Finite Group Databasebeta

← #29654193

Subgroups — #29654193

Order 15616.

1230 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1230 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
2#21456621C212yesmaximal, complemented
3#21444821C214yesmaximal, complemented
4#21444787C214yesmaximal, complemented
5#21444334C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C61×#2583C211yesmaximal, 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#16840095—22no—
17#16840075C2214yescomplemented
18#16840010—24no—
19#16839912—22no—
20#16839901C2214yescomplemented
21#16839726C2212yes—
22#16839495C2218yescomplemented
23#16839488C2214yescomplemented
24C2×C22⋊Q8×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#398C2212yescomplemented
27C61×#395C2212yescomplemented
28C61×#389—22no—
29C61×#385—21no—
30C61×#385—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×#11758201—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#14519710—24no—
58#14519701—24no—
59#14519697—24no—
60C22⋊#92418—22no—
61C22⋊#92418C2314yes—
62C22×Q8×C61C2311yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61D411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—22no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61D412yes—
74C2×C4⋊C4×C61C2311yescharacteristic
75C2×C4⋊C4×C61C2312yes—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—21no—
78C2×C42×C61—21no—
79#95×C61—22no—
80#95×C61—22no—
81#95×C61C2312yes—
82C24×C61C2×D411yescharacteristic
83C23×D61—22no—
84C2×Q8×C61—22no—
85C2×Q8×C61—81no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—24no—
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—81no—
95C22×C4×C61—81no—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2411yescharacteristic
102C2×C22⋊D61—24no—
103C2×C22⋊D61—24no—
104C2×C22⋊D61—48no—
105C22×Dic61—22no—
106#11758222—416no—
107#11758222—48no—
108#11758222—48no—
109C2×D244—24no—
110C2×D244—24no—
111C2×D244—22no—
112C2×D244—44no—
113C2×D244—82no—
114C2×D244—82no—
115C2×D244C2×D414yes—
116C2×C4×D61—28no—
117C2×C4×D61—48no—
118C2×Dic122—24no—
119C2×Dic122—44no—
120C4⋊C4×C61—24no—
121C4⋊C4×C61—24no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—42no—
124C4⋊C4×C61—44no—
125C4⋊C4×C61—82no—
126C22⋊C4×C61—24no—
127C22⋊C4×C61—41no—
128C22⋊C4×C61—41no—
129C42×C61—81no—
130#11758201—84no—
131#11758198—28no—
132#11758198—24no—
133#11758198—24no—
134C4⋊Dic61—24no—
135C4⋊Dic61—22no—
136C4⋊Dic61C2×D414yes—
137#11758194—82no—
138C2×#92418—24no—
139C23×C61—22no—
140C23×C61—22no—
141C23×C61—41no—
142C23×C61#12011yescharacteristic
143C23×C61#12011yescharacteristic
144C23×C61C22×D411yescharacteristic
145C22×D61—24no—
146C22×D61—44no—
147C22×D61—44no—
148C22×D61—44no—
149Q8×C61—44no—
150C2×C4×C61—24no—
151C2×C4×C61—24no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—24no—
155C2×C4×C61—22no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—42no—
168C2×C4×C61—81no—
169C2×C4×C61#12012yes—
170C2×C4×C61AGL(1,ℤ/8)14yes—
171C2×C4×C61C22×D412yes—
172C22⋊D61—48no—
173C22⋊D61—48no—
174C2×Dic61—24no—
175C2×Dic61—44no—
176D244—44no—
177D244—48no—
178D244—82no—
179D244—84no—
180C4×D61—416no—
181Dic122—48no—
182#92418—44no—
183#18325—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—44no—
192C22×C61#45612yes—
193C22×C61C2×#12011yescharacteristic
194C22×C61C2×AGL(1,ℤ/8)12yes—
195C2×D61—44no—
196C2×D61—44no—
197C2×D61—44no—
198C2×D61—88no—
199C4×C61—24no—
200C4×C61—22no—
201C4×C61—22no—
202C4×C61—44no—
203C4×C61—42no—
204C4×C61—81no—
205C4×C61—81no—
206Dic61—44no—
207C2×#537—611no—
208#3193—612no—
209#2583D6111yescharacteristic, complemented
210#1610—614no—
211#1604—614no—
212#1507—612no—
213#1463—611no—
214C2×C61—22no—
215C2×C61—22no—
216C2×C61—41no—
217C2×C61#300311yescharacteristic
218C2×C61C2×#45611yescharacteristic
219C2×C61#302611yescharacteristic
220D61—84no—
221C22×#41—1221no—
222#537—614no—
223#537—1224no—
224#537—1222no—
225C2×C22⋊Q8—1222no—
226C2×C22⋊Q8C2×D6111yescharacteristic
227C2×#121—612no—
