Finite Group Databasebeta

← #29652942

Subgroups — #29652942

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic
2C2×#16839918C212yesmaximal, complemented
3#21444792C214yesmaximal, complemented
4#21444775C214yesmaximal, complemented
5#21444390C212yesmaximal, complemented
6#21444137C211yesmaximal, characteristic
7C61×#2409C211yesmaximal, characteristic
8C22×#11758202—22no—
9#16840471—24no—
10#16840469—22no—
11D4×Dic122—22no—
12#16840393C2212yes—
13#16840388—21no—
14#16840388C2212yes—
15#16840319—21no—
16C2×#14519756—21no—
17C2×#14519756C2212yes—
18#16840237—21no—
19#16839918—24no—
20#16839918C2218yes—
21C2×#14519701—22no—
22C2×C22⋊#92418C2212yes—
23#16839498C2218yes—
24#16839488C2212yes—
25#16839486C2212yes—
26C23×Q8×C61C2211yescharacteristic
27C2×C22⋊Q8×C61—21no—
28C2×C4×D4×C61C2211yescharacteristic
29C61×#395C2212yes—
30C61×#393C2212yes—
31C61×#391—21no—
32C61×#386—21no—
33C61×#385—21no—
34C61×#380C2211yescharacteristic
35C22×Dic122—21no—
36C22×Dic122D412yes—
37C2×#11758202—28no—
38C2×#11758202—48no—
39C22×#92418—22no—
40#14519777—21no—
41#14519777—21no—
42#14519776—24no—
43#14519769—22no—
44#14519768—24no—
45#14519756—22no—
46#14519756—22no—
47#14519756—22no—
48#14519756—22no—
49#14519756—22no—
50#14519756—22no—
51#14519756D414yes—
52#14519756C2314yes—
53#14519754—24no—
54#14519754—24no—
55#14519754—24no—
56#14519754—24no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59D4×Dic61—24no—
60D4×Dic61—24no—
61#14519736—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61D412yes—
64C2×#11758196—22no—
65C2×#11758196—22no—
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61—22no—
68#14519701—28no—
69C22⋊#92418—22no—
70C22⋊#92418C2314yes—
71C22×Q8×C61—21no—
72C22×Q8×C61—22no—
73C22×Q8×C61—22no—
74C22×Q8×C61—44no—
75C22×Q8×C61C2312yes—
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—21no—
78C23×C4×C61D411yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C22⋊Q8×C61—44no—
81C4×D4×C61—22no—
82C4×D4×C61—22no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C22⋊C4×C61—21no—
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61D411yescharacteristic
89C2×C22⋊C4×C61C2311yescharacteristic
90C2×C42×C61D411yescharacteristic
91#95×C61—22no—
92#95×C61—22no—
93#95×C61C2312yes—
94C24×C61C2×D411yescharacteristic
95C24×C61C2×D411yescharacteristic
96C2×Q8×C61—24no—
97C2×Q8×C61—28no—
98C2×Q8×C61—24no—
99C2×Q8×C61—44no—
100C2×Q8×C61—44no—
101C2×Q8×C61—42no—
102C2×Q8×C61—48no—
103C2×Q8×C61—41no—
104C2×Q8×C61—41no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61Q1614yescomplemented
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—42no—
113C22×C4×C61—44no—
114C22×C4×C61—42no—
115C22×C4×C61—42no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—44no—
131C2×Dic122—82no—
132C2×Dic122C2×D414yes—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—42no—
135C4⋊C4×C61—44no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—44no—
139C22⋊C4×C61—42no—
140C42×C61—21no—
141C42×C61—21no—
142C22⋊Dic61—24no—
143C22⋊Dic61—24no—
144C22⋊Dic61—24no—
145#11758202—416no—
146C4⋊Dic61—22no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61C2×D414yes—
152#11758196—24no—
153#11758196—24no—
154#11758196—24no—
