Finite Group Databasebeta

← #29832877

Subgroups — #29832877

Order 15616.

1176 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459601C211yesmaximal, characteristic
2#21454300C212yesmaximal, complemented
3#21447514C211yesmaximal, characteristic, complemented
4#21447507C214yesmaximal, complemented
5#21447502C212yesmaximal, complemented
6#356×D61C214yesmaximal, complemented
7C61×#2135C211yesmaximal, characteristic, complemented
8C2×#14519825C411yescharacteristic
9C2×#14519824C2211yescharacteristic
10#16840528—22no—
11#16840425—22no—
12#16840414—22no—
13#16840410—24no—
14#16840338C411yescharacteristic
15#16840337C2212yes—
16#16840332C412yes—
17#16840311C412yes—
18#16840308C2211yescharacteristic
19C2×#14519775C411yescharacteristic
20C4⋊Q8×D61C2212yes—
21#16840230C411yescharacteristic
22#16839746C2212yes—
23#16839743—24no—
24C2×#14519695—22no—
25#101×D61C2218yes—
26#16839561C2214yes—
27#16839560C2214yes—
28#16839558C2214yes—
29C61×#536C2211yescharacteristic
30C61×#413C2212yes—
31C61×#356C2214yes—
32#14519825—44no—
33#14519824—44no—
34C2×#11758226—21no—
35C2×#11758226C2×C412yes—
36C2×#11758222—22no—
37C2×#11758222C2×C411yescharacteristic
38C2×#11758222C2×C411yescharacteristic
39C2×#11758222D412yes—
40C2×#11758224C2×C412yes—
41C2×#11758224D414yes—
42C22×Dic122C2×C411yescharacteristic
43C22×Dic122D412yes—
44C2×Q8×D61—21no—
45C2×Q8×D61C2×C412yes—
46C2×Q8×D61C2312yes—
47#14519785—44no—
48#14519783—44no—
49C2×#11758194—22no—
50M16×D61—28no—
51#14519777C2×C412yes—
52#14519776—24no—
53#14519775—22no—
54#14519775C2×C412yes—
55#14519775C2×C412yes—
56#14519775C2311yescharacteristic
57#14519773C2×C411yescharacteristic
58#14519766C2311yescharacteristic
59#14519765C2×C412yes—
60#14519764C2×C411yescharacteristic
61#14519762—24no—
62#14519761—22no—
63#14519759—24no—
64#14519757—22no—
65#14519755—24no—
66#14519754—24no—
67#14519750—24no—
68#14519749—22no—
69#14519742—21no—
70#14519739—21no—
71C4×C22⋊D61—22no—
72#14519737—22no—
73#14519737—24no—
74#14519734C2×C412yes—
75C2×#11758196—22no—
76C4⋊C4×D61—24no—
77C2×C4×Dic61C2×C411yescharacteristic
78C42×D61C2×C412yes—
79#14519696C2314yes—
80#14519695—28no—
81#14519694—22no—
82C2×#41×C61C2311yescharacteristic
83C22×Q8×C61C2×C411yescharacteristic
84C2×M16×C61—22no—
85C4⋊Q8×C61C2312yes—
86#124×C61C2×C412yes—
87#123×C61—22no—
88C22⋊Q8×C61—22no—
89#117×C61C2×C411yescharacteristic
90#101×C61C2314yes—
91#100×C61—22no—
92#41×C61—42no—
93C2×Q8×C61—21no—
94C2×Q8×C61—42no—
95C2×Q8×C61C22×C412yes—
96C2×Q8×C61C2411yescharacteristic
97C2×D4×C61C22×C411yescharacteristic
98C2×D4×C61C2×D412yes—
99C22×C4×C61—21no—
100C22×C4×C61C22×C411yescharacteristic
101C22×C4×C61C2×D412yes—
102C2×C22⋊D61—21no—
103C2×C22⋊D61—22no—
104C2×C22⋊D61C22⋊C414yes—
105C2×C22⋊D61C22⋊C412yes—
106C22×Dic61—21no—
107C22×Dic61C22⋊C412yes—
108C22×Dic61C22⋊C414yes—
109#11758226—44no—
110#11758226—44no—
111Q8×D61—44no—
112Q8×D61—44no—
113Q8×D61—44no—
114#11758224—44no—
115#11758224—48no—
116#11758222—24no—
117#11758222—24no—
118#11758222—24no—
119#11758222—48no—
120#11758222—42no—
121C2×D244—22no—
122C2×D244C22⋊C411yescharacteristic
123C2×D244C22×C411yescharacteristic
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—21no—
127C2×C4×D61—22no—
128C2×C4×D61C22×C412yes—
129C2×C4×D61C2×D414yes—
130C2×Dic122—22no—
131C2×Dic122—42no—
132C2×Dic122—42no—
133C2×Dic122—42no—
134C2×Dic122C22⋊C411yescharacteristic
135C2×Dic122C22⋊C412yes—
