Finite Group Databasebeta

← #29725154

Subgroups — #29725154

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
2#21447376C212yesmaximal, complemented
3#21447223C212yesmaximal
4#21446756C211yesmaximal, characteristic, complemented
5#21446577C211yesmaximal, characteristic, complemented
6#21446439C212yesmaximal
7#21446390C212yesmaximal, complemented
8#21445946C211yesmaximal, characteristic, complemented
9#21445816C211yesmaximal, characteristic
10C4×#14519697C211yesmaximal, characteristic
11Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
12C2×#14519753—21no—
13C2×C4×D244C2211yescharacteristic
14C2×#14519710C2212yes—
15C2×C4⋊#92418C2211yescharacteristic
16C2×#14519697C2211yescharacteristic
17C2×#14519697C2211yescharacteristic
18C2×#14519697C2212yes—
19C2×C4×#92418C2211yescharacteristic
20#16839684—22no—
21#16839664C2212yes—
22#16839660C2212yes—
23#16839632C2211yescharacteristic
24#16839619—21no—
25#16839524C2211yescharacteristic
26#16839524C2211yescharacteristic
27#16839524C2212yes—
28#16839524C2212yes—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839517C2212yes—
32#16839517C2212yes—
33C2×C4×Q8×C61—21no—
34C2×C4×Q8×C61C2211yescharacteristic
35C61×#389—22no—
36C61×#389—21no—
37C61×#384C2212yes—
38C61×#384C2211yescharacteristic
39C4×C4⋊C4×C61C2211yescharacteristic
40C4×C4⋊C4×C61C2212yes—
41C22×D244C2311yescharacteristic
42C22×C4×D61—21no—
43C22×#92418C2311yescharacteristic
44C22×#92418C2311yescharacteristic
45C22×#92418C2312yes—
46#14519753—44no—
47C4×D244—22no—
48C4×D244—22no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61C2311yescharacteristic
52C2×C4⋊Dic61C2312yes—
53C2×#11758196—22no—
54C2×#11758198—21no—
55C2×#11758198—22no—
56C2×#11758198—22no—
57#14519710—24no—
58C4⋊#92418—22no—
59#14519697—22no—
60#14519697—24no—
61#14519697—22no—
62C4×#92418—22no—
63#14519684—22no—
64C22×Q8×C61D411yescharacteristic
65C4×Q8×C61—22no—
66C4×Q8×C61—22no—
67C4×Q8×C61—44no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61D412yes—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C4⋊C4×C61C2311yescharacteristic
75C2×C4⋊C4×C61C2312yes—
76C2×C4⋊C4×C61C2311yescharacteristic
77C2×C42×C61—22no—
78C2×C42×C61—21no—
79C2×C42×C61C2312yes—
80C2×C42×C61C2311yescharacteristic
81#95×C61—22no—
82#95×C61—21no—
83C23×D61—21no—
84C2×Q8×C61—22no—
85C2×Q8×C61—22no—
86C2×Q8×C61D814yes—
87C22×C4×C61—21no—
88C22×C4×C61—22no—
89C22×C4×C61—21no—
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2411yescharacteristic
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—
106C4⋊C4×C61—24no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—44no—
114C4⋊C4×C61—42no—
115C4⋊C4×C61—81no—
116C4⋊C4×C61—82no—
117C4⋊C4×C61#4112yes—
118C42×C61—22no—
119C42×C61—22no—
120C42×C61—44no—
121C42×C61—42no—
122C42×C61#4112yes—
123C42×C61#4112yes—
124#11758198—42no—
125#11758198—44no—
126#11758198—82no—
127C4⋊Dic61—22no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—22no—
131C4⋊Dic61—42no—
132C4⋊Dic61—81no—
133#11758196—44no—
134C2×#92418—21no—
135C2×#92418—21no—
136C2×#92418—22no—
137C2×#92418—21no—
138C2×#92418—21no—
139C2×#92418—22no—
140C2×#92418—22no—
141C2×#92418—21no—
142C2×#92418—21no—
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—22no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61#12312yes—
169C2×C4×C61#12312yes—
170C2×C4×C61C2×D814yes—
171C2×C4×C61C2×D812yes—
172C2×C4×C61#13712yes—
173C2×C4×C61C2×#4111yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C2×C4×C61C2×#4111yescharacteristic
177C2×C4×C612−1+411yescharacteristic
178C2×C4×C612−1+411yescharacteristic
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—
190#92418—42no—
191#92418—41no—
192#92418—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—42no—
212C4×C61—82no—
213C4×C61—81no—
214C4×C61#48014yes—
215C4×C61#52911yescharacteristic
216C4×C61#54111yescharacteristic
217C4×C61#55211yescharacteristic
218C4×C61#55611yescharacteristic
219Dic61—81no—
220Dic61—82no—
221#2538—611no—
222Q8×C4⋊C4D6111yescharacteristic, complemented
223#2099—612no—
224#2071—612no—
225#1984—611no—
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—21no—
242C2×C4×Q8C2×D6111yescharacteristic
243C2×C4×D4—611no—
244C2×#107—612no—
245C2×C4⋊C8—611no—
246C2×#102—612no—
247C2×#102—611no—
248C2×#102—611no—
249C2×C4×C8—611no—
250#396—1222no—
251#390—612no—
252#389—21no—
253#389—22no—
254#389—612no—
255#384—611no—
256#384C2×D6111yescharacteristic
257#384C2×D6112yes—