155#11758196—24no—
156C4×Dic61—24no—
157C2×#92418—24no—
158C2×#92418—44no—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61#12012yes—
166C23×C61C2×Q1612yes—
167C23×C61C2×Q1612yescomplemented
168C23×C61#13712yes—
169C23×C61C22×D411yescharacteristic
170Q8×C61—28no—
171Q8×C61—44no—
172Q8×C61—44no—
173Q8×C61—44no—
174D4×C61—28no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—24no—
178C2×C4×C61—22no—
179C2×C4×C61—24no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—48no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—81no—
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61C2×Q1612yescomplemented
197C2×C4×C61C22×D412yes—
198C2×Dic61—24no—
199C2×Dic61—24no—
200C2×Dic61—82no—
201C2×Dic61—82no—
202C2×Dic61—82no—
203C2×Dic61—82no—
204Dic122—44no—
205Dic122—44no—
206Dic122—82no—
207Dic122—82no—
208#92418—44no—
209#18197—611nomaximal, Sylow
210C22×C61—22no—
211C22×C61—21no—
212C22×C61—22no—
213C22×C61—21no—
214C22×C61—22no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—41no—
219C22×C61—41no—
220C22×C61—81no—
221C22×C61#45114yes—
222C22×C61C2×#12011yescharacteristic
223C22×C61C22×Q1611yescharacteristic
224C22×C61C2×#13711yescharacteristic
225C4×C61—22no—
226C4×C61—22no—
227C4×C61—21no—
228C4×C61—41no—
229C4×C61—41no—
230C4×C61—44no—
231C4×C61—82no—
232C4×C61B(4,𝔽2)12yes—
233Dic61—44no—
234Dic61—44no—
235#3501—611no—
236C2×#451—612no—
237#2409D6111yescharacteristic
238#1605—614no—
239#1602—614no—
240#1520—612no—
241#1459—611no—
242C2×C61—21no—
243C2×C61—21no—
244C2×C61—24no—
245C2×C61—42no—
246C2×C61C2×#45111yescharacteristic
247C2×C61#302211yescharacteristic
248C2×C61C2×B(4,𝔽2)11yescharacteristic
249C23×Q8C2×D6111yescharacteristic
250C22×Q16—1222no—
251#554—1222no—
252#554—1224no—
253D4×Q8—1222no—
254#544—612no—
255#543—612no—
256#543—1221no—
257#533—1221no—
258C2×C22⋊Q8—21no—
259C2×C22⋊Q8—612no—
260C2×C22⋊Q8—1221no—
261#518—1221no—
262C2×C4×D4C2×D6111yescharacteristic
263#451—618no—
264#451—1224no—
265C2×#103—1222no—
266C2×C22⋊C8—612no—
267#395C2×D6112yes—
268#393C2×D6112yes—
269#391—21no—
270#386—21no—
271#385—21no—
272#380C2×D6111yescharacteristic
273#328—618no—
274#324—612no—
275#323—612no—
276C61#1819711yescharacteristic, complemented, Sylow
277C22×Q8—21no—
278C22×Q8—22no—
279C22×Q8—22no—
280C22×Q8—44no—
281C22×Q8—612no—
282C22×Q8—1221no—
283C22×Q8C22×D6112yes—
284C22×D4C22⋊D6111yescharacteristic
285C23×C4—21no—
286C23×C4C22⋊D6111yescharacteristic
287C23×C4C22×D6111yescharacteristic
288C2×Q16—1228no—
289C2×Q16—2448no—
290C22×C8—1222no—
291C4⋊Q8—1221no—
292C4⋊Q8—1221no—
293C4⋊Q8—1224no—
294#125—1222no—
295#125—1224no—
296C22⋊Q8—44no—
297C22⋊Q8—614no—
298C22⋊Q8—614no—
299C22⋊Q8—1224no—
300C22⋊Q8—1224no—
301C22⋊Q8—1224no—
302C22⋊Q8—1224no—
303C22⋊Q8—1222no—
304C22⋊Q8—1222no—
305C22⋊Q8—1222no—
306C22⋊Q8—1222no—
307C22⋊Q8—1222no—
308C22⋊Q8—1222no—
309C4×Q8—1222no—
310C4×Q8—1222no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—1224no—
314C4×D4—1224no—
315#117—1222no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—1222no—
326C2×C22⋊C4—1222no—
327C2×C22⋊C4C22⋊D6111yescharacteristic
328C2×C22⋊C4C22×D6111yescharacteristic
329C2×C42C22⋊D6111yescharacteristic
330#103—1228no—
