Finite Group Databasebeta

← #29736161

Subgroups — #29736161

Order 15616.

1262 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic
2#21449459C212yesmaximal, complemented
3#21447172C214yesmaximal, complemented
4#21446803C212yesmaximal, complemented
5#21446099C214yesmaximal
6C4×#14519692C211yesmaximal, characteristic
7C61×#2409C211yesmaximal, characteristic
8#16840471—24no—
9#16840469—22no—
10D4×Dic122—22no—
11#16840393—21no—
12#16840388—21no—
13#16840388C2212yes—
14#16840319—21no—
15C2×#14519775—22no—
16C2×#14519756—21no—
17C2×#14519756C2212yes—
18#16840237C2212yes—
19C2×#14519692C2212yes—
20#16839681—24no—
21#16839669—22no—
22#16839635C2218yes—
23C4×C22⋊Dic61C2212yes—
24#16839529C2212yes—
25#16839528C2218yes—
26C23×Q8×C61C2211yescharacteristic
27C2×C22⋊Q8×C61—21no—
28C2×C4×D4×C61C2211yescharacteristic
29C61×#395—21no—
30C61×#393—21no—
31C61×#391—21no—
32C61×#386C2212yes—
33C61×#385C2212yes—
34C61×#380C2211yescharacteristic
35C22×Dic122—21no—
36C22×Dic122D412yes—
37#14519777—21no—
38#14519777—21no—
39#14519776—24no—
40#14519775—48no—
41#14519769—22no—
42#14519768—24no—
43#14519756—22no—
44#14519756—22no—
45#14519756—22no—
46#14519756—22no—
47#14519756—22no—
48#14519756—22no—
49#14519756—22no—
50#14519756—22no—
51#14519754—24no—
52#14519754—24no—
53#14519754—24no—
54#14519754—24no—
55C4×Dic122—22no—
56C4×Dic122—22no—
57D4×Dic61—24no—
58D4×Dic61—24no—
59#14519736—22no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61D412yes—
62C2×#11758196—22no—
63C2×#11758196—28no—
64C2×#11758196D414yes—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61C2314yes—
67C2×C22⋊Dic61C2314yes—
68C2×C4×Dic61—22no—
69#14519692Q818yescomplemented
70#14519684—22no—
71#14519684—28no—
72C22×Q8×C61—22no—
73C22×Q8×C61—21no—
74C22×Q8×C61—22no—
75C22×Q8×C61—21no—
76C22×Q8×C61—44no—
77C22×D4×C61C2311yescharacteristic
78C23×C4×C61D411yescharacteristic
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C22⋊Q8×C61—44no—
82C4×D4×C61—22no—
83C4×D4×C61—22no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61D411yescharacteristic
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61D411yescharacteristic
89C2×C22⋊C4×C61C2312yes—
90C2×C22⋊C4×C61C2311yescharacteristic
91C2×C42×C61D411yescharacteristic
92#95×C61—21no—
93#95×C61—22no—
94#95×C61—22no—
95C24×C61C2×D411yescharacteristic
96C24×C61C2411yescharacteristic
97C2×Q8×C61—24no—
98C2×Q8×C61—28no—
99C2×Q8×C61—44no—
100C2×Q8×C61—42no—
101C2×Q8×C61—48no—
102C2×Q8×C61—44no—
103C2×Q8×C61—41no—
104C2×Q8×C61—42no—
105C2×Q8×C61—41no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61C2×Q814yes—
109C22×C4×C61—22no—
110C22×C4×C61—24no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—42no—
115C22×C4×C61—44no—
116C22×C4×C61—42no—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×Dic61—24no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61C2×D414yes—
128C22×Dic61C2×D414yes—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—44no—
134C2×Dic122—82no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—42no—
137C4⋊C4×C61—44no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—24no—
141C22⋊C4×C61—44no—
142C42×C61—21no—
143C42×C61—21no—
144C22⋊Dic61—24no—
145C22⋊Dic61—24no—
146C22⋊Dic61—24no—
147C22⋊Dic61—24no—
148C22⋊Dic61C2×Q818yescomplemented
