Finite Group Databasebeta

← #29652852

Subgroups — #29652852

Order 15616.

1198 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 167 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#21459590C211yesmaximal, characteristic, complemented
2#21455910C212yesmaximal, complemented
3#21444855C214yesmaximal, complemented
4#21444753C214yesmaximal, complemented
5#21444326C212yesmaximal, complemented
6#21444163C211yesmaximal, characteristic, complemented
7C61×#2398C211yesmaximal, characteristic, complemented
8C2×#14519825—21no—
9#16840434—22no—
10Q8×D244—22no—
11#16840424—24no—
12#16840343—21no—
13#16840343C2214yes—
14#16840312—22no—
15#16840312C2212yes—
16#16840080C2214yes—
17#16840052—22no—
18#16839957—22no—
19#16839896C2214yes—
20#16839872—24no—
21#16839819C2212yes—
22#16839495C2218yes—
23#16839488C2214yes—
24C2×C4×Q8×C61C2211yescharacteristic
25C2×C4×D4×C61C2211yescharacteristic
26C61×#400C2212yes—
27C61×#397—21no—
28C61×#394C2212yes—
29C61×#393—21no—
30C4×C22⋊C4×C61—22no—
31C61×#376C2211yescharacteristic
32#14519825—48no—
33C2×#11758226—24no—
34C2×#11758222—24no—
35C2×#11758222—22no—
36C2×Q8×D61—24no—
37C2×#11758199—24no—
38#14519777—22no—
39#14519777D412yes—
40#14519774—22no—
41#14519774D412yes—
42#14519765—24no—
43#14519765—22no—
44#14519757—24no—
45#14519757—28no—
46#14519752—24no—
47#14519752—28no—
48#14519751—22no—
49#14519751—24no—
50#14519751C2314yes—
51C4×D244—24no—
52C4×D244—24no—
53C4×D244—22no—
54C4×D244D414yes—
55#14519708—24no—
56C4⋊#92418—22no—
57#14519698—24no—
58#14519697—24no—
59C22⋊#92418C2314yes—
60#14519688—22no—
61C22×Q8×C61D411yescharacteristic
62C22×D4×C61C2311yescharacteristic
63C23×C4×C61—21no—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C4×D4×C61—22no—
67C4×D4×C61—22no—
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C4⋊C4×C61C2311yescharacteristic
70C2×C4⋊C4×C61C2312yes—
71C2×C22⋊C4×C61—22no—
72C2×C22⋊C4×C61—22no—
73C2×C22⋊C4×C61—21no—
74C2×C42×C61—21no—
75C2×C42×C61D412yes—
76C2×C42×C61C2311yescharacteristic
77#95×C61—21no—
78#95×C61—22no—
79#95×C61—22no—
80#95×C61C2312yes—
81C24×C61—21no—
82C2×Q8×C61—22no—
83C2×Q8×C61—24no—
84C2×D4×C61—22no—
85C2×D4×C61—42no—
86C22×C4×C61—21no—
87C22×C4×C61—22no—
88C22×C4×C61—44no—
89C22×C4×C61—41no—
90C22×C4×C61—41no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2411yescharacteristic
96C2×C22⋊D61—24no—
97#11758226—416no—
98Q8×D61—416no—
99#11758222—48no—
100#11758222—48no—
101#11758222—48no—
102C2×D244—24no—
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—24no—
106C2×D244C2×D414yes—
107C2×C4×D61—28no—
108C2×C4×D61—24no—
109C2×Dic122—24no—
110C2×Dic122—22no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61—22no—
114C22⋊C4×C61—44no—
115C22⋊C4×C61—42no—
116C22⋊C4×C61—44no—
117C42×C61—22no—
118C42×C61—22no—
119C42×C61—42no—
120C42×C61C2×D412yes—
121C42×C61C2×D412yes—
122#11758199—48no—
123#11758198—28no—
124#11758198—24no—
125#11758198—24no—
126C4⋊Dic61—24no—
127C4⋊Dic61—22no—
128C4⋊Dic61C2×D414yes—
129C2×#92418—24no—
130C23×C61—22no—
131C23×C61—42no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—42no—
135C23×C61C22×D411yescharacteristic
136C22×D61—24no—
137Q8×C61—28no—
138D4×C61—24no—
139D4×C61—42no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—41no—
146C2×C4×C61—42no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—82no—
153C2×C4×C61—82no—
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12012yes—
156C2×C4×C61#12011yescharacteristic
157C2×C4×C61AGL(1,ℤ/8)14yes—
158C2×C4×C61C22×D412yes—
159C22⋊D61—48no—
160C2×Dic61—24no—
161D244—44no—
162D244—44no—
163D244—48no—
164D244—44no—
165D244—84no—
166C4×D61—48no—
167C4×D61—416no—
168Dic122—44no—
169Dic122—48no—
170#92418—44no—
171#18186—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—44no—
175C22×C61—42no—
176C22×C61—82no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61C2×#12011yescharacteristic
180C22×C61C2×AGL(1,ℤ/8)12yes—
181C2×D61—44no—
182C2×D61—44no—
183C2×D61—44no—
184C4×C61—22no—
185C4×C61—24no—
186C4×C61—21no—
187C4×C61—22no—
188C4×C61—42no—
189C4×C61—82no—
190C4×C61—81no—
191C4×C61#45412yes—
192Dic61—44no—