228C2×#121—1222no—
229C2×#121—1221no—
230C22×C4⋊C4C2×D6111yescharacteristic
231#484—1222no—
232#480—614no—
233#468—1224no—
234#450—1222no—
235#448—614no—
236#407—612no—
237#398C2×D6112yescomplemented
238#395C2×D6112yescomplemented
239#389—22no—
240#385—21no—
241#385—21no—
242C4×C22⋊C4C2×D6111yescharacteristic
243#327—618no—
244#324—614no—
245C61#1832511yescharacteristic, complemented, Sylow
246C2×#41—1224no—
247C2×#41—1224no—
248C2×#41—2448no—
249C22×Q8—1221no—
250C22×Q8C22×D6111yescharacteristic
251C22×D4—612no—
252C22×D4—1221no—
253C22×D4—1221no—
254C22×D4—1222no—
255C23×C4—1222no—
256C23×C4C22⋊D6111yescharacteristic
257C23×C4C22⋊D6111yescharacteristic
258C23×C4C22×D6111yescharacteristic
259C2×SD16—1224no—
260C2×M16—1222no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—1228no—
264C22⋊Q8—1228no—
265#121—614no—
266#121—614no—
267#121—1228no—
268#121—1228no—
269#121—1224no—
270#121—1222no—
271#121—1222no—
272#121—1224no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—612no—
279C2×C4⋊C4—1221no—
280C2×C4⋊C4C22⋊D6112yes—
281C2×C4⋊C4C22×D6111yescharacteristic
282C2×C4⋊C4C22×D6112yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—1222no—
286C2×C22⋊C4—1222no—
287C2×C42—21no—
288#107—1224no—
289#103—1224no—
290#102—1224no—
291C22⋊C8—614no—
292C22⋊C8—1222no—
293#95—22no—
294#95—22no—
295#95C22×D6112yes—
296C24—1222no—
297C24C2×C22⋊D6111yescharacteristic
298#41—24416no—
299#41—2448no—
300#41—2448no—
301C2×Q8—22no—
302C2×Q8—81no—
303C2×Q8—1224no—
304C2×Q8—2444no—
305C2×D4—614no—
306C2×D4—1224no—
307C2×D4—1224no—
308C2×D4—1224no—
309C2×D4—1224no—
310C2×D4—1222no—
311C2×D4—4882no—
312C2×D4—4882no—
313C2×D4—2444no—
314C2×D4—2448no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—1228no—
326C22×C4—1222no—
327C22×C4—2448no—
328C22×C4D4×D6112yes—
329C22×C4D4×D6112yes—
330C22×C4C2×C22⋊D6112yes—
331C22×C4C2×C22⋊D6112yes—
332C22×C4C2×C22⋊D6111yescharacteristic
333C22×C4C23×D6111yescharacteristic
334SD16—4884no—
335M16—4882no—
336C2×C8—1224no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—82no—
343C4⋊C4—614no—
344C4⋊C4—1224no—
345C4⋊C4—1222no—
346C22⋊C4—24no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—1228no—
350C22⋊C4—1224no—
351C22⋊C4—1224no—
352C42—81no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—1224no—
357C23—2444no—
358C23—2444no—
359C23—2444no—
360C23#1451974511yescharacteristic
361C23#1451974511yescharacteristic
362C23C22×C22⋊D6111yescharacteristic
363Q8—44no—
364Q8—2448no—
365D4—4882no—
366D4—4884no—
367D4—2444no—
368D4—2448no—
369D4—2448no—
370D4—2448no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—1224no—
391C2×C4—24416no—
392C2×C4—2444no—
393C2×C4#1451974512yes—
394C2×C4#1451980214yes—
395C2×C4C2×D4×D6112yes—
396C8—2444no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—4888no—
406C22—2444no—
407C22—2444no—
408C22—2444no—
409C22#1683994412yes—
410C22C2×#1451974511yescharacteristic
411C22C2×#1451980212yes—
412C4—22no—
413C4—22no—
414C4—24no—
415C4—44no—
416C4—42no—
417C4—81no—
418C4—81no—
419C4—2444no—
420C2—22no—
421C2—22no—
422C2—41no—
423C2—4884no—
424C2#2145504511yescharacteristic
425C2C2×#1683994411yescharacteristic
426C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2583C2
socleC22×C61C2×#120
prime radicalC2×#16840343C2
hypercenter#2583D61
hypocenterC61#18325
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
2. #21456621C2semidirect
3. #21444821C2semidirect
4. #21444787C2semidirect
5. #21444334C2semidirect
6. #21444149C2semidirect
7. C61×#2583C2semidirect
17. #16840075C22semidirect
20. #16839901C22semidirect
22. #16839495C22semidirect
23. #16839488C22semidirect
26. C61×#398C22semidirect
27. C61×#395C22semidirect
209. #2583D61semidirect
237. #398C2×D61semidirect
238. #395C2×D61semidirect
245. C61#18325semidirect

Maximal subgroups

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

subgroupindex
C2×#168403432
#214566212
#214448212
#214447872
#214443342
#214441492
C61×#25832
#1832561