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155#11758196—24no—
156#11758196—24no—
157#11758196—24no—
158#11758196—24no—
159#11758196—416no—
160C4×Dic61—24no—
161C4×Dic61—44no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61—41no—
168C23×C61C4⋊Q812yescomplemented
169C23×C61C22×D411yescharacteristic
170C23×C61C22×D412yes—
171C23×C61C22×Q812yes—
172C23×C612−1+412yes—
173Q8×C61—48no—
174Q8×C61—44no—
175Q8×C61—44no—
176D4×C61—28no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—24no—
180C2×C4×C61—22no—
181C2×C4×C61—28no—
182C2×C4×C61—24no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—44no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—81no—
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12012yes—
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61C4⋊Q812yescomplemented
201C2×Dic61—24no—
202C2×Dic61—24no—
203C2×Dic61—28no—
204C2×Dic61—44no—
205C2×Dic61—44no—
206C2×Dic61—44no—
207C2×Dic61—44no—
208C2×Dic61—82no—
209C2×Dic61—82no—
210Dic122—44no—
211Dic122—82no—
212Dic122—82no—
213Dic122—82no—
214#25852—611nomaximal, Sylow
215C22×C61—22no—
216C22×C61—21no—
217C22×C61—22no—
218C22×C61—21no—
219C22×C61—22no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61—81no—
226C22×C61C2×#12011yescharacteristic
227C22×C61C2×C4⋊Q811yescharacteristic
228C22×C61#53611yescharacteristic
229C22×C61D4×Q814yes—
230C4×C61—22no—
231C4×C61—22no—
232C4×C61—21no—
233C4×C61—41no—
234C4×C61—44no—
235C4×C61—41no—
236C4×C61—82no—
237C4×C61B(4,𝔽2)12yes—
238Dic61—48no—
239Dic61—44no—
240Dic61—44no—
241#3501—611no—
242#2434—612no—
243#2409D6111yescharacteristic
244#2062—614no—
245#1996—612no—
246#1867—614no—
247C4×#99—611no—
248C2×C61—24no—
249C2×C61—21no—
250C2×C61—21no—
251C2×C61—42no—
252C2×C61#243411yescharacteristic
253C2×C61C2×B(4,𝔽2)11yescharacteristic
254C2×C61#302811yescharacteristic
255C23×Q8C2×D6111yescharacteristic
256#554—1222no—
257#554—1224no—
258D4×Q8—1222no—
259#544—1221no—
260#543—612no—
261#543—1221no—
262#533—1221no—
263C2×C4⋊Q8—1222no—
264C2×C22⋊Q8—21no—
265C2×C22⋊Q8—612no—
266C2×C22⋊Q8—1221no—
267#518—612no—
268C2×C4×D4C2×D6111yescharacteristic
269C2×#99—612no—
270#395—21no—
271#395—1224no—
272#393—21no—
273#391—21no—
274#391—1222no—
275#386C2×D6112yes—
276#385—618no—
277#385C2×D6112yes—
278#380C2×D6111yescharacteristic
279C4×C22⋊C4—612no—
280#342—612no—
281#342—618no—
282C61#2585211yescharacteristic, complemented, Sylow
283C22×Q8—22no—
284C22×Q8—22no—
285C22×Q8—21no—
286C22×Q8—21no—
287C22×Q8—44no—
288C22×Q8—612no—
289C22×Q8—1221no—
290C22×D4C22×D6111yescharacteristic
291C23×C4C22⋊D6111yescharacteristic
292C23×C4C22×D6112yes—
293C23×C4C22×D6111yescharacteristic
294C4⋊Q8—1221no—
295C4⋊Q8—1221no—
296C4⋊Q8—1224no—
297C4⋊Q8—2448no—
298#125—1222no—
299#125—1224no—
300C22⋊Q8—44no—
301C22⋊Q8—1224no—
302C22⋊Q8—1224no—
303C22⋊Q8—1224no—
304C22⋊Q8—1224no—
305C22⋊Q8—1222no—
306C22⋊Q8—1222no—
307C22⋊Q8—1222no—
308C22⋊Q8—1222no—
309C22⋊Q8—1222no—
310C22⋊Q8—1222no—
311C22⋊Q8—1222no—
312C22⋊Q8—1222no—
313C4×Q8—1222no—
314C4×Q8—1222no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—1224no—
318C4×D4—1224no—
319#117—1222no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—612no—
323C2×C4⋊C4—614no—
324C2×C4⋊C4—1222no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—1228no—
