Finite Group Databasebeta

← #29692131

Subgroups — #29692131

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic
2#21448283C211yesmaximal, characteristic
3#21446768C211yesmaximal, characteristic, complemented
4#21446575C211yesmaximal, characteristic, complemented
5#21446559C212yesmaximal, complemented
6#21446515C212yesmaximal
7#21446439C212yesmaximal
8#21446404C212yesmaximal, complemented
9#21445816C211yesmaximal, characteristic
10#21445672C211yesmaximal, characteristic, complemented
11C4×C4⋊Q8×C61C211yesmaximal, characteristic, complemented
12C2×C4×D244C2211yescharacteristic
13C2×#14519734—21no—
14#16840049C418yescomplemented
15C2×#14519710C2212yes—
16C2×#14519708C2212yes—
17C2×C4⋊#92418C2211yescharacteristic
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519697C2212yes—
20C2×#14519697C2212yes—
21C2×#14519688C2212yes—
22#16839664C2211yescharacteristic
23#16839664C2211yescharacteristic
24#16839660C2211yescharacteristic
25#16839660C2211yescharacteristic
26#16839653—21no—
27#16839653—21no—
28C4×C4⋊Dic61C2211yescharacteristic
29C4×#11758198—21no—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839517C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×C4×Q8×C61—21no—
37C61×#391—21no—
38C61×#389—21no—
39C61×#389C2212yes—
40C61×#384C2211yescharacteristic
41C4×C4⋊C4×C61—21no—
42C4×C4⋊C4×C61C2212yes—
43C43×C61C2211yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C22×#92418C2312yes—
47C22×#92418C2312yes—
48C4×D244—22no—
49C4×D244C2×C414yes—
50#14519734—44no—
51C2×C4⋊Dic61C2311yescharacteristic
52C2×C4⋊Dic61C2311yescharacteristic
53C2×C4⋊Dic61C2311yescharacteristic
54C2×#11758196—21no—
55C2×#11758196—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×C4×Dic61—21no—
62#14519710—42no—
63#14519708C2×C418yes—
64C4⋊#92418—22no—
65C4⋊#92418C2×C414yes—
66#14519697—42no—
67#14519697C2×C418yes—
68#14519688—42no—
69#14519684—21no—
70#14519684—21no—
71#14519684—21no—
72C22×Q8×C61—21no—
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61C2×C414yes—
75C4×Q8×C61—44no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61C2312yes—
81C2×C4⋊C4×C61C2312yes—
82C2×C42×C61—22no—
83C2×C42×C61D411yescharacteristic
84C2×C42×C61D411yescharacteristic
85C2×C42×C61D411yescharacteristic
86C2×C42×C61D411yescharacteristic
87C2×C42×C61C2311yescharacteristic
88C2×C42×C61C2311yescharacteristic
89C2×C42×C61C2311yescharacteristic
90#95×C61—22no—
91#95×C61—22no—
92C23×D61—21no—
93C2×Q8×C61—24no—
94C2×Q8×C61—42no—
95C2×Q8×C61—42no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2411yescharacteristic
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244C22×C412yes—
114C2×C4×D61—24no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C4⋊C4×C61—24no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—42no—
122C4⋊C4×C61—44no—
123C4⋊C4×C61—42no—
124C4⋊C4×C61C22×C414yes—
125C42×C61—22no—
126C42×C61—22no—
127C42×C61—22no—
128C42×C61—21no—
129C42×C61—21no—
130C42×C61—21no—
131C42×C61—21no—
132C42×C61—41no—
133C42×C61—44no—
134C42×C61C22×C412yes—
135C42×C61#4112yes—
136#11758198—24no—
137#11758198—42no—
138#11758198—42no—
139#11758198—42no—
140#11758198—42no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61C22×C412yes—
147#11758196—42no—
148#11758196—42no—
149C4×Dic61—42no—
150C2×#92418—22no—
151C2×#92418—22no—
152C2×#92418—22no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418C22×C414yes—
