Finite Group Databasebeta

← #29653095

Subgroups — #29653095

Order 15616.

914 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455854C211yesmaximal, characteristic, complemented
2#21455846C211yesmaximal, characteristic
3#21451181C211yesmaximal, characteristic, complemented
4#21444850C214yesmaximal, complemented
5#21444782C214yesmaximal, complemented
6#21444383C211yesmaximal, characteristic, complemented
7#21444379C211yesmaximal, characteristic
8#21444143C211yesmaximal, characteristic, complemented
9C61×#3507C211yesmaximal, characteristic
10C2×C4×Dic122C2211yescharacteristic
11#16840095C2212yes—
12#16840078C2212yes—
13#16840054—21no—
14#16840044—21no—
15#16839969—21no—
16#16839959—21no—
17#16839917C2212yes—
18#16839900C2212yes—
19C4×#11758202—22no—
20C4×#11758200—22no—
21#16839815C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839701C2212yes—
24#16839682C2212yes—
25#16839669—21no—
26#16839658—21no—
27#16839646—22no—
28#16839628C2211yescharacteristic
29#16839499C2214yes—
30#16839496C2214yes—
31#16839488C2214yes—
32C61×#559—21no—
33C61×#557—21no—
34C61×#556—21no—
35C61×#556—22no—
36C61×#554—21no—
37C61×#552—22no—
38C61×#539C2214yes—
39C61×#536—21no—
40C61×#533C2211yescharacteristic
41C61×#532C2211yescharacteristic
42C61×#529—22no—
43C61×#520—21no—
44C2×C4×Q8×C61C2211yescharacteristic
45C22×Dic122C2311yescharacteristic
46C2×#11758202—22no—
47C2×#11758200—22no—
48C4×Dic122—21no—
49C4×Dic122—21no—
50C4×Dic122—21no—
51C4×Dic122—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×#11758196—22no—
55C2×#11758196—21no—
56C2×C4×Dic61—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519701—22no—
62#14519700—22no—
63#14519700—22no—
64#14519698—22no—
65C22⋊#92418C2312yes—
66C22⋊#92418C2312yes—
67C4×#92418—21no—
68C4×#92418—21no—
69#14519684—22no—
70#14519684—22no—
71#14519684—22no—
72#14519684—21no—
73#14519684C2312yes—
74C2×#41×C61—21no—
75C22×Q8×C61D411yescharacteristic
76C4⋊Q8×C61—22no—
77C4⋊Q8×C61D411yescharacteristic
78C4⋊Q8×C61D411yescharacteristic
79C4⋊Q8×C61D411yescharacteristic
80#127×C61D411yescharacteristic
81#126×C61—24no—
82#126×C61—24no—
83#125×C61—22no—
84#125×C61—24no—
85#125×C61—21no—
86#124×C61—22no—
87#124×C61—22no—
88#124×C61—21no—
89#123×C61—24no—
90#123×C61—24no—
91C22⋊Q8×C61—22no—
92C22⋊Q8×C61—22no—
93C22⋊Q8×C61—24no—
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61C2312yes—
96#121×C61—21no—
97#121×C61C2312yes—
98C4×Q8×C61—22no—
99C4×Q8×C61—22no—
100C4×Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×Q8×C61—21no—
103C4×Q8×C61—21no—
104C4×Q8×C61—21no—
105C4×Q8×C61D412yes—
106C4×D4×C61—22no—
107C4×D4×C61—22no—
108C4×D4×C61—24no—
109C4×D4×C61—21no—
110C4×D4×C61D412yes—
111#117×C61—22no—
112#117×C61—24no—
113#117×C61—21no—
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61C2312yes—
117C2×C42×C61D412yes—
118C2×C42×C61C2311yescharacteristic
119#41×C61—42no—
120#41×C61—42no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61C2×D412yes—
128C2×D4×C61—22no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C2×D412yes—
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—42no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61C2×D412yes—
159C4⋊C4×C61C2×D412yes—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C42×C61—24no—
167C42×C61—22no—
168C42×C61—22no—
169C42×C61—22no—
170C42×C61C2×D411yescharacteristic
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174#11758202—44no—
175#11758200—44no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—42no—
179#11758196—44no—
180#11758196—41no—
181#11758196—41no—
182C4×Dic61—41no—
183C4×Dic61—41no—
184C2×#92418—22no—
185C2×#92418—22no—
186C23×C61—22no—
187C23×C61C22×D411yescharacteristic
188Q8×C61—42no—
189Q8×C61—42no—
190Q8×C61—42no—
191Q8×C61—42no—
192Q8×C61—42no—
193D4×C61—42no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—44no—
208C2×C4×C61—42no—
209C2×C4×C61—84no—
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#13512yes—
215C2×C4×C61#13712yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226Dic122—22no—
227Dic122—22no—
228Dic122—41no—
229Dic122—41no—
230#92418—42no—
231#92418—42no—
232#18215—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61C2×#12011yescharacteristic
239C22×C61C2×#13511yescharacteristic
240C22×C61C2×#13711yescharacteristic
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—42no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61—42no—
250C4×C61#45212yes—
251C4×C61#45212yes—
252Dic61—42no—
253Dic61—82no—
254Dic61—81no—
255#3507D6111yescharacteristic
256#3104—611no—
257#3103—611no—
258#2581—611no—
259#1615—614no—
260#1603—614no—
261#1518—611no—
