Finite Group Databasebeta

← #29731959

Subgroups — #29731959

Order 15616.

798 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840053C211yesmaximal, characteristic
2C2×#16839889C211yesmaximal, characteristic, complemented
3#21451027C211yesmaximal, characteristic, complemented
4#21450742C211yesmaximal, characteristic
5#21447319C212yesmaximal, complemented
6#21447202C212yesmaximal
7#21446779C211yesmaximal, characteristic, complemented
8#21446558C212yesmaximal, complemented
9#21446517C212yesmaximal
10C4×#14519697C211yesmaximal, characteristic
11C61×#1947C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519757—21no—
14C2×C4×Dic122C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16#16840053—24no—
17#16839889—24no—
18C2×#14519709C2212yes—
19C2×#14519706C2212yes—
20C2×#14519699C2212yes—
21C2×#14519697C2211yescharacteristic
22C2×#14519697C2211yescharacteristic
23C2×#14519687C2212yes—
24C2×C4×#92418C2211yescharacteristic
25#16839684—22no—
26#16839683—22no—
27#16839668—21no—
28#16839663C2212yes—
29#16839661C2212yes—
30#16839661C2212yes—
31#16839661C2212yes—
32#16839632—21no—
33#16839619—21no—
34#16839524C2211yescharacteristic
35#16839524C2211yescharacteristic
36#16839524C2212yes—
37#16839524C2212yes—
38C2×#107×C61C2212yes—
39C2×C4⋊C8×C61C2211yescharacteristic
40C61×#384C2211yescharacteristic
41C4×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C22×Dic122—21no—
44C22×Dic122D411yescharacteristic
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C22×#92418C2311yescharacteristic
48C22×#92418C2311yescharacteristic
49#14519777—22no—
50#14519777—22no—
51#14519757—44no—
52C4×Dic122—22no—
53C4×Dic122—22no—
54C4×D244—22no—
55C4×D244—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—22no—
59C2×C4⋊Dic61—22no—
60C2×C4⋊Dic61D411yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×#11758196D412yes—
64C2×#11758198—21no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61C2312yes—
68#14519709—24no—
69#14519706—24no—
70#14519699—24no—
71#14519697—22no—
72#14519697Q814yescomplemented
73#14519687—24no—
74C4×#92418—22no—
75#14519684—22no—
76#14519684—22no—
77#14519684—22no—
78C22×C8×C61C2312yes—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61C2312yes—
82C2×C4⋊C4×C61C2311yescharacteristic
83C2×C4⋊C4×C61C2311yescharacteristic
84C2×C42×C61—21no—
85C2×C42×C61C2311yescharacteristic
86#107×C61—24no—
87C4⋊C8×C61D414yes—
88#95×C61—21no—
89C23×D61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2411yescharacteristic
96C22×Dic61—21no—
97C22×Dic61—22no—
98C22×Dic61C2×D412yes—
99C22×Dic61C2×D412yes—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—21no—
103C2×D244—21no—
104C2×D244—21no—
105C2×D244C2×Q812yes—
106C2×C4×D61—42no—
107C2×C4×D61—42no—
108C2×C4×D61—42no—
109C2×Dic122—22no—
110C2×Dic122—22no—
111C2×Dic122—22no—
112C2×Dic122—42no—
113C2×Dic122—81no—
114C2×Dic122—81no—
115C2×C8×C61—22no—
116C2×C8×C61—22no—
117C2×C8×C61C2×D414yes—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—42no—
123C4⋊C4×C61—81no—
124C4⋊C4×C61C2×Q812yes—
125C4⋊C4×C61#4112yes—
126C42×C61—42no—
127C42×C61C2×D412yes—
128C42×C61#4112yes—
129#11758198—44no—
130#11758198—42no—
131#11758198—82no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—22no—
137C4⋊Dic61—44no—
138C4⋊Dic61—42no—
139C4⋊Dic61—82no—
140C4⋊Dic61—82no—
141#11758196—24no—
142C4×Dic61—22no—
143C4×Dic61—22no—
144C2×#92418—21no—
145C2×#92418—21no—
146C2×#92418—21no—
147C2×#92418—21no—
148C2×#92418—21no—
149C2×#92418C2×Q812yes—
