Finite Group Databasebeta

← #29713309

Subgroups — #29713309

Order 15616.

1144 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840319C211yesmaximal, characteristic
2C2×#16840093C212yesmaximal
3C2×#16839660C211yesmaximal, characteristic
4#21447153C212yesmaximal, complemented
5#21447066C212yesmaximal, complemented
6#21446124C212yesmaximal
7#21446090C214yesmaximal, complemented
8C61×#1817C211yesmaximal, characteristic
9#16840319—28no—
10C2×#14519777C2211yescharacteristic
11C2×#14519777C2211yescharacteristic
12C2×#14519769—21no—
13C2×#14519756—21no—
14C2×#14519756C2212yes—
15C2×C4×Dic122—21no—
16C2×C4×Dic122C2212yes—
17C2×#14519736—21no—
18C22×C4⋊Dic61C2211yescharacteristic
19C22×C4⋊Dic61C2211yescharacteristic
20C22×#11758196—21no—
21#16840093—28no—
22C2×#14519709C2214yes—
23C2×#14519702C2214yes—
24C2×#14519702C2214yes—
25#16839660—21no—
26#16839660—21no—
27#16839660C2212yes—
28#16839660C2212yes—
29C2×#14519684—21no—
30#16839525C2214yes—
31C22×C42×C61C2211yescharacteristic
32C2×C22⋊C8×C61C2212yes—
33C61×#336C2212yes—
34C61×#336C2212yes—
35C22×Dic122—21no—
36C22×Dic122C2312yes—
37C23×Dic61—21no—
38C23×Dic61—21no—
39#14519777—24no—
40#14519777—24no—
41#14519769—28no—
42#14519756—28no—
43#14519756—24no—
44#14519756D418yes—
45C4×Dic122—28no—
46C4×Dic122—24no—
47C4×Dic122—24no—
48#14519736—28no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—24no—
54C2×C4⋊Dic61—24no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2312yes—
59C2×#11758196—22no—
60C2×#11758196—22no—
61C2×#11758196—22no—
62C2×#11758196—22no—
63C2×#11758196—44no—
64C2×C22⋊Dic61—22no—
65C2×C4×Dic61—22no—
66#14519709—28no—
67#14519702—28no—
68#14519702—28no—
69#14519684—22no—
70#14519684—22no—
71C23×C4×C61D411yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C22×C8×C61C2314yes—
75C2×C42×C61—24no—
76C2×C42×C61D411yescharacteristic
77C2×C42×C61D411yescharacteristic
78C2×C42×C61C2311yescharacteristic
79C2×C42×C61C2311yescharacteristic
80C22⋊C8×C61—24no—
81C24×C61C2×D411yescharacteristic
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—22no—
87C22×C4×C61—22no—
88C22×C4×C61—41no—
89C22×C4×C61SD1614yes—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D414yes—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2411yescharacteristic
97C22×Dic61—24no—
98C22×Dic61—22no—
99C22×Dic61—22no—
100C22×Dic61—22no—
101C22×Dic61—22no—
102C22×Dic61—42no—
103C22×Dic61—42no—
104C22×Dic61—42no—
105C22×Dic61—42no—
106C22×Dic61—42no—
107C2×Dic122—24no—
108C2×Dic122—24no—
109C2×Dic122—24no—
110C2×Dic122—22no—
111C2×Dic122—42no—
112C2×Dic122—42no—
113C2×Dic122C2×D414yes—
114C2×C8×C61—24no—
115C2×C8×C61—24no—
116C2×C8×C61—24no—
117C42×C61—22no—
118C42×C61—22no—
119C42×C61—22no—
120C42×C61—22no—
121C22⋊Dic61—28no—
122C4⋊Dic61—24no—
123C4⋊Dic61—24no—
124C4⋊Dic61—24no—
125C4⋊Dic61—24no—
126C4⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4⋊Dic61—24no—
129C4⋊Dic61—42no—
130C4⋊Dic61C2×D414yes—
131C4⋊Dic61#4114yes—
132#11758196—28no—
133#11758196—28no—
134#11758196—44no—