193#3497—611no—
194#3109—612no—
195#2398D6111yescharacteristic, complemented
196#1616—614no—
197#1598—614no—
198#1505—612no—
199#1467—611no—
200C2×C61—22no—
201C2×C61—82no—
202C2×C61#300311yescharacteristic
203C2×C61C2×#45411yescharacteristic
204C2×C61#302511yescharacteristic
205D61—84no—
206C2×2−1+4—1221no—
207#550—1222no—
208D4×Q8—1222no—
209#548—1224no—
210#537—614no—
211#537—1221no—
212#532—612no—
213#532—1222no—
214C2×C4×Q8C2×D6111yescharacteristic
215C2×C4×D4C2×D6111yescharacteristic
216#481—614no—
217#476—1222no—
218#459—1222no—
219#447—614no—
220#442—1224no—
221#432—612no—
222#400C2×D6112yes—
223#397—21no—
224#394C2×D6112yes—
225#393—21no—
226C4×C22⋊C4—22no—
227#376C2×D6111yescharacteristic
228#327—618no—
229#324—614no—
230C61#1818611yescharacteristic, complemented, Sylow
2312−1+4—2448no—
232C2×#41—1224no—
233C2×#41—1222no—
234C2×#41—1224no—
235C22×Q8—1224no—
236C22×Q8C22⋊D6111yescharacteristic
237C22×D4C22×D6111yescharacteristic
238C23×C4—21no—
239C2×D8—1224no—
240C4⋊Q8—612no—
241C4⋊Q8—1222no—
242#127—612no—
243#127—1222no—
244#124—1224no—
245#124—1222no—
246C22⋊Q8—1228no—
247C22⋊Q8—1224no—
248#121—614no—
249#121—1228no—
250#121—1224no—
251#121—1222no—
252#121—1224no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—614no—
258C4×D4—1224no—
259C4×D4—1224no—
260C4×D4—1222no—
261C2×C4⋊C4C22⋊D6111yescharacteristic
262C2×C4⋊C4C22×D6111yescharacteristic
263C2×C4⋊C4C22×D6112yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C42—21no—
268C2×C42C22⋊D6112yes—
269C2×C42C22×D6111yescharacteristic
270#106—1224no—
271C4⋊C8—1222no—
272#102—1224no—
273#102—1224no—
274C22⋊C8—614no—
275#97—1222no—
276#95—21no—
277#95—22no—
278#95—22no—
279#95C22×D6112yes—
280C24—21no—
281#41—2448no—
282#41—2448no—
283#41—24416no—
284#41—2448no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—1224no—
288C2×Q8—1222no—
289C2×Q8—24416no—
290C2×D4—22no—
291C2×D4—42no—
292C2×D4—614no—
293C2×D4—1224no—
294C2×D4—1224no—
295C2×D4—1224no—
296C2×D4—1222no—
297C2×D4—1222no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—1228no—
304C22×C4—1224no—
305C22×C4C2×C22⋊D6111yescharacteristic
306C22×C4C2×C22⋊D6111yescharacteristic
307C22×C4C2×C22⋊D6112yes—
308C22×C4C2×C22⋊D6112yes—
309C22×C4C23×D6111yescharacteristic
310D8—2448no—
311C2×C8—1224no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—614no—
316C4⋊C4—1224no—
317C4⋊C4—1222no—
318C22⋊C4—44no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—1228no—
322C22⋊C4—1224no—
323C22⋊C4—1224no—
324C42—22no—
325C42—22no—
326C42—42no—
327C42D4×D6112yes—
328C42D4×D6112yes—
329C23—22no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—1224no—
335C23C22×C22⋊D6111yescharacteristic
336Q8—28no—
337Q8—2444no—
338Q8—2448no—
339D4—24no—
340D4—42no—
341D4—4884no—
342D4—2444no—
343D4—2444no—
344D4—2448no—
345D4—2448no—
346D4—2444no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—1224no—
362C2×C4—2448no—
363C2×C4—24416no—
364C2×C4#1451974511yescharacteristic
365C2×C4#1451974512yes—
366C2×C4#1451974511yescharacteristic
367C2×C4#1451980214yes—
368C2×C4C2×D4×D6112yes—
369C8—2444no—
370C22—21no—
371C22—21no—
372C22—44no—
373C22—42no—
374C22—82no—
375C22—81no—
376C22—81no—
377C22—2444no—
378C22—2444no—
379C22—2444no—
380C22C2×#1451974511yescharacteristic
381C22C2×#1451980212yes—
382C4—21no—
383C4—22no—
384C4—22no—
385C4—24no—
386C4—42no—
387C4—82no—
388C4—81no—
389C4—2444no—
390C4#1683993412yes—
391C2—22no—
392C2—82no—
393C2—4884no—
394C2#2145504511yescharacteristic
395C2C2×#1683993411yescharacteristic
396C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2398C2
socleC22×C61C2×#120
hypercenter#2398D61
hypocenterC61#18186
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. #21459590C2semidirect
2. #21455910C2semidirect
3. #21444855C2semidirect
4. #21444753C2semidirect
5. #21444326C2semidirect
6. #21444163C2semidirect
7. C61×#2398C2semidirect
195. #2398D61semidirect
230. C61#18186semidirect

Maximal subgroups

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

subgroupindex
#214595902
#214559102
#214448552
#214447532
#214443262
#214441632
C61×#23982
#1818661