2×Dic61—21no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—44no—
161C2×Dic61—44no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61#9912yes—
166C2×Dic61#10412yes—
167C2×Dic61AGL(1,ℤ/8)12yes—
168C2×Dic61AGL(1,ℤ/8)12yes—
169D244—82no—
170#92420—44no—
171#92420—82no—
172#7704—611nomaximal, Sylow
173C22×C61—22no—
174C22×C61—22no—
175C22×C61—22no—
176C22×C61—41no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61#40811yescharacteristic
188C22×C61C2×#9911yescharacteristic
189C22×C61C2×#10011yescharacteristic
190C22×C61#41811yescharacteristic
191C22×C61#41811yescharacteristic
192C22×C61C2×#10411yescharacteristic
193C22×C61#42111yescharacteristic
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C4×C61—22no—
202C4×C61—41no—
203C4×C61—81no—
204Dic61—22no—
205Dic61—22no—
206Dic61—22no—
207Dic61—22no—
208Dic61—81no—
209#2856—611no—
210#1817—611no—
211#1749—611no—
212#1270—612no—
213#1270—612no—
214C2×C61—42no—
215C2×C61—161no—
216C2×C61#147111yescharacteristic
217C2×C61#148411yescharacteristic
218C2×C61#149611yescharacteristic
219C2×C61#150711yescharacteristic
220C2×C61#151011yescharacteristic
221C2×C61#153011yescharacteristic
222C2×C61#155511yescharacteristic
223D61—161no—
224C2×#127—611no—
225C2×#121—611no—
226C2×#121—1221no—
227C2×#121—1221no—
228C2×#121#692311yescharacteristic
229C22×C42—611no—
230C2×C22⋊C8—1221no—
231C2×C22⋊C8—1221no—
232C2×#97—1221no—
233#390—611no—
234#390—611no—
235#390—1221no—
236#390—1221no—
237#384—611no—
238#336—612no—
239#336—612no—
240#336—1221no—
241C61#770411yescharacteristic, complemented, Sylow
242C22×D4—21no—
243C22×D4—611no—
244C22×D4—1221no—
245C22×D4—1221no—
246C22×D4—1221no—
247C22×D4C2×#692311yescharacteristic
248C23×C4—611no—
249C23×C4—611no—
250C23×C4C2×#692311yescharacteristic
251C22×C8—1222no—
252C22×C8—1222no—
253#127—2442no—
254#121—42no—
255#121—2442no—
256#121—2444no—
257#121—2444no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4C2×#692311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C42—611no—
269C2×C42—611no—
270C2×C42—611no—
271C2×C42—611no—
272C2×C42—2442no—
273C22⋊C8—1224no—
274C22⋊C8—2442no—
275#97—2442no—
276C24—21no—
277C24—1221no—
278C24C22⋊#692311yescharacteristic
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—1222no—
286C2×D4—2441no—
287C2×D4—2441no—
288C2×D4—2442no—
289C2×D4—2442no—
290C2×D4—2442no—
291C2×D4—2442no—
292C2×D4—2442no—
293C2×D4—2442no—
294C2×D4—2442no—
295C2×D4—2442no—
296C2×D4—2442no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—611no—
302C22×C4—611no—
303C22×C4—611no—
304C22×C4—611no—
305C22×C4—1222no—
306C22×C4—1222no—
307C22×C4—1222no—
308C22×C4—1222no—
309C22×C4—1221no—
310C22×C4—2441no—
311C22×C4—2441no—
312C22×C4C22⋊#692311yescharacteristic
313C22×C4C22×#692311yescharacteristic
314C2×C8—1224no—
315C2×C8—2442no—
316C2×C8—2442no—
317C2×C8—2442no—
318C2×C8—2442no—
319C2×C8—2442no—
320C4⋊C4—41no—
321C4⋊C4—2441no—
322C4⋊C4—2442no—
323C4⋊C4—2442no—
324C22⋊C4—42no—
325C22⋊C4—2442no—
326C22⋊C4—2442no—
327C22⋊C4—2442no—
328C22⋊C4—2442no—
329C22⋊C4—2442no—
330C42—1222no—
331C42—1222no—
332C42—2441no—
333C42—2441no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—2441no—
348C23—2441no—
349C23—2441no—
350C23—2441no—
351C23—2441no—
352C23—2441no—
353C23—2441no—
354C23#1451983712yes—
355C23C2×C22⋊#692311yescharacteristic
356D4—82no—
357D4—84no—
358D4—4882no—
359D4—4884no—
360D4—4884no—
361D4—4884no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—44no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—612no—
372C2×C4—612no—
373C2×C4—612no—
374C2×C4—612no—
375C2×C4—1221no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—1221no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—2444no—
384C2×C4—2444no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4#1451986412yes—
389C8—4882no—
390C8—2444no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—4881no—
406C22—4881no—
407C22—4881no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22#1684073511yescharacteristic
413C22D61⋊#9911yescharacteristic
414C22C2×#1451983711yescharacteristic
415C22#1684075411yescharacteristic
416C22#1684075411yescharacteristic
417C22D61⋊#10411yescharacteristic
418C22#1684075911yescharacteristic
419C4—22no—
420C4—41no—
421C4—81no—
422C4—1222no—
423C4—1222no—
424C4—1222no—
425C4—1222no—
426C4—4881no—
427C2—42no—
428C2—161no—
429C2—9761no—
430C2#2146109511yescharacteristic
431C2#2146113811yescharacteristic
432C2#2146117811yescharacteristic
433C2#2146121411yescharacteristic
434C2#2146122511yescharacteristic
435C2#2146128811yescharacteristic
436C2#2146134611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#121×C61C4
socleC23×C61C2×C22⋊C4
prime radical#21454162C2
cosocleC2×C4×Dic61C23
hypercenterC2×#121#6923
hypocenterC61#7704
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #21461709C2semidirect
2. #21461545C2semidirect
3. #21460636C2semidirect
4. #21460635C2semidirect
241. C61#7704semidirect

Maximal subgroups

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

subgroupindex
#214617092
#214615452
#214606362
#214606352
#214541622
#770461