327C2×C4⋊C4C22⋊D6111yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—614no—
330C2×C22⋊C4—614no—
331C2×C22⋊C4—1222no—
332C2×C22⋊C4C22⋊D6111yescharacteristic
333C2×C22⋊C4C22×D6112yes—
334C2×C22⋊C4C22×D6111yescharacteristic
335C2×C42—1222no—
336C2×C42C22⋊D6111yescharacteristic
337#99—618no—
338#95—21no—
339#95—22no—
340#95—22no—
341#95—1228no—
342#95—1222no—
343C24C2×C22⋊D6111yescharacteristic
344C24C23×D6111yescharacteristic
345C2×Q8—24no—
346C2×Q8—28no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—41no—
352C2×Q8—42no—
353C2×Q8—41no—
354C2×Q8—1222no—
355C2×Q8—1222no—
356C2×Q8—1222no—
357C2×Q8—1222no—
358C2×Q8—4882no—
359C2×Q8—2444no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4Q8×D6114yes—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—614no—
372C22×C4—614no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1224no—
376C22×C4D4×D6112yes—
377C22×C4D4×D6112yes—
378C22×C4C2×C22⋊D6111yescharacteristic
379C22×C4C2×C22⋊D6111yescharacteristic
380C22×C4C2×C22⋊D6111yescharacteristic
381C22×C4C2×C22⋊D6111yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—1224no—
387C4⋊C4—1222no—
388C4⋊C4—1224no—
389C4⋊C4—1222no—
390C4⋊C4—1222no—
391C4⋊C4—1222no—
392C4⋊C4—1224no—
393C4⋊C4—1222no—
394C4⋊C4—1224no—
395C4⋊C4—1222no—
396C4⋊C4—24416no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—24no—
400C22⋊C4—44no—
401C22⋊C4—618no—
402C22⋊C4—1224no—
403C22⋊C4—1224no—
404C22⋊C4—1224no—
405C22⋊C4—1224no—
406C42—21no—
407C42—21no—
408C42—1224no—
409C42—2444no—
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23#1451977512yescomplemented
417C23C2×D4×D6112yes—
418C23C22×C22⋊D6111yescharacteristic
419C23C2×Q8×D6112yes—
420C23#1451982412yes—
421Q8—44no—
422Q8—48no—
423Q8—44no—
424Q8—4882no—
425Q8—4882no—
426Q8—4882no—
427Q8—2444no—
428D4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—1224no—
450C2×C4—1224no—
451C2×C4—1228no—
452C2×C4—4882no—
453C2×C4—4882no—
454C2×C4—2444no—
455C2×C4—2444no—
456C2×C4—2444no—
457C2×C4—2444no—
458C2×C4#1451974511yescharacteristic
459C2×C4#1451974512yes—
460C2×C4#1451974511yescharacteristic
461C2×C4#1451977512yescomplemented
462C22—21no—
463C22—22no—
464C22—21no—
465C22—22no—
466C22—22no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—81no—
473C22C2×#1451974511yescharacteristic
474C22C2×#1451977511yescharacteristic
475C22#1684033311yescharacteristic
476C22#1684042514yes—
477C4—21no—
478C4—22no—
479C4—22no—
480C4—44no—
481C4—41no—
482C4—41no—
483C4—82no—
484C4—2448no—
485C4—2444no—
486C4—2444no—
487C4#1683994812yes—
488C2—21no—
489C2—24no—
490C2—21no—
491C2—42no—
492C2#2144946411yescharacteristic
493C2C2×#1683994811yescharacteristic
494C2#2145522411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#2409C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#2409D61
hypocenterC61#25852
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. #21449459C2semidirect
3. #21447172C2semidirect
4. #21446803C2semidirect
69. #14519692Q8semidirect
148. C22⋊Dic61C2×Q8semidirect
168. C23×C61C4⋊Q8semidirect
200. C2×C4×C61C4⋊Q8semidirect
282. C61#25852semidirect
416. C23#14519775semidirect
461. C2×C4#14519775semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214494592
#214471722
#214468032
#214460992
C4×#145196922
C61×#24092
#2585261