150C23×C61C22×D411yescharacteristic
151C22×D61—22no—
152C22×D61—41no—
153C22×D61—41no—
154C22×D61—41no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61#12112yes—
170C2×C4×C61#12112yes—
171C2×C4×C61AGL(1,ℤ/8)12yes—
172C2×C4×C61#13712yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×Q811yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C2×C4×C61C2×#4111yescharacteristic
177C2×C4×C612−1+411yescharacteristic
178C2×C4×C612−1+411yescharacteristic
179C2×Dic61—22no—
180C2×Dic61—22no—
181C2×Dic61—22no—
182C2×Dic61—22no—
183C2×Dic61—22no—
184C2×Dic61—22no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191D244—22no—
192D244—22no—
193D244—41no—
194D244—41no—
195C4×D61—84no—
196Dic122—44no—
197Dic122—84no—
198C8×C61—24no—
199#92418—22no—
200#92418—41no—
201#25477—611nomaximal, Sylow
202C22×C61C2×#12111yescharacteristic
203C22×C61#54811yescharacteristic
204C22×C61D4×Q811yescharacteristic
205C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
206C22×C61C2×#13711yescharacteristic
207C22×C61#57611yescharacteristic
208C22×C61#57711yescharacteristic
209C2×D61—42no—
210C2×D61—42no—
211C2×D61—42no—
212C2×D61—81no—
213C2×D61—81no—
214C2×D61—81no—
215C2×D61—81no—
216C4×C61—22no—
217C4×C61—42no—
218C4×C61—42no—
219C4×C61—82no—
220C4×C61#46814yes—
221C4×C61#53311yescharacteristic
222C4×C61#54111yescharacteristic
223C4×C61#54711yescharacteristic
224C4×C61D4×Q811yescharacteristic
225Dic61—42no—
226Dic61—42no—
227Dic61—81no—
228Dic61—82no—
229C2×#476—611no—
230C2×#446—611no—
231#2567—611no—
232#2538—611no—
233#2088—612no—
234#2067—612no—
235#1990—611no—
236#1947D6111yescharacteristic, complemented
237#1943—612no—
238#1936—612no—
239C4×#102—611no—
240C2×C61#242911yescharacteristic
241C2×C61C2×#46811yescharacteristic
242C2×C61#306211yescharacteristic
243C2×C61#336611yescharacteristic
244C2×C61#336911yescharacteristic
245C2×C61#338411yescharacteristic
246C2×C61#338511yescharacteristic
247D61—82no—
248C2×C4⋊Q8—611no—
249C2×C22⋊Q8—1221no—
250C2×C4×Q8—611no—
251C2×C4×D4—611no—
252#476—1224no—
253#446—1224no—
254C2×#107C2×D6112yes—
255C2×#106—612no—
256C2×C4⋊C8—612no—
257C2×C4⋊C8C2×D6111yescharacteristic
258C2×#102—611no—
259C2×#102—612no—
260C2×#102—611no—
261C2×#97—612no—
262C2×C4×C8—611no—
263#396—1222no—
264#395—1222no—
265#391—1221no—
266#390—612no—
267#389—612no—
268#389—612no—
269#389—612no—
270#384—1221no—
271#384C2×D6111yescharacteristic
272#380—1221no—
273C4×C4⋊C4C2×D6111yescharacteristic
274#340—612no—
275#340—611no—
276#340—612no—
277#340—611no—
278#340C2×D6112yes—
279C61#2547711yescharacteristic, complemented, Sylow
280C22×Q8—611no—
281C22×Q8—1221no—
282C22×D4—611no—
283C23×C4—1221no—
284C22×C8—611no—
285C22×C8—611no—
286C22×C8C22×D6112yes—
287C4⋊Q8—1222no—
288C4⋊Q8—1222no—
289C22⋊Q8—2444no—
290C4×Q8—1222no—
291C4×Q8—1222no—
292C4×D4—1222no—
293C4×D4—1222no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—612no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—612no—
300C2×C4⋊C4—1222no—
301C2×C4⋊C4—1222no—
302C2×C4⋊C4—1222no—
303C2×C4⋊C4—1221no—
304C2×C4⋊C4C22×D6112yes—
305C2×C4⋊C4C22×D6111yescharacteristic
306C2×C4⋊C4C22×D6111yescharacteristic
307C2×C22⋊C4—1222no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1221no—
310C2×C42—21no—
311C2×C42—612no—
312C2×C42C22×D6111yescharacteristic
313#107—24no—
314#106—1224no—
315C4⋊C8—1224no—
316C4⋊C8D24414yes—
317#102—614no—
318#102—1224no—
319#102—1222no—
320#97—1224no—
321C4×C8—1222no—
322#95—21no—
323#95—1222no—
324#95—1222no—
325#95—1222no—