331C22⋊C8—614no—
332C22⋊C8—1222no—
333#95—22no—
334#95—22no—
335#95C22×D6112yes—
336C24C2×C22⋊D6111yescharacteristic
337C24C2×C22⋊D6111yescharacteristic
338C2×Q8—24no—
339C2×Q8—28no—
340C2×Q8—24no—
341C2×Q8—44no—
342C2×Q8—42no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—614no—
348C2×Q8—1222no—
349C2×Q8—1222no—
350C2×Q8—1222no—
351C2×Q8—4882no—
352C2×Q8—2444no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4#1175820214yescomplemented
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—1222no—
365C22×C4—1222no—
366C22×C4—1222no—
367C22×C4—1222no—
368C22×C4D4×D6112yes—
369C22×C4D4×D6112yes—
370C22×C4C2×C22⋊D6111yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372C22×C4C2×C22⋊D6111yescharacteristic
373C22×C4C2×C22⋊D6111yescharacteristic
374C22×C4C23×D6111yescharacteristic
375Q16—24416no—
376C2×C8—1224no—
377C2×C8—2444no—
378C4⋊C4—22no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—614no—
382C4⋊C4—1224no—
383C4⋊C4—1222no—
384C4⋊C4—1224no—
385C4⋊C4—1222no—
386C4⋊C4—1222no—
387C4⋊C4—1222no—
388C4⋊C4—1224no—
389C4⋊C4—1222no—
390C4⋊C4—1224no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—44no—
394C22⋊C4—42no—
395C22⋊C4—1224no—
396C22⋊C4—1224no—
397C22⋊C4—1224no—
398C42—21no—
399C42—21no—
400C42—1224no—
401C23—21no—
402C23—21no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23#1451974512yes—
408C23C2×#1175820212yes—
409C23C2×#1175820212yescomplemented
410C23#1451980712yes—
411C23C22×C22⋊D6111yescharacteristic
412Q8—28no—
413Q8—44no—
414Q8—44no—
415Q8—44no—
416Q8—4882no—
417Q8—4882no—
418Q8—2444no—
419Q8—2444no—
420D4—28no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—48no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—1224no—
441C2×C4—1224no—
442C2×C4—4882no—
443C2×C4—4882no—
444C2×C4—4882no—
445C2×C4—4882no—
446C2×C4#1451974511yescharacteristic
447C2×C4#1451974511yescharacteristic
448C2×C4C2×#1175820212yescomplemented
449C2×C4C2×D4×D6112yes—
450C8—2444no—
451C22—21no—
452C22—22no—
453C22—21no—
454C22—22no—
455C22—22no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—41no—
460C22—41no—
461C22—81no—
462C22#1683991714yes—
463C22C2×#1451974511yescharacteristic
464C22C22×#1175820211yescharacteristic
465C22C2×#1451980711yescharacteristic
466C4—22no—
467C4—21no—
468C4—22no—
469C4—41no—
470C4—44no—
471C4—41no—
472C4—82no—
473C4—2444no—
474C4—2444no—
475C4#1683994812yes—
476C2—21no—
477C2—21no—
478C2—24no—
479C2—42no—
480C2C2×#1683991711yescharacteristic
481C2#2145519711yescharacteristic
482C2C2×#1683994811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2409C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#2409D61
hypocenterC61#18197
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. C2×#16839918C2semidirect
3. #21444792C2semidirect
4. #21444775C2semidirect
5. #21444390C2semidirect
107. C2×D4×C61Q16semidirect
167. C23×C61C2×Q16semidirect
196. C2×C4×C61C2×Q16semidirect
276. C61#18197semidirect
355. C2×D4#11758202semidirect
409. C23C2×#11758202semidirect
448. C2×C4C2×#11758202semidirect

Maximal subgroups

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

subgroupindex
#214596222
C2×#168399182
#214447922
#214447752
#214443902
#214441372
C61×#24092
#1819761