Finite Group Databasebeta

← #29654186

Subgroups — #29654186

Order 15616.

1256 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1256 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
1C2×#16840343C211yesmaximal, characteristic, complemented
2#21456616C212yesmaximal, complemented
3#21444855C214yesmaximal, complemented
4#21444753C214yesmaximal, complemented
5#21444334C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C61×#2591C211yesmaximal, characteristic, complemented
8C22×#11758222—21no—
9#16840343—24no—
10#16840343—22no—
11#16840343C2214yes—
12C2×#14519757—22no—
13C2×#14519752—22no—
14C2×#14519751—21no—
15C2×#14519751C2212yes—
16#16840080C2214yescomplemented
17#16840078—22no—
18#16840013—24no—
19#16839896C2214yescomplemented
20#16839895—22no—
21#16839726C2212yes—
22#16839495C2218yescomplemented
23#16839488C2214yescomplemented
24C2×#121×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#400C2212yescomplemented
27C61×#394C2212yescomplemented
28C61×#388—22no—
29C61×#386—21no—
30C61×#386—21no—
31C4×C22⋊C4×C61C2211yescharacteristic
32C2×#11758222—24no—
33C2×#11758222—24no—
34C2×#11758222—48no—
35C22×Dic122—21no—
36C22×D244—21no—
37C22×D244—21no—
38C22×D244D412yes—
39C22×C22⋊D61—22no—
40C22×C4×D61—22no—
41C2×#11758199—24no—
42C2×#11758194—22no—
43#14519757—28no—
44#14519757—28no—
45#14519752—28no—
46#14519752—28no—
47#14519751—24no—
48#14519751—24no—
49#14519751—22no—
50#14519751—22no—
51#14519751D414yes—
52#14519751C2314yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61D412yes—
55C2×#11758198—22no—
56C2×#11758198—22no—
57#14519708—24no—
58#14519698—24no—
59#14519697—24no—
60C22⋊#92418—22no—
61C22⋊#92418C2314yes—
62C22×D4×C61—21no—
63C22×D4×C61C2311yescharacteristic
64C23×C4×C61D411yescharacteristic
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67#121×C61—22no—
68#121×C61—22no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61D412yes—
72C2×C4⋊C4×C61C2311yescharacteristic
73C2×C4⋊C4×C61C2312yes—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—22no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—22no—
78C2×C42×C61—21no—
79#95×C61—22no—
80#95×C61—22no—
81#95×C61C2312yes—
82C24×C61—21no—
83C24×C61C2×D411yescharacteristic
84C23×D61—22no—
85C2×D4×C61—22no—
86C2×D4×C61—44no—
87C2×D4×C61—41no—
88C2×D4×C61—41no—
89C2×D4×C61—81no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—24no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—81no—
98C22×C4×C61—81no—
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D412yes—
104C22×C4×C61C2411yescharacteristic
105C2×C22⋊D61—24no—
106C2×C22⋊D61—24no—
107C2×C22⋊D61—48no—
108C22×Dic61—22no—
109#11758222—416no—
110#11758222—48no—
111#11758222—48no—
112C2×D244—24no—
113C2×D244—24no—
114C2×D244—22no—
115C2×D244—44no—
116C2×D244—82no—
117C2×D244—82no—
118C2×D244C2×D414yes—
119C2×C4×D61—28no—
120C2×C4×D61—48no—
121C2×Dic122—24no—
122C2×Dic122—44no—
123C4⋊C4×C61—24no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—22no—
126C22⋊C4×C61—42no—
127C22⋊C4×C61—42no—
128C22⋊C4×C61—44no—
129C22⋊C4×C61—82no—
130C42×C61—81no—
131#11758199—84no—
132#11758198—28no—
133#11758198—24no—
134#11758198—24no—
135C4⋊Dic61—24no—
136C4⋊Dic61—22no—
137C4⋊Dic61C2×D414yes—
138#11758194—82no—
139C2×#92418—24no—
140C23×C61—22no—
141C23×C61—22no—
142C23×C61—41no—
143C23×C61—42no—
144C23×C61—42no—
145C23×C61—41no—
146C23×C61—42no—
147C23×C61#12011yescharacteristic
148C23×C61#12011yescharacteristic
149C23×C61C22×D411yescharacteristic
150C22×D61—24no—
151C22×D61—44no—
152C22×D61—44no—
153C22×D61—44no—
154D4×C61—44no—
155D4×C61—82no—
156D4×C61—82no—
157C2×C4×C61—24no—
158C2×C4×C61—24no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—42no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—42no—
170C2×C4×C61—42no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—81no—
174C2×C4×C61#12012yes—
175C2×C4×C61AGL(1,ℤ/8)14yes—
176C2×C4×C61C22×D412yes—
177C22⋊D61—48no—
178C22⋊D61—48no—
179C2×Dic61—24no—
180C2×Dic61—44no—
181D244—44no—
182D244—48no—
183D244—82no—
184D244—84no—
185C4×D61—416no—
186Dic122—48no—
187#92418—44no—
188#18324—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—44no—
197C22×C61—84no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61#45412yes—
202C22×C61C2×#12011yescharacteristic
203C22×C61C2×AGL(1,ℤ/8)12yes—
204C2×D61—44no—
205C2×D61—44no—
206C2×D61—44no—
207C2×D61—88no—
208C4×C61—24no—
209C4×C61—22no—
210C4×C61—22no—
211C4×C61—42no—
212C4×C61—82no—
213C4×C61—81no—
214Dic61—44no—
215C2×#537—611no—
216#3192—612no—
217#2591D6111yescharacteristic, complemented