326C24—1221no—
327C2×Q8—1222no—
328C2×Q8—1222no—
329C2×Q8—1222no—
330C2×Q8—4881no—
331C2×Q8—4881no—
332C2×Q8—2442no—
333C2×D4—612no—
334C2×D4—1221no—
335C2×D4—1221no—
336C2×D4—1221no—
337C2×D4—1221no—
338C2×D4—1221no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—612no—
343C22×C4—612no—
344C22×C4—1221no—
345C22×C4—1222no—
346C22×C4—2442no—
347C22×C4—2442no—
348C22×C4—2442no—
349C22×C4D4×D6111yescharacteristic
350C22×C4D4×D6111yescharacteristic
351C22×C4C23×D6111yescharacteristic
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—612no—
355C2×C8—1221no—
356C2×C8—1221no—
357C2×C8—1221no—
358C2×C8—1221no—
359C2×C8—1221no—
360C2×C8C2×D24414yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—42no—
366C4⋊C4—81no—
367C4⋊C4—1224no—
368C4⋊C4—1222no—
369C4⋊C4—1222no—
370C4⋊C4—1221no—
371C4⋊C4—1221no—
372C4⋊C4—1222no—
373C4⋊C4—4882no—
374C4⋊C4—4882no—
375C4⋊C4—2444no—
376C4⋊C4—2442no—
377C4⋊C4#1175822412yes—
378C4⋊C4Q8×D6112yes—
379C22⋊C4—4882no—
380C22⋊C4—2444no—
381C22⋊C4—2442no—
382C42—42no—
383C42—1222no—
384C42—1222no—
385C42C2×D24412yes—
386C42#1175822612yes—
387C23—1222no—
388C23—2441no—
389C23—2441no—
390C23—2441no—
391C23—2441no—
392C23—2441no—
393C23—2441no—
394C23C2×D4×D6111yescharacteristic
395Q8—4884no—
396Q8—2444no—
397D4—1222no—
398D4—1222no—
399D4—2441no—
400D4—2441no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—1222no—
413C2×C4—1222no—
414C2×C4—1222no—
415C2×C4—1222no—
416C2×C4—1222no—
417C2×C4—1222no—
418C2×C4—4884no—
419C2×C4—2441no—
420C2×C4—2442no—
421C2×C4—2441no—
422C2×C4—2441no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4#1451975212yes—
426C2×C4#1451975212yes—
427C2×C4#1451980112yes—
428C2×C4#1451980912yes—
429C2×C4C22×D24411yescharacteristic
430C2×C4C2×Q8×D6111yescharacteristic
431C2×C4C2×#1175822411yescharacteristic
432C2×C4C2×#1175822611yescharacteristic
433C2×C4#1451982411yescharacteristic
434C2×C4#1451982411yescharacteristic
435C8—24no—
436C8—1222no—
437C8—2441no—
438C22—4881no—
439C22—4881no—
440C22—4881no—
441C22—4881no—
442C22—2442no—
443C22—2442no—
444C22—2442no—
445C22C2×#1451975211yescharacteristic
446C22#1684041811yescharacteristic
447C22#1684042511yescharacteristic
448C22C2×#1451980111yescharacteristic
449C22C2×#1451980911yescharacteristic
450C22#1684056611yescharacteristic
451C22#1684057111yescharacteristic
452C4—22no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—4881no—
457C4—4882no—
458C4—2442no—
459C4—2442no—
460C4#1684000914yes—
461C4#1684031711yescharacteristic
462C4#1684037511yescharacteristic
463C4#1684041611yescharacteristic
464C4Q8×D24411yescharacteristic
465C2—4882no—
466C2#2144941611yescharacteristic
467C2C2×#1684000911yescharacteristic
468C2#2145550011yescharacteristic
469C2#2145853011yescharacteristic
470C2#2145856311yescharacteristic
471C2#2145872211yescharacteristic
472C2#2145872811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1947C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1947D61
hypocenterC61#25477
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
2. C2×#16839889C2semidirect
3. #21451027C2semidirect
5. #21447319C2semidirect
7. #21446779C2semidirect
8. #21446558C2semidirect
11. C61×#1947C2semidirect
72. #14519697Q8semidirect
236. #1947D61semidirect
279. C61#25477semidirect

Maximal subgroups

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

subgroupindex
C2×#168400532
C2×#168398892
#214510272
#214507422
#214473192
#214472022
#214467792
#214465582
#214465172
C4×#145196972
C61×#19472
#2547761