136C2×Dic122C22×C411yescharacteristic
137C2×Dic122C22×C412yes—
138C2×Dic122C2×D414yes—
139M16×C61—24no—
140M16×C61—42no—
141C2×C8×C61—42no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—24no—
144C22⋊C4×C61—21no—
145C22⋊C4×C61—24no—
146C42×C61C22×C412yes—
147C22⋊Dic61—22no—
148#11758198—22no—
149#11758198—22no—
150#11758198—24no—
151C4⋊Dic61—24no—
152#11758196—22no—
153#11758196—24no—
154#11758196—24no—
155#11758196—22no—
156C4×Dic61—21no—
157C4×Dic61—22no—
158C4×Dic61C22×C411yescharacteristic
159C4×Dic61C22×C411yescharacteristic
160#11758194—24no—
161#11758194—42no—
162C2×#92418—42no—
163#11758189—48no—
164C8×D61—48no—
165C23×C61C2×C22⋊C412yes—
166C23×C61C2×C22⋊C411yescharacteristic
167C22×D61—21no—
168C22×D61C2×C22⋊C412yes—
169Q8×C61—41no—
170Q8×C61—42no—
171Q8×C61—42no—
172D4×C61—41no—
173D4×C61—42no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—42no—
180C2×C4×C61C2×C22⋊C411yescharacteristic
181C2×C4×C61C23×C411yescharacteristic
182C2×C4×C61C22×D412yes—
183C22⋊D61—24no—
184C22⋊D61—28no—
185C22⋊D61—44no—
186C22⋊D61—42no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—44no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61C2×C22⋊C412yes—
194C2×Dic61C2×C22⋊C414yes—
195D244—41no—
196D244—44no—
197D244#9914yes—
198C4×D61—28no—
199C4×D61—24no—
200C4×D61—44no—
201C4×D61—44no—
202C4×D61—44no—
203C4×D61—42no—
204Dic122—22no—
205Dic122—22no—
206Dic122—22no—
207Dic122—42no—
208Dic122—44no—
209Dic122—44no—
210Dic122—41no—
211C8×C61—44no—
212#92418—44no—
213#33372—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—22no—
216C22×C61—41no—
217C22×C61—42no—
218C22×C61C22×C22⋊C411yescharacteristic
219C2×D61—42no—
220C2×D61—42no—
221C2×D61—41no—
222C2×D61C2×#9914yes—
223C4×C61—22no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—42no—
227C4×C61C2×#9912yes—
228Dic61—22no—
229Dic61—22no—
230Dic61—24no—
231Dic61—41no—
232Dic61—41no—
233#3499—611no—
234#2889—612no—
235#2135—611no—
236#2135D6111yescharacteristic, complemented
237#2133—614no—
238#2132—612no—
239C2×#356—614no—
240C2×C61—21no—
241C2×C61—41no—
242C2×C61—42no—
243C2×C61C22×#9911yescharacteristic
244D61—24no—
245D61—81no—
246C2×2−1+4—611no—
247C2×2−1+4—611no—
248#568—1222no—
249D4×Q8—1222no—
250#547—1222no—
251#547—1224no—
252#536—612no—
253#536—612no—
254#536—611no—
255#536C2×D6111yescharacteristic
256#532—611no—
257#532—612no—
258C2×C4⋊Q8—611no—
259C2×C4⋊Q8—612no—
260C4×#41—611no—
261#413—612no—
262#413—1224no—
263#413C2×D6112yes—
264C2×#101—618no—
265C2×#101—1222no—
266#356—614no—
267#356—614no—
268#356C2×D6114yes—
269#355—614no—
270C61#3337211yescharacteristic, complemented, Sylow
2712−1+4—2444no—
2722−1+4—2444no—
273C2×#41—612no—
274C2×#41—614no—
275C2×#41—611no—
276C2×#41—612no—
277C2×#41—612no—
278C2×#41—611no—
279C2×#41—1222no—
280C2×#41—1221no—
281C2×#41C22×D6111yescharacteristic
282C22×Q8—612no—
283C22×Q8—611no—
284C22×Q8—612no—
285C22×Q8—612no—
286C22×Q8—1221no—
287C22×Q8C4×D6111yescharacteristic
288#131—2444no—
289#131—2444no—
290C2×M16—22no—
291C2×M16—1228no—
292C2×M16—1222no—
293C4⋊Q8—612no—
294C4⋊Q8—612no—
295C4⋊Q8—611no—
296C4⋊Q8—612no—
297C4⋊Q8—1224no—
298C4⋊Q8—1222no—
299C4⋊Q8C22×D6112yes—
300#127—611no—
301#124—611no—
302#124—611no—
303#124—612no—
304#124C4×D6112yes—
305#123—22no—
306#123—1222no—
307#123—1224no—
308#123—1224no—
309C22⋊Q8—22no—
310C22⋊Q8—1224no—
311C22⋊Q8—1224no—
312C22⋊Q8—1222no—
313#121—1224no—
314#121—1222no—
315C4×Q8—1221no—
