Finite Group Databasebeta

← #29708644

Subgroups — #29708644

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839788C211yesmaximal, characteristic, complemented
2C2×#16839761C211yesmaximal, characteristic
3C2×#16839759C211yesmaximal, characteristic
4#21448437C211yesmaximal, characteristic, complemented
5#21448412C211yesmaximal, characteristic
6#21447209C214yesmaximal, complemented
7#21447202C214yesmaximal, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C22×#107×C61C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14#16839788—24no—
15C2×#14519706C2214yes—
16#16839761—24no—
17#16839759—24no—
18C2×#14519702C2212yes—
19C2×#14519700C2212yes—
20C2×#14519699C2212yes—
21C2×#14519697C2212yes—
22#16839667C2212yes—
23#16839663C2212yes—
24#16839661—22no—
25#16839661C2214yes—
26#16839641—22no—
27#16839622—22no—
28#16839619—22no—
29#16839524C2214yes—
30C23×C8×C61C2211yescharacteristic
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×C4⋊C4×C61C2211yescharacteristic
33C2×#107×C61—24no—
34C2×#107×C61C2214yes—
35C2×#11758222—22no—
36C2×#11758222—22no—
37C2×#11758222—22no—
38C2×#11758222—22no—
39C2×#11758222—22no—
40C2×#11758222Q814yes—
41C22×Dic122C2311yescharacteristic
42C22×D244C2311yescharacteristic
43C22×C22⋊D61—21no—
44C22×C4×D61—21no—
45C2×#11758194—22no—
46C2×#11758194—22no—
47C2×#11758194—22no—
48C22×#92418C2312yes—
49#14519736—22no—
50#14519736—22no—
51#14519736—22no—
52#14519736—22no—
53C2×C4⋊Dic61—24no—
54C2×C4⋊Dic61D412yes—
55C2×C4⋊Dic61C2312yes—
56C2×C22⋊Dic61—21no—
57C2×C22⋊Dic61D412yes—
58C2×#11758198—24no—
59C2×C4×Dic61—21no—
60C2×C4×Dic61C2312yes—
61#14519706—28no—
62#14519702—24no—
63#14519700—24no—
64#14519699—24no—
65#14519697—24no—
66#14519684—24no—
67#14519684—24no—
68C23×C4×C61—21no—
69C23×C4×C61—21no—
70C23×C4×C61C2311yescharacteristic
71C22×C8×C61—22no—
72C22×C8×C61—22no—
73C22×C8×C61D414yes—
74C22×C8×C61C2312yes—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61D414yes—
81C2×C4⋊C4×C61C2312yes—
82C2×C4⋊C4×C61C2312yes—
83#107×C61—28no—
84C24×C61C2×D411yescharacteristic
85C23×D61—21no—
86C22×C4×C61—22no—
87C22×C4×C61—24no—
88C22×C4×C61—22no—
89C22×C4×C61—42no—
90C22×C4×C61—82no—
91C22×C4×C61—82no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×Q812yes—
96C22×C4×C61#4112yes—
97C22×C4×C61#4112yes—
98C22×C4×C61#4112yes—
99C22×C4×C61C2411yescharacteristic
100C2×C22⋊D61—24no—
101C2×C22⋊D61—42no—
102C2×C22⋊D61—42no—
103C22×Dic61—21no—
104C22×Dic61—22no—
105C22×Dic61C2×D414yes—
106#11758222—28no—
107#11758222—28no—
108#11758222—44no—
109#11758222—44no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244C2×Q812yes—
116C2×C4×D61—24no—
117C2×C4×D61—42no—
118C2×C4×D61—42no—
119C2×Dic122—21no—
120C2×Dic122—21no—
121C2×Dic122—21no—
122C2×Dic122—21no—
123C2×Dic122—21no—
124C2×Dic122C2×Q812yes—
125C2×C8×C61—28no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61C2×D414yes—
129C4⋊C4×C61—24no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—42no—
134C4⋊C4×C61C2×D414yes—
135C22⋊Dic61—24no—
136C22⋊Dic61—42no—
137#11758198—84no—
138C4⋊Dic61—24no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—84no—
142C4×Dic61—22no—
143C4×Dic61—22no—
144C4×Dic61—42no—
145#11758194—44no—
146C2×#92418—22no—
147C2×#92418—22no—
148C2×#92418—22no—
149C23×C61—22no—
150C23×C61—22no—
151C23×C61#12112yes—
152C23×C61#12112yes—
153C23×C61C22⋊Q812yes—
154C23×C61C22×D411yescharacteristic