218#1616—614no—
219#1598—614no—
220#1507—612no—
221#1463—611no—
222C2×C61—22no—
223C2×C61—22no—
224C2×C61—41no—
225C2×C61—161no—
226C2×C61#300311yescharacteristic
227C2×C61C2×#45411yescharacteristic
228C2×C61#302611yescharacteristic
229D61—84no—
230C22×#41—1221no—
231#537—614no—
232#537—1224no—
233#537—1222no—
234C2×C22⋊Q8—1222no—
235C2×#121—612no—
236C2×#121—1222no—
237C2×#121—1221no—
238C2×#121C2×D6111yescharacteristic
239C22×C4⋊C4C2×D6111yescharacteristic
240#481—614no—
241#481—1222no—
242#469—1224no—
243#447—614no—
244#447—1222no—
245#407—612no—
246#400C2×D6112yescomplemented
247#394C2×D6112yescomplemented
248#388—22no—
249#386—21no—
250#386—21no—
251C4×C22⋊C4C2×D6111yescharacteristic
252#327—618no—
253#324—614no—
254C61#1832411yescharacteristic, complemented, Sylow
255C2×#41—1224no—
256C2×#41—1224no—
257C2×#41—2448no—
258C22×Q8—1221no—
259C22×D4—21no—
260C22×D4—612no—
261C22×D4—1221no—
262C22×D4—1221no—
263C22×D4—1222no—
264C22×D4C22×D6111yescharacteristic
265C23×C4—1222no—
266C23×C4C22⋊D6111yescharacteristic
267C23×C4C22⋊D6111yescharacteristic
268C23×C4C22×D6111yescharacteristic
269C2×D8—1224no—
270C2×M16—1222no—
271C22⋊Q8—1228no—
272C22⋊Q8—1228no—
273#121—22no—
274#121—22no—
275#121—614no—
276#121—614no—
277#121—1228no—
278#121—1228no—
279#121—1224no—
280#121—1222no—
281#121—1222no—
282#121—1224no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—612no—
286C2×C4⋊C4—1221no—
287C2×C4⋊C4C22⋊D6112yes—
288C2×C4⋊C4C22×D6111yescharacteristic
289C2×C4⋊C4C22×D6112yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—1222no—
295C2×C22⋊C4—1222no—
296C2×C42—21no—
297#106—1224no—
298#102—1224no—
299#102—1224no—
300C22⋊C8—614no—
301C22⋊C8—1222no—
302#95—22no—
303#95—22no—
304#95C22×D6112yes—
305C24—21no—
306C24—1222no—
307C24C2×C22⋊D6111yescharacteristic
308#41—24416no—
309#41—2448no—
310#41—2448no—
311C2×Q8—1224no—
312C2×Q8—2444no—
313C2×D4—22no—
314C2×D4—44no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—81no—
318C2×D4—614no—
319C2×D4—1224no—
320C2×D4—1224no—
321C2×D4—1224no—
322C2×D4—1224no—
323C2×D4—1222no—
324C2×D4—4882no—
325C2×D4—4882no—
326C2×D4—2444no—
327C2×D4—2448no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—1228no—
338C22×C4—1222no—
339C22×C4—2448no—
340C22×C4D4×D6112yes—
341C22×C4D4×D6112yes—
342C22×C4C2×C22⋊D6112yes—
343C22×C4C2×C22⋊D6112yes—
344C22×C4C2×C22⋊D6111yescharacteristic
345C22×C4C23×D6111yescharacteristic
346D8—4884no—
347M16—4882no—
348C2×C8—1224no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—614no—
353C4⋊C4—1224no—
354C4⋊C4—1222no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—44no—
358C22⋊C4—82no—
359C22⋊C4—1228no—
360C22⋊C4—1224no—
361C22⋊C4—1224no—
362C42—81no—
363C23—22no—
364C23—22no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—41no—
369C23—42no—
370C23—1224no—
371C23—2444no—
372C23—2444no—
373C23—2444no—
374C23#1451974511yescharacteristic
375C23#1451974511yescharacteristic
376C23C22×C22⋊D6111yescharacteristic
377Q8—2448no—
378D4—44no—
379D4—82no—
380D4—82no—
381D4—4882no—
382D4—4884no—
383D4—2444no—
384D4—2448no—
385D4—2448no—
386D4—2448no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—81no—
404C2×C4—1224no—
405C2×C4—24416no—
406C2×C4—2444no—
407C2×C4#1451974512yes—
408C2×C4#1451980214yes—
409C2×C4C2×D4×D6112yes—
410C8—2444no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—84no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—4888no—
424C22—2444no—
425C22—2444no—
426C22—2444no—
427C22#1683993412yes—
428C22C2×#1451974511yescharacteristic
429C22C2×#1451980212yes—
430C4—22no—
431C4—22no—
432C4—24no—
433C4—42no—
434C4—82no—
435C4—81no—
436C4—2444no—
437C2—22no—
438C2—22no—
439C2—41no—
440C2—161no—
441C2—4884no—
442C2#2145504511yescharacteristic
443C2C2×#1683993411yescharacteristic
444C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2591C2
socleC22×C61C2×#120
hypercenter#2591D61
hypocenterC61#18324
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. C2×#16840343C2semidirect
2. #21456616C2semidirect
3. #21444855C2semidirect
4. #21444753C2semidirect
5. #21444334C2semidirect
6. #21444149C2semidirect
7. C61×#2591C2semidirect
16. #16840080C22semidirect
19. #16839896C22semidirect
22. #16839495C22semidirect
23. #16839488C22semidirect
26. C61×#400C22semidirect
27. C61×#394C22semidirect
217. #2591D61semidirect
246. #400C2×D61semidirect
247. #394C2×D61semidirect
254. C61#18324semidirect

Maximal subgroups

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

subgroupindex
C2×#168403432
#214566162
#214448552
#214447532
#214443342
#214441492
C61×#25912
#1832461