Finite Group Databasebeta

← #29658228

Subgroups — #29658228

Order 15616.

932 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C211yesmaximal, characteristic, complemented
2#21456741C211yesmaximal, characteristic, complemented
3#21456212C211yesmaximal, characteristic, complemented
4#21445026C214yesmaximal, complemented
5#21445019C214yesmaximal, complemented
6#21444481C211yesmaximal, characteristic, complemented
7#21444240C211yesmaximal, characteristic, complemented
8#21444236C211yesmaximal, characteristic, complemented
9C61×#3548C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840158—21no—
12#16840134—21no—
13#16840121—22no—
14#16840114—21no—
15#16840112—21no—
16#16840112C2212yes—
17#16840108C2212yes—
18#16840090—22no—
19#16840047C2214yes—
20#16839967C2214yes—
21#16839900—22no—
22#16839815C2211yescharacteristic
23#16839815C2211yescharacteristic
24#16839763—22no—
25C4×#11758194C2211yescharacteristic
26#16839511C2212yes—
27#16839506C2212yes—
28Q8⋊#92418C2214yes—
29D4⋊#92418C2214yes—
30C61×#562—22no—
31C61×#554—21no—
32C61×#554C2212yes—
33C61×#550—21no—
34C61×#550C2212yes—
35C61×#547—21no—
36C61×#547—21no—
37C61×#540—22no—
38C61×#537—22no—
39C61×#537—22no—
40C61×#533C2211yescharacteristic
41C61×#532C2211yescharacteristic
42C61×#532C2211yescharacteristic
43C22×Dic122—21no—
44C2×#11758200—24no—
45C2×#11758194—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777C2312yes—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52#14519708—24no—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519701—24no—
56#14519701—24no—
57#14519700—24no—
58#14519698—24no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61#14519688—21no—
62C4×#92418—21no—
63C4×#92418—21no—
64C4×#92418C2312yes—
65C2×#41×C61—22no—
66C2×#41×C61—22no—
67C2×#41×C61—21no—
68C2×#41×C61D412yes—
69C4⋊Q8×C61D411yescharacteristic
70C4⋊Q8×C61D411yescharacteristic
71C4⋊Q8×C61D411yescharacteristic
72C4⋊Q8×C61C2311yescharacteristic
73#127×C61—22no—
74#127×C61D411yescharacteristic
75#127×C61C2311yescharacteristic
76#125×C61—22no—
77#125×C61—21no—
78#124×C61—21no—
79#124×C61D412yes—
80#123×C61—24no—
81#123×C61—24no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61D412yes—
88#121×C61—24no—
89#121×C61—24no—
90#121×C61—22no—
91#121×C61—24no—
92#121×C61—24no—
93#121×C61—21no—
94#121×C61—21no—
95#121×C61—21no—
96#121×C61—22no—
97#121×C61D412yes—
98C4×Q8×C61—21no—
99C4×Q8×C61C2312yes—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61C2312yes—
106#117×C61—21no—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61—22no—
109C2×C42×C61C2311yescharacteristic
110#41×C61—44no—
111#41×C61—82no—
112#41×C61—84no—
113#41×C61—84no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C2×D412yes—
117C2×Q8×C61C2×D412yes—
118C2×D4×C61—24no—
119C2×D4×C61—24no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61C2×D412yes—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—24no—
129C22×C4×C61—24no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×Dic61—41no—
136C2×Dic122—22no—
137C2×Dic122—41no—
138C2×Dic122—82no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C2×D412yes—
150C4⋊C4×C61C2×D412yes—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C42×C61—22no—
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2411yescharacteristic
163#11758200—84no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—41no—
167C4⋊Dic61—41no—
168#11758194—42no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61#12011yescharacteristic
175Q8×C61—24no—
176Q8×C61—42no—
177Q8×C61—42no—
178Q8×C61—81no—
179D4×C61—24no—
180D4×C61—42no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—81no—
185D4×C61—82no—
186D4×C61—82no—
187D4×C61—84no—
188D4×C61—84no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—44no—
197C2×C4×C61—44no—
198C2×C4×C61—84no—
199C2×C4×C61—84no—
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×Dic61—42no—
207C2×Dic61—82no—
208Dic122—44no—
209Dic122—82no—
210#92418—42no—
211#92418—42no—
212#92418—42no—
213#18723—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—42no—
218C22×C61—42no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61C2×#12011yescharacteristic
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61—42no—
228C4×C61—42no—
229C4×C61—42no—
230C4×C61—42no—
231C4×C61#45512yes—
232C4×C61#57712yes—
233C4×C61#57712yes—
234Dic61—82no—
235#3548D6111yescharacteristic
236#3229—611no—
237#3206—611no—
