Finite Group Databasebeta

← #29725158

Subgroups — #29725158

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450737C211yesmaximal, characteristic
2Q8×C4⋊Dic61C211yesmaximal, characteristic, complemented
3#21447378C212yesmaximal, complemented
4#21447224C212yesmaximal
5#21446578C211yesmaximal, characteristic, complemented
6#21446440C212yesmaximal
7#21446391C212yesmaximal, complemented
8#21445947C211yesmaximal, characteristic, complemented
9#21445820C211yesmaximal, characteristic
10C4×#14519699C211yesmaximal, characteristic
11C61×#1984C211yesmaximal, characteristic, complemented
12C2×#14519753—21no—
13C2×Q8×Dic61—21no—
14C2×C4×D244C2211yescharacteristic
15C2×#14519711C2212yes—
16C2×#14519699C2211yescharacteristic
17C2×#14519699C2211yescharacteristic
18C2×#14519699C2212yes—
19#16839684—22no—
20#16839664C2212yes—
21#16839661—22no—
22#16839661—21no—
23#16839660C2212yes—
24#16839632C2212yes—
25#16839632C2211yescharacteristic
26#16839632C2211yescharacteristic
27#16839619—21no—
28C4×C4⋊Dic61C2211yescharacteristic
29C4×C4⋊Dic61C2212yes—
30#16839525C2211yescharacteristic
31#16839525C2211yescharacteristic
32#16839525C2212yes—
33#16839525C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36C2×C4×Q8×C61C2211yescharacteristic
37C2×C4⋊C8×C61C2211yescharacteristic
38C2×C4×C8×C61C2211yescharacteristic
39C61×#336C2212yes—
40C61×#336C2212yes—
41C22×D244C2311yescharacteristic
42C22×C4×D61—21no—
43#14519753—44no—
44Q8×Dic61—44no—
45C4×D244—22no—
46C4×D244—22no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2312yes—
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×#11758196—22no—
59C2×#11758198—21no—
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×C4×Dic61—22no—
63C2×C4×Dic61—21no—
64#14519711—24no—
65#14519699—22no—
66#14519699—24no—
67#14519699—22no—
68#14519684—22no—
69#14519684—22no—
70#14519684—21no—
71C22×Q8×C61D411yescharacteristic
72C22×C8×C61C2311yescharacteristic
73C22×C8×C61C2311yescharacteristic
74C22×C8×C61C2312yes—
75C4×Q8×C61—22no—
76C4×Q8×C61—22no—
77C2×C4⋊C4×C61D412yes—
78C2×C4⋊C4×C61D411yescharacteristic
79C2×C42×C61C2312yes—
80C2×C42×C61C2311yescharacteristic
81C4⋊C8×C61—22no—
82C4×C8×C61—22no—
83C23×D61—21no—
84C2×Q8×C61—22no—
85C2×Q8×C61—22no—
86C2×Q8×C61D814yes—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2411yescharacteristic
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—22no—
95C22×Dic61—21no—
96C22×Dic61—22no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244#4112yes—
103C2×C4×D61—42no—
104C2×C4×D61—42no—
105C2×C4×D61—42no—
106C2×C8×C61—21no—
107C2×C8×C61—21no—
108C2×C8×C61—22no—
109C2×C8×C61—22no—
110C2×C8×C61—21no—
111C2×C8×C61—21no—
112C2×C8×C61—22no—
113C2×C8×C61—21no—
114C2×C8×C61—21no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—22no—
117C42×C61—22no—
118C42×C61—22no—
119C42×C61#4112yes—
120C42×C61#4112yes—
121#11758198—42no—
122#11758198—44no—
123#11758198—82no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—22no—
131C4⋊Dic61—21no—
132C4⋊Dic61—22no—
133C4⋊Dic61—44no—
134C4⋊Dic61—42no—
135C4⋊Dic61—42no—
136C4⋊Dic61—81no—
137C4⋊Dic61—82no—
138C4⋊Dic61—81no—
139C4⋊Dic61#4112yes—
140#11758196—44no—
141C4×Dic61—44no—
142C4×Dic61—42no—
143C23×C61C22×D411yescharacteristic
144C22×D61—22no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C22×D61—41no—
151Q8×C61—28no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61#12312yes—