156C23×C61C22×D411yescharacteristic
157C22×D61—22no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164Q8×C61—48no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—24no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61C4×D412yes—
185C2×C4×C61C4×D412yes—
186C2×C4×C61AGL(1,ℤ/8)12yes—
187C2×C4×C61AGL(1,ℤ/8)12yes—
188C2×C4×C61#13712yes—
189C2×C4×C61C23×C411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61H(5,𝔽2)11yescharacteristic
192C2×C4×C612−1+411yescharacteristic
193C2×C4×C612−1+411yescharacteristic
194C2×C4×C612−1+411yescharacteristic
195C2×Dic61—22no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204D244—22no—
205D244—22no—
206D244—41no—
207D244—41no—
208C4×D61—48no—
209#92418—24no—
210#92418—42no—
211#22117—611nomaximal, Sylow
212C22×C61C2×C4×D411yescharacteristic
213C22×C61#53611yescharacteristic
214C22×C61#53711yescharacteristic
215C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
216C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
217C22×C61C2×#13711yescharacteristic
218C22×C61#57511yescharacteristic
219C2×D61—42no—
220C2×D61—42no—
221C2×D61—42no—
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C4×C61—22no—
227C4×C61—22no—
228C4×C61—44no—
229C4×C61—41no—
230C4×C61—41no—
231C4×C61—41no—
232C4×C61—82no—
233C4×C61#44014yes—
234C4×C61#51911yescharacteristic
235C4×C61#52011yescharacteristic
236C4×C61#54111yescharacteristic
237C4×C61#54211yescharacteristic
238Dic61—42no—
239Dic61—81no—
240Dic61—81no—
241C2×#476—611no—
242C4×C4⋊Q8D6111yescharacteristic, complemented
243#2293—611no—
244#1987—611no—
245#1947—611no—
246#1944—612no—
247#1936—612no—
248#1925—612no—
249#1920—612no—
250#1802—611no—
251#1767—611no—
252C2×C61#232311yescharacteristic
253C2×C61C2×#44011yescharacteristic
254C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
255C2×C61#323511yescharacteristic
256C2×C61#323811yescharacteristic
257C2×C61#325211yescharacteristic
258C2×C61#325311yescharacteristic
259D61—82no—
260C2×C4⋊Q8C2×D6111yescharacteristic
261C2×C4×Q8—21no—
262C2×C4×D4—611no—
263C2×#117—1221no—
264#476—618no—
265C2×#107—612no—
266C2×#106—612no—
267C2×C4⋊C8—611no—
268C2×C4⋊C8—611no—
269C2×#102—612no—
270C2×#102—612no—
271C2×#97—612no—
272#391—21no—
273#390—611no—
274#390—611no—
275#389—21no—
276#389—611no—
277#389—611no—
278#389C2×D6112yes—
279#388—1221no—
280#388—1221no—
281#384C2×D6111yescharacteristic
282C4×C4⋊C4—21no—
283C4×C4⋊C4—611no—
284C4×C4⋊C4C2×D6112yes—
285C4×C22⋊C4—1221no—
286C43C2×D6111yescharacteristic
287#340—612no—
288#340—612no—
289#340—612no—
290#340—612no—
291#336—612no—
292C61#2211711yescharacteristic, complemented, Sylow
293C22×Q8—21no—
294C22×D4—611no—
295C23×C4—1221no—
296C22×C8—612no—
297C22×C8—612no—
298C4⋊Q8—22no—
299C4⋊Q8C4×D6114yes—
300C4×Q8—44no—
301C4×D4—614no—
302C4×D4—1222no—
303#117—2444no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—611no—
310C2×C4⋊C4—611no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4C22×D6112yes—
314C2×C4⋊C4C22×D6112yes—
315C2×C22⋊C4—1221no—
316C2×C22⋊C4—1221no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C42—22no—
321C2×C42—1221no—
322C2×C42C22⋊D6111yescharacteristic
323C2×C42C22⋊D6111yescharacteristic
324C2×C42C22⋊D6111yescharacteristic
325C2×C42C22⋊D6111yescharacteristic
326C2×C42C22×D6111yescharacteristic
327C2×C42C22×D6111yescharacteristic
328C2×C42C22×D6111yescharacteristic
329#107—2442no—
330#106—618no—