135#11758196—44no—
136C4×Dic61—28no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61#12012yes—
142C23×C61C2×SD1612yes—
143C23×C61#13712yes—
144C23×C61C22×D411yescharacteristic
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—28no—
154C2×C4×C61—28no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—44no—
159C2×C4×C61#12012yes—
160C2×C4×C61C2×SD1612yes—
161C2×C4×C61C2×SD1612yes—
162C2×C4×C61#13712yes—
163C2×C4×C61C22×D412yes—
164C2×C4×C61C2×#4112yes—
165C2×C4×C612−1+412yes—
166C2×Dic61—24no—
167C2×Dic61—24no—
168C2×Dic61—24no—
169C2×Dic61—48no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—84no—
180Dic122—28no—
181Dic122—48no—
182Dic122—44no—
183C8×C61—44no—
184#24063—611nomaximal, Sylow
185C22×C61—22no—
186C22×C61—22no—
187C22×C61—22no—
188C22×C61—22no—
189C22×C61—22no—
190C22×C61—22no—
191C22×C61#44814yes—
192C22×C61C2×#12011yescharacteristic
193C22×C61#52511yescharacteristic
194C22×C61#53611yescharacteristic
195C22×C61C22×SD1611yescharacteristic
196C22×C61C2×#13711yescharacteristic
197C22×C61C2×#13711yescharacteristic
198C22×C61#57511yescharacteristic
199C4×C61—22no—
200C4×C61—22no—
201C4×C61—22no—
202C4×C61—22no—
203C4×C61—41no—
204C4×C61#54714yes—
205Dic61—44no—
206Dic61—44no—
207Dic61—44no—
208Dic61—82no—
209C2×#533—611no—
210C2×#483—612no—
211C2×#389—611no—
212#2058—612no—
213#2043—612no—
214#1872—612no—
215#1865—614no—
216#1817D6111yescharacteristic
217C2×C61—24no—
218C2×C61#240611yescharacteristic
219C2×C61C2×#44811yescharacteristic
220C2×C61#300711yescharacteristic
221C2×C61#310111yescharacteristic
222C2×C61#311311yescharacteristic
223C2×C61#322911yescharacteristic
224C2×C61#323611yescharacteristic
225#533—1228no—
226C2×C4⋊Q8—611no—
227C2×C4⋊Q8—611no—
228C2×#125—1221no—
229C2×C22⋊Q8—612no—
230C2×C22⋊Q8—1221no—
231C2×C4×Q8—612no—
232C2×C4×Q8—1221no—
233C2×#117—1221no—
234C22×C4⋊C4—611no—
235C22×C4⋊C4—611no—
236C22×C4⋊C4—1221no—
237C22×C42C2×D6111yescharacteristic
238#483—1228no—
239C2×#106—614no—
240C2×#103—614no—
241C2×#103—614no—
242C2×C22⋊C8C2×D6112yes—
243#389—612no—
244#389—612no—
245#389—1221no—
246#389—1221no—
247C2×#95—1221no—
248#340—614no—
249#336C2×D6112yes—
250#336C2×D6112yes—
251C61#2406311yescharacteristic, complemented, Sylow
252C22×Q8—612no—
253C22×Q8—1221no—
254C23×C4—1221no—
255C23×C4—1221no—
256C23×C4D24411yescharacteristic
257C23×C4D24411yescharacteristic
258C23×C4C22×D6111yescharacteristic
259C22×C8C22×D6114yes—
260C4⋊Q8—1224no—
261C4⋊Q8—1224no—
262#125—1228no—
263C22⋊Q8—618no—
264C22⋊Q8—1228no—
265C22⋊Q8—1224no—
266C4×Q8—1224no—
267C4×Q8—1224no—
268C4×Q8—1228no—
269#117—1228no—
270C2×C4⋊C4—612no—
271C2×C4⋊C4—612no—
272C2×C4⋊C4—612no—
273C2×C4⋊C4—1224no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4—1222no—
278C2×C4⋊C4—1222no—
279C2×C4⋊C4—1221no—
280C2×C4⋊C4—1224no—
281C2×C4⋊C4—1221no—
282C2×C4⋊C4—1222no—
283C2×C4⋊C4—1222no—
284C2×C4⋊C4—2444no—
285C2×C22⋊C4—1222no—
286C2×C42—24no—
287C2×C42—1222no—
288C2×C42D24411yescharacteristic
289C2×C42D24411yescharacteristic