160C2×C4×C61#12312yes—
161C2×C4×C61C2×D814yes—
162C2×C4×C61C2×D812yes—
163C2×C4×C61#13712yes—
164C2×C4×C61C2×#4111yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C2×C4×C61C2×#4111yescharacteristic
168C2×C4×C612−1+411yescharacteristic
169C2×C4×C612−1+411yescharacteristic
170C2×Dic61—22no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—42no—
175C2×Dic61—41no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—42no—
181C2×Dic61—41no—
182C2×Dic61—42no—
183C2×Dic61—41no—
184C2×Dic61—42no—
185D244—22no—
186D244—22no—
187D244—41no—
188D244—41no—
189C4×D61—84no—
190C8×C61—42no—
191C8×C61—41no—
192C8×C61—41no—
193#24990—611nomaximal, Sylow
194C22×C61C2×#12311yescharacteristic
195C22×C61#54811yescharacteristic
196C22×C61#54811yescharacteristic
197C22×C61C22×D811yescharacteristic
198C22×C61C2×#13711yescharacteristic
199C22×C61#57711yescharacteristic
200C22×C61#57711yescharacteristic
201C2×D61—42no—
202C2×D61—42no—
203C2×D61—42no—
204C2×D61—81no—
205C2×D61—81no—
206C2×D61—81no—
207C2×D61—81no—
208C4×C61—24no—
209C4×C61—22no—
210C4×C61—42no—
211C4×C61#48014yes—
212C4×C61#52911yescharacteristic
213C4×C61#54111yescharacteristic
214C4×C61#55211yescharacteristic
215C4×C61#55611yescharacteristic
216Dic61—42no—
217Dic61—81no—
218Dic61—82no—
219Dic61—82no—
220Dic61—81no—
221#2538—611no—
222Q8×C4⋊C4—611no—
223#2099—612no—
224#2071—612no—
225#1984D6111yescharacteristic, complemented
226#1947—611no—
227#1925—612no—
228#1918—612no—
229#1834—611no—
230#1802—611no—
231C4×#102—611no—
232C2×C61#242111yescharacteristic
233C2×C61C2×#48011yescharacteristic
234C2×C61#309911yescharacteristic
235C2×C61#335811yescharacteristic
236C2×C61#335811yescharacteristic
237C2×C61#336911yescharacteristic
238C2×C61#337211yescharacteristic
239D61—82no—
240C2×C22⋊Q8—1221no—
241C2×C4×Q8—1221no—
242C2×C4×Q8C2×D6111yescharacteristic
243C2×C4×D4—611no—
244C2×#107—612no—
245C2×C4⋊C8C2×D6111yescharacteristic
246C2×#102—612no—
247C2×#102—611no—
248C2×#102—611no—
249C2×C4×C8C2×D6111yescharacteristic
250#396—1222no—
251#390—612no—
252#389—612no—
253#389—1221no—
254#389—1222no—
255#384—611no—
256#384—612no—
257#384—611no—
258#380—1221no—
259C4×C4⋊C4—612no—
260C4×C4⋊C4—611no—
261#340—612no—
262#340—612no—
263#340—612no—
264#340—612no—
265#340—611no—
266#340—611no—
267#336C2×D6112yes—
268#336C2×D6112yes—
269C61#2499011yescharacteristic, complemented, Sylow
270C22×Q8D24411yescharacteristic
271C22×D4—611no—
272C23×C4—1221no—
273C22×C8C22×D6111yescharacteristic
274C22×C8C22×D6111yescharacteristic
275C22×C8C22×D6112yes—
276C22⋊Q8—2444no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—2444no—
280C4×D4—1222no—
281C4×D4—1222no—
282C2×C4⋊C4—611no—
283C2×C4⋊C4—611no—
284C2×C4⋊C4—612no—
285C2×C4⋊C4—611no—
286C2×C4⋊C4—612no—
287C2×C4⋊C4—1222no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4—1222no—
291C2×C4⋊C4—1221no—
292C2×C4⋊C4—1221no—
293C2×C4⋊C4—1222no—
294C2×C4⋊C4D24411yescharacteristic
295C2×C4⋊C4D24412yes—
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—1221no—
298C2×C22⋊C4—1222no—
299C2×C42—1222no—
300C2×C42—1221no—
301C2×C42C22×D6112yes—
302C2×C42C22×D6111yescharacteristic
303#107—1224no—
304C4⋊C8—22no—
305#102—1222no—
306#102—1224no—
307#102—1222no—
308C4×C8—22no—
309#95—1222no—
310#95—1221no—
311#95—1222no—
312C24—1221no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8D48814yes—