155C22×D61—22no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—42no—
159C22×D61—42no—
160C2×C4×C61—24no—
161C2×C4×C61—44no—
162C2×C4×C61—42no—
163C2×C4×C61—44no—
164C2×C4×C61—84no—
165C2×C4×C61—84no—
166C2×C4×C61#12114yes—
167C2×C4×C61#12112yes—
168C2×C4×C61#12112yes—
169C2×C4×C61C22⋊Q814yes—
170C2×C4×C61C22⋊Q814yes—
171C2×C4×C61C22⋊Q812yes—
172C2×C4×C61#12614yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C22×Q811yescharacteristic
176C2×C4×C61C2×#4111yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C22⋊D61—48no—
180C2×Dic61—24no—
181C2×Dic61—22no—
182C2×Dic61—24no—
183C2×Dic61—24no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189D244—22no—
190D244—22no—
191D244—41no—
192D244—41no—
193C4×D61—48no—
194Dic122—22no—
195Dic122—22no—
196Dic122—41no—
197Dic122—41no—
198C8×C61—24no—
199#92418—42no—
200#23611—611nomaximal, Sylow
201C22×C61—24no—
202C22×C61—22no—
203C22×C61—22no—
204C22×C61—24no—
205C22×C61#39614yes—
206C22×C61C2×#12111yescharacteristic
207C22×C61C2×#12111yescharacteristic
208C22×C61C2×C22⋊Q811yescharacteristic
209C22×C61#57612yes—
210C22×C61#57812yes—
211C2×D61—44no—
212C2×D61—42no—
213C2×D61—81no—
214C2×D61—81no—
215C2×D61—82no—
216C4×C61—44no—
217C4×C61—82no—
218C4×C61#39614yes—
219C4×C61C2×#12111yescharacteristic
220C4×C61C2×C22⋊Q811yescharacteristic
221C4×C61C2×C22⋊Q811yescharacteristic
222C4×C61C2×#12611yescharacteristic
223Dic61—42no—
224Dic61—44no—
225Dic61—82no—
226C22×#107D6111yescharacteristic, complemented
227C2×#424—611no—
228C2×#417—611no—
229C2×#416—611no—
230#2316—611no—
231#2311—611no—
232#2068—614no—
233#2067—614no—
234#1736—611no—
235C2×C61—24no—
236C2×C61C2×#39611yescharacteristic
237C2×C61#304111yescharacteristic
238C2×C61#304411yescharacteristic
239C2×C61#306112yes—
240C2×C61#308312yes—
241D61—82no—
242C22×#41—611no—
243C22×M16—611no—
244C23×C8C2×D6111yescharacteristic
245C2×#117—611no—
246C2×#117—611no—
247C22×C4⋊C4C2×D6111yescharacteristic
248C22×C4⋊C4C2×D6111yescharacteristic
249C2×#107—24no—
250C2×#107C2×D6114yes—
251#424—1224no—
252C2×C4⋊C8—614no—
253#417—1224no—
254#416—1224no—
255C2×#103—612no—
256C2×#103—612no—
257C2×#102—612no—
258C2×#102—612no—
259#391—612no—
260#390—612no—
261#389—614no—
262#389—1222no—
263#386—1222no—
264#380—1222no—
265#380—1222no—
266#340—614no—
267C61#2361111yescharacteristic, complemented, Sylow
268C2×#41—614no—
269C2×#41—1222no—
270C2×#41—1222no—
271C2×#41—1222no—
272C2×#41—1222no—
273C2×#41—1222no—
274C22×Q8—611no—
275C22×D4—611no—
276C22×D4—1221no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—1221no—
280C23×C4C22×D6111yescharacteristic
281C2×M16—1222no—
282C2×M16—1222no—
283C2×M16—1222no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—612no—
287C22×C8D24414yes—
288C22×C8C22×D6112yes—
289#117—1222no—
290#117—1222no—
291#117—1222no—
292#117—1222no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—612no—
299C2×C4⋊C4—612no—
300C2×C4⋊C4—1224no—
301C2×C4⋊C4C22⋊D6114yes—
302C2×C4⋊C4C22×D6112yes—
303C2×C4⋊C4C22×D6112yes—
304C2×C22⋊C4—612no—
305C2×C22⋊C4—1224no—
306C2×C22⋊C4—1221no—
307C2×C42—612no—
308C2×C42—1221no—
309#107—28no—
310C4⋊C8—1228no—
311#103—1224no—
312#103—1224no—
313#102—1224no—
314#102—1224no—
315#95—1224no—
316#95—1224no—
317C24—1221no—
318C24D4×D6111yescharacteristic
319#41—1228no—
320#41—1228no—
321#41—2444no—
322#41—2444no—
323C2×Q8—612no—
324C2×Q8—1221no—