238#3141—611no—
239#1643—614no—
240#1642—614no—
241#1543—611no—
242#1485—611no—
243#1484—611no—
244C2×C61—41no—
245C2×C61—82no—
246C2×C61—82no—
247C2×C61#300911yescharacteristic
248C2×C61#301011yescharacteristic
249C2×C61C2×#45511yescharacteristic
250#562—22no—
251#554—21no—
252#554C2×D6112yes—
253#550—21no—
254#550C2×D6112yes—
255#547—21no—
256#547—21no—
257#540—22no—
258#537—22no—
259#537—22no—
260#533C2×D6111yescharacteristic
261#532C2×D6111yescharacteristic
262#532C2×D6111yescharacteristic
263C2×C4⋊Q8—611no—
264#498—1221no—
265#492—1221no—
266#489—1222no—
267#487—612no—
268#487—1221no—
269#487—1221no—
270#486—612no—
271#483—1222no—
272#475—614no—
273#461—614no—
274#448—1222no—
275#431—611no—
276#431—611no—
277#417—1222no—
278C4×M16—611no—
279#333—612no—
280#331—612no—
281Q8⋊C8—614no—
282D4⋊C8—614no—
283C61#1872311yescharacteristic, complemented, Sylow
284C2×#41—22no—
285C2×#41—22no—
286C2×#41—21no—
287C2×#41C22⋊D6112yes—
288C22×Q8—1221no—
289C2×SD16—1224no—
290C2×M16—1221no—
291C4⋊Q8—612no—
292C4⋊Q8—1221no—
293C4⋊Q8—1221no—
294C4⋊Q8—1221no—
295C4⋊Q8C22⋊D6111yescharacteristic
296C4⋊Q8C22⋊D6111yescharacteristic
297C4⋊Q8C22⋊D6111yescharacteristic
298C4⋊Q8C22×D6111yescharacteristic
299#127—22no—
300#127C22⋊D6111yescharacteristic
301#127C22×D6111yescharacteristic
302#125—22no—
303#125—21no—
304#124—21no—
305#124C22⋊D6112yes—
306#123—24no—
307#123—24no—
308C22⋊Q8—22no—
309C22⋊Q8—24no—
310C22⋊Q8—22no—
311C22⋊Q8—24no—
312C22⋊Q8—21no—
313C22⋊Q8C22⋊D6112yes—
314#121—24no—
315#121—24no—
316#121—22no—
317#121—24no—
318#121—24no—
319#121—21no—
320#121—21no—
321#121—21no—
322#121—22no—
323#121C22⋊D6112yes—
324C4×Q8—21no—
325C4×Q8C22×D6112yes—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4C22×D6112yes—
332#117—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C42C22×D6111yescharacteristic
338#106—1224no—
339C4⋊C8—612no—
340C4⋊C8—612no—
341#103—1224no—
342#103—1224no—
343#103—1224no—
344#102—1224no—
345C22⋊C8—1221no—
346C22⋊C8—1221no—
347#97—1221no—
348C4×C8—612no—
349C4×C8—1221no—
350C4×C8—1221no—
351#41—44no—
352#41—82no—
353#41—84no—
354#41—84no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—1222no—
358C2×Q8—4882no—
359C2×Q8—2441no—
360C2×Q8C2×C22⋊D6112yes—
361C2×Q8C2×C22⋊D6112yes—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4C2×C22⋊D6112yes—
371C2×D4C2×C22⋊D6112yes—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—2441no—
377C22×C4C2×C22⋊D6111yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379C22×C4C2×C22⋊D6111yescharacteristic
380SD16—4884no—
381M16—2442no—
382C2×C8—1222no—
383C2×C8—1222no—
384C2×C8—1222no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—1222no—
396C4⋊C4—1222no—
397C4⋊C4—2441no—
398C4⋊C4—2441no—
399C4⋊C4C2×C22⋊D6112yes—
400C4⋊C4C2×C22⋊D6112yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C42—22no—
409C42C2×C22⋊D6111yescharacteristic
410C42C2×C22⋊D6111yescharacteristic
411C42C2×C22⋊D6111yescharacteristic
412C42C23×D6111yescharacteristic
413C23—22no—
414C23—22no—
415C23#1451974611yescharacteristic
416Q8—24no—
417Q8—42no—
418Q8—42no—
419Q8—81no—
420Q8—4882no—
421Q8—2444no—
422D4—24no—
423D4—42no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—81no—
428D4—82no—
429D4—82no—
430D4—84no—
431D4—84no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—4882no—
444C2×C4—2442no—
445C2×C4#1451974611yescharacteristic
446C2×C4#1451974611yescharacteristic
447C2×C4#1451974611yescharacteristic
448C2×C4C22×C22⋊D6111yescharacteristic
449C2×C4C22×C22⋊D6111yescharacteristic
450C2×C4C22×C22⋊D6111yescharacteristic
451C8—2442no—
452C8—2442no—
453C8—2442no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22C2×#1451974611yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—4882no—
472C4#1683994012yes—
473C4#1684057012yes—
474C4#1684057312yes—
475C2—41no—
476C2—82no—
477C2—82no—
478C2#2145509611yescharacteristic
479C2#2145510511yescharacteristic
480C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3548C2
socleC22×C61C2×#120
hypercenter#3548D61
hypocenterC61#18723
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
2. #21456741C2semidirect
3. #21456212C2semidirect
4. #21445026C2semidirect
5. #21445019C2semidirect
6. #21444481C2semidirect
7. #21444240C2semidirect
8. #21444236C2semidirect
283. C61#18723semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214567412
#214562122
#214450262
#214450192
#214444812
#214442402
#214442362
C61×#35482
#1872361