290C2×C42C22×D6111yescharacteristic
291C2×C42C22×D6111yescharacteristic
292#106—1228no—
293#103—1228no—
294#103—1228no—
295C22⋊C8—24no—
296#95—1222no—
297#95—1222no—
298C24C2×D24411yescharacteristic
299C2×Q8—614no—
300C2×Q8—1224no—
301C2×Q8—1222no—
302C2×Q8—1224no—
303C2×Q8—1224no—
304C2×Q8—2442no—
305C2×Q8—2442no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—1224no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4—1222no—
317C22×C4—1222no—
318C22×C4—2442no—
319C22×C4—2442no—
320C22×C4—2442no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4#1175819014yes—
324C22×C4C2×D24411yescharacteristic
325C22×C4C2×D24411yescharacteristic
326C22×C4C2×D24411yescharacteristic
327C22×C4C2×D24411yescharacteristic
328C22×C4C2×D24411yescharacteristic
329C22×C4D4×D6114yes—
330C22×C4C23×D6111yescharacteristic
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C4⋊C4—614no—
335C4⋊C4—614no—
336C4⋊C4—1224no—
337C4⋊C4—1228no—
338C4⋊C4—1224no—
339C4⋊C4—1224no—
340C4⋊C4—1224no—
341C4⋊C4—1222no—
342C4⋊C4—1222no—
343C4⋊C4—1228no—
344C4⋊C4—1224no—
345C4⋊C4—2442no—
346C4⋊C4—2444no—
347C4⋊C4—2444no—
348C22⋊C4—1228no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—1228no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23#1451974712yes—
359C23C2×#1175819012yes—
360C23#1451981012yes—
361C23C22×D24411yescharacteristic
362Q8—1228no—
363Q8—2448no—
364Q8—2444no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—44no—
379C2×C4—1224no—
380C2×C4—1224no—
381C2×C4—1224no—
382C2×C4—4884no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2448no—
387C2×C4—2442no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4#1451974712yes—
394C2×C4C2×#1175819012yes—
395C2×C4C2×#1175819012yes—
396C2×C4#1451981012yes—
397C2×C4C2×D4×D6112yes—
398C2×C4#41×D6112yes—
399C2×C4#1451982412yes—
400C8—44no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22#1683990314yes—
408C22C2×#1451974711yescharacteristic
409C22#1684027411yescharacteristic
410C22#1684033811yescharacteristic
411C22C22×#1175819011yescharacteristic
412C22C2×#1451981011yescharacteristic
413C22C2×#1451981011yescharacteristic
414C22#1684056211yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—41no—
420C4—4882no—
421C4—2444no—
422C4—2444no—
423C4—2444no—
424C4#1684041014yes—
425C2—24no—
426C2#2144915611yescharacteristic
427C2C2×#1683990311yescharacteristic
428C2#2145508111yescharacteristic
429C2#2145582811yescharacteristic
430C2#2145595011yescharacteristic
431C2#2145696511yescharacteristic
432C2#2145702711yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1817C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1817D61
hypocenterC61#24063
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
4. #21447153C2semidirect
5. #21447066C2semidirect
7. #21446090C2semidirect
251. C61#24063semidirect

Maximal subgroups

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

subgroupindex
C2×#168403192
C2×#168400932
C2×#168396602
#214471532
#214470662
#214461242
#214460902
C61×#18172
#2406361