331C4⋊C8—614no—
332C4⋊C8—1222no—
333#102—618no—
334#102—2442no—
335#97—2442no—
336#95—22no—
337#95—22no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341C24—1221no—
342C2×Q8—24no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×D4—612no—
346C2×D4—1221no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—1221no—
354C22×C4—1224no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4C2×C22⋊D6111yescharacteristic
360C22×C4C2×C22⋊D6111yescharacteristic
361C22×C4C2×C22⋊D6111yescharacteristic
362C22×C4C2×C22⋊D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365C22×C4C23×D6111yescharacteristic
366C2×C8—614no—
367C2×C8—1222no—
368C2×C8—1222no—
369C2×C8—1222no—
370C2×C8—1222no—
371C2×C8—1222no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—612no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—2442no—
386C4⋊C4—2442no—
387C4⋊C4C2×C4×D6114yes—
388C22⋊C4—1224no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C22⋊C4—2442no—
392C22⋊C4—2442no—
393C42—22no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—22no—
399C42—22no—
400C42—41no—
401C42—44no—
402C42—2442no—
403C42C2×C4×D6112yes—
404C42#1175822212yes—
405C23—1222no—
406C23—2441no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23—2441no—
411C23—2441no—
412C23C22×C22⋊D6111yescharacteristic
413Q8—48no—
414D4—1222no—
415D4—1222no—
416D4—2441no—
417D4—2441no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—1222no—
438C2×C4—2441no—
439C2×C4—2441no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2448no—
447C2×C4C4×C22⋊D6112yes—
448C2×C4C4×C22⋊D6112yes—
449C2×C4#1451980212yes—
450C2×C4#1451980312yes—
451C2×C4#1451980812yes—
452C2×C4C22×C4×D6111yescharacteristic
453C2×C4C2×#1175822211yescharacteristic
454C2×C4#1451982311yescharacteristic
455C2×C4#1451982411yescharacteristic
456C2×C4#1451982511yescharacteristic
457C2×C4#1451982511yescharacteristic
458C8—1224no—
459C8—2442no—
460C22—4881no—
461C22—4881no—
462C22—4881no—
463C22—4881no—
464C22—2442no—
465C22—2442no—
466C22—2442no—
467C22C2×C4×C22⋊D6111yescharacteristic
468C22#1684033611yescharacteristic
469C22#1684034011yescharacteristic
470C22C2×#1451980211yescharacteristic
471C22C2×#1451980311yescharacteristic
472C22C2×#1451980811yescharacteristic
473C22#1684056011yescharacteristic
474C4—22no—
475C4—22no—
476C4—41no—
477C4—44no—
478C4—41no—
479C4—41no—
480C4—82no—
481C4—4881no—
482C4—4881no—
483C4—2442no—
484C4#1683986014yes—
485C4#1684023811yescharacteristic
486C4#1684024411yescharacteristic
487C4#1684037011yescharacteristic
488C4#1684038411yescharacteristic
489C2—4882no—
490C2#2144849511yescharacteristic
491C2C2×#1683986011yescharacteristic
492C2C4×#1451980311yescharacteristic
493C2#2145701711yescharacteristic
494C2#2145704011yescharacteristic
495C2#2145718611yescharacteristic
496C2#2145719711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×C4⋊Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenterC4×C4⋊Q8D61
hypocenterC61#22117
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21446768C2semidirect
4. #21446575C2semidirect
5. #21446559C2semidirect
8. #21446404C2semidirect
10. #21445672C2semidirect
11. C4×C4⋊Q8×C61C2semidirect
14. #16840049C4semidirect
242. C4×C4⋊Q8D61semidirect
292. C61#22117semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
#214482832
#214467682
#214465752
#214465592
#214465152
#214464392
#214464042
#214458162
#214456722
C4×C4⋊Q8×C612
#2211761