258#380—1221no—
259C4×C4⋊C4C2×D6112yes—
260C4×C4⋊C4C2×D6111yescharacteristic
261#340—612no—
262#340—612no—
263#340—612no—
264#340—612no—
265#340—611no—
266#340—611no—
267#336—612no—
268#336—612no—
269C61#2499011yescharacteristic, complemented, Sylow
270C22×Q8C22⋊D6111yescharacteristic
271C22×D4—611no—
272C23×C4—1221no—
273C22×C8—611no—
274C22×C8—611no—
275C22×C8—612no—
276C22⋊Q8—2444no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—44no—
280C4×D4—1222no—
281C4×D4—1222no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—611no—
287C2×C4⋊C4—612no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4—1222no—
291C2×C4⋊C4C22⋊D6111yescharacteristic
292C2×C4⋊C4C22⋊D6112yes—
293C2×C4⋊C4C22×D6112yes—
294C2×C4⋊C4C22×D6111yescharacteristic
295C2×C4⋊C4C22×D6111yescharacteristic
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—1221no—
298C2×C22⋊C4—1222no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42C22×D6112yes—
302C2×C42C22×D6111yescharacteristic
303#107—1224no—
304C4⋊C8—1222no—
305#102—1224no—
306#102—1222no—
307#102—1222no—
308C4×C8—1222no—
309#95—22no—
310#95—21no—
311#95—1222no—
312C24—1221no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8#1175819914yes—
316C2×D4—612no—
317C2×D4—1221no—
318C2×D4—1221no—
319C2×D4—1221no—
320C2×D4—1221no—
321C2×D4—1221no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—1221no—
326C22×C4—1222no—
327C22×C4—2442no—
328C22×C4—2442no—
329C22×C4—2442no—
330C22×C4C2×C22⋊D6112yes—
331C22×C4C2×C22⋊D6111yescharacteristic
332C22×C4C2×C22⋊D6112yes—
333C22×C4C2×C22⋊D6111yescharacteristic
334C22×C4C23×D6111yescharacteristic
335C2×C8—1221no—
336C2×C8—1222no—
337C2×C8—1221no—
338C2×C8—1221no—
339C2×C8—1222no—
340C2×C8—1221no—
341C2×C8—1222no—
342C2×C8—1221no—
343C2×C8—1221no—
344C4⋊C4—21no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—81no—
354C4⋊C4—82no—
355C4⋊C4—1221no—
356C4⋊C4—1221no—
357C4⋊C4—1222no—
358C4⋊C4—1222no—
359C4⋊C4—4881no—
360C4⋊C4—2444no—
361C4⋊C4—2442no—
362C4⋊C4#1175822612yes—
363C22⋊C4—4882no—
364C22⋊C4—2442no—
365C22⋊C4—2444no—
366C42—22no—
367C42—22no—
368C42—44no—
369C42—42no—
370C42#1175822212yes—
371C42#1175822212yes—
372C23—1222no—
373C23—2441no—
374C23—2441no—
375C23—2441no—
376C23—2441no—
377C23—2441no—
378C23—2441no—
379C23C22×C22⋊D6111yescharacteristic
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—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—4884no—
402C2×C4—2441no—
403C2×C4—2442no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2442no—
407C2×C4—2442no—
408C2×C4#1451976012yes—
409C2×C4#1451976012yes—
410C2×C4C2×#1175819914yes—
411C2×C4C2×#1175819912yes—
412C2×C4#1451980812yes—
413C2×C4#41×D6111yescharacteristic
414C2×C4C2×#1175822211yescharacteristic
415C2×C4C2×#1175822211yescharacteristic
416C2×C4C2×#1175822611yescharacteristic
417C2×C4#1451982411yescharacteristic
418C2×C4#1451982511yescharacteristic
419C8—2441no—
420C8—2441no—
421C8—2442no—
422C22—4881no—
423C22—4881no—
424C22—4881no—
425C22—4881no—
426C22—2442no—
427C22—2442no—
428C22—2442no—
429C22C2×#1451976011yescharacteristic
430C22#1684041911yescharacteristic
431C22#1684042311yescharacteristic
432C22C22×#1175819911yescharacteristic
433C22C2×#1451980811yescharacteristic
434C22#1684057011yescharacteristic
435C22#1684057311yescharacteristic
436C4—24no—
437C4—22no—
438C4—42no—
439C4—42no—
440C4—82no—
441C4—81no—
442C4—4882no—
443C4—4881no—
444C4#1684007514yes—
445C4#1684029911yescharacteristic
446C4#1684037011yescharacteristic
447C4#1684045311yescharacteristic
448C4#1684048511yescharacteristic
449C2—4882no—
450C2#2144931611yescharacteristic
451C2C2×#1684007511yescharacteristic
452C2#2145580811yescharacteristic
453C2#2145845011yescharacteristic
454C2#2145845411yescharacteristic
455C2#2145855911yescharacteristic
456C2#2145859611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingQ8×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterQ8×C4⋊C4D61
hypocenterC61#24990
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21447376C2semidirect
4. #21446756C2semidirect
5. #21446577C2semidirect
7. #21446390C2semidirect
8. #21445946C2semidirect
11. Q8×C4⋊C4×C61C2semidirect
222. Q8×C4⋊C4D61semidirect
269. C61#24990semidirect

Maximal subgroups

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

subgroupindex
#214507372
#214473762
#214472232
#214467562
#214465772
#214464392
#214463902
#214459462
#214458162
C4×#145196972
Q8×C4⋊C4×C612
#2499061