316C2×D4—612no—
317C2×D4—1221no—
318C2×D4—1221no—
319C2×D4—1221no—
320C2×D4—1221no—
321C2×D4—1221no—
322C22×C4—1221no—
323C22×C4—1221no—
324C22×C4—1222no—
325C22×C4—1221no—
326C22×C4—1222no—
327C22×C4—2442no—
328C22×C4—2442no—
329C22×C4—2442no—
330C22×C4C2×D24412yes—
331C22×C4C2×D24411yescharacteristic
332C22×C4C2×D24412yes—
333C22×C4C2×D24411yescharacteristic
334C22×C4C23×D6111yescharacteristic
335C2×C8—21no—
336C2×C8—22no—
337C2×C8—21no—
338C2×C8—22no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—612no—
347C4⋊C4—1221no—
348C4⋊C4—1222no—
349C4⋊C4—1221no—
350C4⋊C4—1222no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C4⋊C4—1222no—
354C4⋊C4—1221no—
355C4⋊C4—1222no—
356C4⋊C4—4881no—
357C4⋊C4—4881no—
358C4⋊C4—4882no—
359C4⋊C4—2444no—
360C4⋊C4—2444no—
361C4⋊C4—2442no—
362C4⋊C4—2442no—
363C22⋊C4—4882no—
364C22⋊C4—2442no—
365C22⋊C4—2444no—
366C42—22no—
367C42—22no—
368C42—2444no—
369C42—2442no—
370C42#1175822412yes—
371C42#1175822412yes—
372C23—1222no—
373C23—2441no—
374C23—2441no—
375C23—2441no—
376C23—2441no—
377C23—2441no—
378C23—2441no—
379C23C22×D24411yescharacteristic
380Q8—28no—
381D4—1222no—
382D4—1222no—
383D4—2441no—
384D4—2441no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—1222no—
393C2×C4—4884no—
394C2×C4—2441no—
395C2×C4—2442no—
396C2×C4—2441no—
397C2×C4—2442no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4—2442no—
401C2×C4—2442no—
402C2×C4—2441no—
403C2×C4—2442no—
404C2×C4—2442no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4#1451976112yes—
409C2×C4#1451976112yes—
410C2×C4C2×D48814yes—
411C2×C4C2×D48812yes—
412C2×C4#1451981012yes—
413C2×C4#41×D6111yescharacteristic
414C2×C4#41×D6111yescharacteristic
415C2×C4C2×#1175822411yescharacteristic
416C2×C4C2×#1175822411yescharacteristic
417C2×C4#1451982511yescharacteristic
418C2×C4#1451982511yescharacteristic
419C8—41no—
420C8—42no—
421C8—41no—
422C22—4881no—
423C22—4881no—
424C22—4881no—
425C22—4881no—
426C22—2442no—
427C22—2442no—
428C22—2442no—
429C22C2×#1451976111yescharacteristic
430C22#1684042411yescharacteristic
431C22#1684042411yescharacteristic
432C22C22×D48811yescharacteristic
433C22C2×#1451981011yescharacteristic
434C22#1684057411yescharacteristic
435C22#1684057411yescharacteristic
436C4—24no—
437C4—22no—
438C4—42no—
439C4—4882no—
440C4—4881no—
441C4—4882no—
442C4—4881no—
443C4—2442no—
444C4#1684007614yes—
445C4#1684029111yescharacteristic
446C4#1684037311yescharacteristic
447C4#1684044611yescharacteristic
448C4#1684048311yescharacteristic
449C2—4882no—
450C2#2144932011yescharacteristic
451C2C2×#1684007611yescharacteristic
452C2#2145581011yescharacteristic
453C2#2145845511yescharacteristic
454C2#2145845511yescharacteristic
455C2#2145856111yescharacteristic
456C2#2145860211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1984C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1984D61
hypocenterC61#24990
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic61C2semidirect
3. #21447378C2semidirect
5. #21446578C2semidirect
7. #21446391C2semidirect
8. #21445947C2semidirect
11. C61×#1984C2semidirect
225. #1984D61semidirect
269. C61#24990semidirect

Maximal subgroups

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

subgroupindex
#214507372
Q8×C4⋊Dic612
#214473782
#214472242
#214465782
#214464402
#214463912
#214459472
#214458202
C4×#145196992
C61×#19842
#2499061