325C2×Q8—1221no—
326C2×Q8—1221no—
327C2×Q8—1221no—
328C2×Q8—1221no—
329C2×D4—612no—
330C2×D4—1224no—
331C2×D4—1221no—
332C2×D4—1221no—
333C2×D4—1221no—
334C2×D4—1221no—
335C2×D4—1221no—
336C2×D4—2442no—
337C2×D4—2442no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—614no—
345C22×C4—1221no—
346C22×C4—1224no—
347C22×C4—1222no—
348C22×C4—2442no—
349C22×C4—2442no—
350C22×C4C2×D24412yes—
351C22×C4#1175822212yes—
352C22×C4D4×D6111yescharacteristic
353C22×C4#1175822412yes—
354C22×C4Q8×D6112yes—
355C22×C4#1175822612yes—
356C22×C4C2×C22⋊D6112yes—
357C22×C4C23×D6111yescharacteristic
358M16—2444no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—28no—
362C2×C8—1222no—
363C2×C8—1222no—
364C2×C8—1222no—
365C2×C8C2×D24414yes—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—42no—
371C4⋊C4—1224no—
372C4⋊C4—1222no—
373C4⋊C4—1222no—
374C4⋊C4—4884no—
375C4⋊C4C2×C22⋊D6114yes—
376C22⋊C4—1224no—
377C22⋊C4—4884no—
378C22⋊C4—2442no—
379C42—1222no—
380C42—1222no—
381C42—2442no—
382C23—22no—
383C23—22no—
384C23—1222no—
385C23—2441no—
386C23—2441no—
387C23—2442no—
388C23—2442no—
389C23#1451974912yes—
390C23#1451975212yes—
391C23#1451975512yes—
392C23C2×D4×D6111yescharacteristic
393Q8—1222no—
394Q8—1222no—
395Q8—2441no—
396Q8—2441no—
397D4—1222no—
398D4—1222no—
399D4—2448no—
400D4—2441no—
401D4—2441no—
402C2×C4—24no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—1224no—
409C2×C4—1222no—
410C2×C4—1224no—
411C2×C4—1224no—
412C2×C4—2442no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2448no—
418C2×C4#1451974912yes—
419C2×C4#1451975114yes—
420C2×C4#1451975212yes—
421C2×C4#1451975314yes—
422C2×C4#1451975512yes—
423C2×C4#1451975714yes—
424C2×C4#1451977214yes—
425C2×C4C22×C22⋊D6111yescharacteristic
426C2×C4C22×D24411yescharacteristic
427C2×C4C2×Q8×D6111yescharacteristic
428C2×C4C2×#1175822411yescharacteristic
429C2×C4C2×#1175822211yescharacteristic
430C2×C4C2×#1175822611yescharacteristic
431C8—24no—
432C8—2442no—
433C22—24no—
434C22—24no—
435C22—22no—
436C22—22no—
437C22—4882no—
438C22—4881no—
439C22—4881no—
440C22—2444no—
441C22—2442no—
442C22#1683968414yes—
443C22C2×#1451974911yescharacteristic
444C22C2×#1451975211yescharacteristic
445C22C2×#1451975511yescharacteristic
446C22#1684056312yes—
447C22#1684057512yes—
448C4—44no—
449C4—82no—
450C4—4882no—
451C4—2442no—
452C4—2444no—
453C4#1683968414yes—
454C4C2×#1451975111yescharacteristic
455C4C2×#1451975311yescharacteristic
456C4C2×#1451975711yescharacteristic
457C4C2×#1451977211yescharacteristic
458C2—24no—
459C2—4882no—
460C2C2×#1683968411yescharacteristic
461C2#2145531911yescharacteristic
462C2#2145534611yescharacteristic
463C2#2145549112yes—
464C2#2145567512yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#107×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC22×#107D61
hypocenterC61#23611
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#107

Complements

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

NH ≅ G/Ntype
1. C2×#16839788C2semidirect
4. #21448437C2semidirect
6. #21447209C2semidirect
7. #21447202C2semidirect
8. #21445538C2semidirect
9. C22×#107×C61C2semidirect
226. C22×#107D61semidirect
267. C61#23611semidirect

Maximal subgroups

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

subgroupindex
C2×#168397882
C2×#168397612
C2×#168397592
#214484372
#214484122
#214472092
#214472022
#214455382
C22×#107×C612
#2361161