262#1517—611no—
263#1461—611no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61—82no—
267C2×C61C2×#45211yescharacteristic
268C2×C61#302011yescharacteristic
269C2×C61#302511yescharacteristic
270#559—21no—
271#557—21no—
272#556—21no—
273#556—22no—
274#554—21no—
275#552—22no—
276#539C2×D6114yes—
277#536—21no—
278#533C2×D6111yescharacteristic
279#532C2×D6111yescharacteristic
280#529—22no—
281#520—21no—
282C2×C4×Q8—611no—
283C2×C4×Q8C2×D6111yescharacteristic
284#484—612no—
285#481—612no—
286#477—1221no—
287#475—1221no—
288#462—1221no—
289#460—1221no—
290#451—612no—
291#448—612no—
292C4×Q16—1222no—
293C4×SD16—1222no—
294#431—611no—
295#431—611no—
296#400—612no—
297#395—612no—
298#391—1221no—
299#389—1221no—
300#387—1222no—
301#382—611no—
302#328—614no—
303#327—614no—
304#324—614no—
305C61#1821511yescharacteristic, complemented, Sylow
306C2×#41—21no—
307C22×Q8—611no—
308C22×Q8C22⋊D6111yescharacteristic
309C2×Q16—1222no—
310C2×SD16—1222no—
311C4⋊Q8—22no—
312C4⋊Q8C22⋊D6111yescharacteristic
313C4⋊Q8C22⋊D6111yescharacteristic
314C4⋊Q8C22⋊D6111yescharacteristic
315#127C22⋊D6111yescharacteristic
316#126—24no—
317#126—24no—
318#125—22no—
319#125—24no—
320#125—21no—
321#124—22no—
322#124—22no—
323#124—21no—
324#123—24no—
325#123—24no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—24no—
329C22⋊Q8—21no—
330C22⋊Q8C22×D6112yes—
331#121—21no—
332#121C22×D6112yes—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8—21no—
340C4×Q8—1221no—
341C4×Q8—1221no—
342C4×Q8—1221no—
343C4×Q8—1221no—
344C4×Q8C22⋊D6112yes—
345C4×D4—22no—
346C4×D4—22no—
347C4×D4—24no—
348C4×D4—21no—
349C4×D4C22⋊D6112yes—
350#117—22no—
351#117—24no—
352#117—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4C22⋊D6111yescharacteristic
359C2×C4⋊C4C22×D6112yes—
360C2×C42—1221no—
361C2×C42C22⋊D6112yes—
362C2×C42C22×D6111yescharacteristic
363#107—1222no—
364#106—1222no—
365C4⋊C8—1221no—
366C4⋊C8—1221no—
367#103—1222no—
368#103—1222no—
369#103—1222no—
370#102—1222no—
371C22⋊C8—612no—
372C22⋊C8—612no—
373C4×C8—1221no—
374C4×C8—1221no—
375#95—612no—
376#95—1221no—
377#95—1222no—
378#95—1222no—
379#95—1222no—
380#41—42no—
381#41—42no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—1221no—
389C2×Q8—1221no—
390C2×Q8—2442no—
391C2×Q8C2×C22⋊D6112yes—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4C2×C22⋊D6112yes—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—1221no—
399C22×C4—1222no—
400C22×C4—1221no—
401C22×C4C2×C22⋊D6112yes—
402C22×C4C2×C22⋊D6112yes—
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C23×D6111yescharacteristic
406Q16—2444no—
407SD16—2444no—
408C2×C8—1222no—
409C2×C8—1222no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—2444no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C4⋊C4—2442no—
429C4⋊C4C2×C22⋊D6112yes—
430C4⋊C4C2×C22⋊D6112yes—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C42—24no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—2441no—
442C42—2441no—
443C42C2×C22⋊D6111yescharacteristic
444C42C2×C22⋊D6111yescharacteristic
445C42C2×C22⋊D6111yescharacteristic
446C42C2×C22⋊D6111yescharacteristic
447C23—22no—
448C23C22×C22⋊D6111yescharacteristic
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—1222no—
455Q8—1222no—
456Q8—2441no—
457Q8—2441no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—84no—
475C2×C4—1222no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4#1451974611yescharacteristic
484C2×C4#1451974611yescharacteristic
485C2×C4#1451974611yescharacteristic
486C2×C4#1451974611yescharacteristic
487C2×C4#1451979712yes—
488C2×C4#1451980712yes—
489C2×C4C22×C22⋊D6111yescharacteristic
490C2×C4C22×C22⋊D6111yescharacteristic
491C8—2442no—
492C8—2442no—
493C22—21no—
494C22—21no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22C2×#1451974611yescharacteristic
499C22C2×#1451979711yescharacteristic
500C22C2×#1451980711yescharacteristic
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—4882no—
511C4—4881no—
512C4—2442no—
513C4#1683992412yes—
514C4#1683992412yes—
515C2—21no—
516C2—21no—
517C2—82no—
518C2C2×#1683992411yescharacteristic
519C2#2145518311yescharacteristic
520C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3507D61
hypocenterC61#18215
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
1. #21455854C2semidirect
3. #21451181C2semidirect
4. #21444850C2semidirect
5. #21444782C2semidirect
6. #21444383C2semidirect
8. #21444143C2semidirect
305. C61#18215semidirect

Maximal subgroups

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

subgroupindex
#214558542
#214558462
#214511812
#214448502
#214447822
#214443832
#214443792
#214441432
C61×#35072
#1821561