316C4×D4—1224no—
317C4×D4—1222no—
318C4×D4—1221no—
319C4×D4—1222no—
320#117—612no—
321#117C4×D6111yescharacteristic
322C2×C4⋊C4—1224no—
323C2×C4⋊C4—1222no—
324C2×C42—611no—
325C2×C42—612no—
326#101—614no—
327#101—1228no—
328#101C22×D6114yes—
329#100—22no—
330#100—1222no—
331#41—42no—
332#41—1224no—
333#41—1224no—
334#41—1224no—
335#41—2442no—
336#41—2448no—
337#41—2448no—
338#41—2444no—
339#41—2444no—
340#41—2444no—
341C2×Q8—21no—
342C2×Q8—42no—
343C2×Q8—612no—
344C2×Q8—614no—
345C2×Q8—611no—
346C2×Q8—612no—
347C2×Q8—611no—
348C2×Q8—1222no—
349C2×Q8—2442no—
350C2×Q8—2442no—
351C2×Q8—2442no—
352C2×Q8—2444no—
353C2×Q8—2444no—
354C2×Q8—2444no—
355C2×Q8C2×C4×D6112yes—
356C2×Q8C23×D6111yescharacteristic
357C2×D4—612no—
358C2×D4—614no—
359C2×D4—611no—
360C2×D4—611no—
361C2×D4—1222no—
362C2×D4—1222no—
363C2×D4—1221no—
364C2×D4C2×C4×D6111yescharacteristic
365C2×D4D4×D6112yes—
366C22×C4—21no—
367C22×C4—612no—
368C22×C4—614no—
369C22×C4—612no—
370C22×C4—614no—
371C22×C4—1222no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4C2×C4×D6111yescharacteristic
377C22×C4D4×D6112yes—
378M16—24no—
379M16—42no—
380M16—1224no—
381M16—2448no—
382M16—2442no—
383C2×C8—42no—
384C2×C8—2448no—
385C2×C8—2442no—
386C4⋊C4—24no—
387C4⋊C4—21no—
388C4⋊C4—1224no—
389C4⋊C4—1224no—
390C4⋊C4—1224no—
391C4⋊C4—1222no—
392C4⋊C4—1222no—
393C22⋊C4—21no—
394C22⋊C4—24no—
395C22⋊C4—1224no—
396C22⋊C4—1222no—
397C22⋊C4—1222no—
398C22⋊C4—1222no—
399C42—611no—
400C42—611no—
401C42—1222no—
402C42—1221no—
403C42C2×C4×D6112yes—
404C23—612no—
405C23—1221no—
406C23C22⋊C4×D6112yes—
407C23C22⋊C4×D6111yescharacteristic
408Q8—41no—
409Q8—42no—
410Q8—42no—
411Q8—1222no—
412Q8—1222no—
413Q8—1222no—
414Q8—2444no—
415Q8—2442no—
416Q8—2441no—
417Q8—2444no—
418D4—41no—
419D4—42no—
420D4—614no—
421D4—1228no—
422D4—1224no—
423D4—2441no—
424D4—2444no—
425D4—2444no—
426D4—2442no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—42no—
433C2×C4—612no—
434C2×C4—614no—
435C2×C4—1228no—
436C2×C4—1224no—
437C2×C4—1221no—
438C2×C4—1221no—
439C2×C4—1221no—
440C2×C4—2444no—
441C2×C4—2444no—
442C2×C4—2444no—
443C2×C4—2444no—
444C2×C4—2442no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4C22⋊C4×D6111yescharacteristic
448C2×C4C22×C4×D6111yescharacteristic
449C2×C4C2×D4×D6112yes—
450C8—44no—
451C8—2444no—
452C22—21no—
453C22—22no—
454C22—41no—
455C22—42no—
456C22—614no—
457C22—2442no—
458C22—2442no—
459C22—2441no—
460C22C2×C22⋊C4×D6111yescharacteristic
461C4—22no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—1222no—
466C4—1222no—
467C4—1224no—
468C4—2441no—
469C4—2441no—
470C4#99×D6112yes—
471C2—21no—
472C2—41no—
473C2—42no—
474C2—1224no—
475C2—4881no—
476C2C2×#99×D6111yescharacteristic
subgroupG/H
centerC2C2×#99×D61
commutatorC2×C4×C61C23×C4
FrattiniC2×Q8C23×D61
FittingC61×#2135C2
socleC2×C61C22×#99
prime radical#21459601C2
cosocleC2×Q8×C61C24
hypercenter#2135D61
hypocenterC61#33372
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2135

Complements

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

NH ≅ G/Ntype
2. #21454300C2semidirect
3. #21447514C2semidirect
4. #21447507C2semidirect
5. #21447502C2semidirect
6. #356×D61C2semidirect
7. C61×#2135C2semidirect
236. #2135D61semidirect
270. C61#33372semidirect

Maximal subgroups

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

subgroupindex
#214596012
#214543002
#214475142
#214475072
#214475022
#356×D612
C61×#21352
#3337261