Finite Group Databasebeta

← #29656948

Subgroups — #29656948

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457013C211yesmaximal, characteristic
2#21456560C211yesmaximal, characteristic, complemented
3#21455858C211yesmaximal, characteristic, complemented
4#21445089C212yesmaximal, complemented
5#21445033C212yesmaximal, complemented
6#21444991C212yesmaximal, complemented
7#21444953C212yesmaximal, complemented
8#21444529C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic
11C61×#3507C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840168—21no—
14#16840158C2211yescharacteristic
15#16840141C2211yescharacteristic
16#16840126—21no—
17#16840114C2211yescharacteristic
18#16840112C2211yescharacteristic
19#16840097—21no—
20#16840083—22no—
21#16840057C2212yes—
22#16840052C2212yes—
23#16840042C2212yes—
24#16840018—22no—
25#16839972C2212yes—
26#16839918—21no—
27#16839867—22no—
28C4×#11758202—22no—
29#16839819C2211yescharacteristic
30#16839815C2211yescharacteristic
31C4×#11758194C2211yescharacteristic
32#16839511C2212yes—
33#16839508C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C61×#559—21no—
39C61×#557—21no—
40C61×#556—22no—
41C61×#556C2212yes—
42C61×#554C2212yes—
43C61×#552—22no—
44C61×#539—22no—
45C61×#536—21no—
46C61×#533C2211yescharacteristic
47C61×#532C2211yescharacteristic
48C61×#529—22no—
49C61×#520—21no—
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758202—22no—
52C2×#11758202—22no—
53C2×#11758194—21no—
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519769—21no—
57#14519756—21no—
58#14519756—21no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61#14519736—21no—
62#14519710—22no—
63#14519710—22no—
64#14519708—22no—
65#14519708—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519701—22no—
69#14519701—22no—
70#14519700—22no—
71#14519700—22no—
72#14519698—22no—
73#14519698—22no—
74C22⋊#92418—21no—
75C22⋊#92418—21no—
76#14519688—21no—
77#14519688—21no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C2×#41×C61—21no—
82C22×Q8×C61D411yescharacteristic
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61D411yescharacteristic
85C4⋊Q8×C61C2311yescharacteristic
86C4⋊Q8×C61C2311yescharacteristic
87#127×C61D411yescharacteristic
88#126×C61—24no—
89#126×C61—24no—
90#125×C61—24no—
91#125×C61—22no—
92#125×C61—21no—
93#124×C61—22no—
94#124×C61—22no—
95#124×C61—21no—
96#123×C61—24no—
97#123×C61—24no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61D412yes—
103#121×C61—21no—
104#121×C61D412yes—
105C4×Q8×C61—22no—
106C4×Q8×C61—22no—
107C4×Q8×C61—21no—
108C4×Q8×C61—21no—
109C4×Q8×C61—21no—
110C4×Q8×C61D412yes—
111C4×Q8×C61D412yes—
112C4×Q8×C61C2312yes—
113C4×D4×C61—24no—
114C4×D4×C61—22no—
115C4×D4×C61—22no—
116C4×D4×C61—21no—
117C4×D4×C61C2312yes—
118#117×C61—24no—
119#117×C61—21no—
120#117×C61—22no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—22no—
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#41×C61—44no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—42no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61C2×D412yes—
133C2×D4×C61—22no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61C2×D412yes—
137C22×C4×C61—21no—
138C22×C4×C61—24no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C2×Dic122—21no—
145C2×Dic122—21no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C2×D412yes—
159C4⋊C4×C61C2×D412yes—
160C4⋊C4×C61C2×D412yes—
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C42×C61—24no—
168C42×C61—22no—
169C42×C61—22no—
170C42×C61—22no—
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2411yescharacteristic
175C22⋊Dic61—41no—
176#11758202—44no—
177#11758202—44no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182#11758196—41no—
183#11758196—41no—
184C4×Dic61—41no—
185#11758194—42no—
186C2×#92418—21no—
187C2×#92418—22no—
188C2×#92418—21no—
189C2×#92418—22no—
190C23×C61—22no—
191C23×C61#12011yescharacteristic
192Q8×C61—22no—
193Q8×C61—22no—
194Q8×C61—42no—
195Q8×C61—42no—
196Q8×C61—42no—
197Q8×C61—42no—
198D4×C61—24no—
199D4×C61—42no—
200D4×C61—44no—
201D4×C61—81no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—44no—
212C2×C4×C61—81no—
213C2×C4×C61—84no—
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×Dic61—41no—
221C2×Dic61—41no—
222Dic122—42no—
223Dic122—42no—
224#92418—22no—
225#92418—41no—
226#92418—42no—
227#92418—42no—
228#18605—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61C2×#12011yescharacteristic
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—42no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61#57512yes—
246C4×C61#57812yes—
247Dic61—81no—
248Dic61—81no—
249#3507D6111yescharacteristic
250#3235—611no—
251#3185—611no—
252#3104—611no—
253#1652—612no—
254#1644—612no—
255#1638—612no—
256#1632—612no—
257#1554—611no—
258#1491—611no—
259#1484—611no—
260C2×C61—41no—
261C2×C61—82no—
262C2×C61#300511yescharacteristic
263C2×C61#301111yescharacteristic
264C2×C61#301811yescharacteristic
265#559—21no—
266#557—21no—
267#556—22no—
268#556C2×D6112yes—
269#554C2×D6112yes—
270#552—22no—
271#539—22no—
272#536—21no—
273#533—611no—
274#533C2×D6111yescharacteristic
275#532C2×D6111yescharacteristic
276#529—22no—
277#520—21no—
278C2×C4×Q8C2×D6111yescharacteristic
279#501—1221no—
280#498—611no—
281#494—611no—
282#490—1221no—
283#487—611no—
284#487—611no—
285#484—1221no—
286#482—1222no—
287#477—612no—
288#476—612no—
289#474—612no—
290#470—1222no—
291#462—612no—
292#451—1221no—
293#441—1222no—
294C4×Q16—1222no—
295#432—611no—
296#431—611no—
297C4×M16—611no—
298#333—612no—
299#332—612no—
300Q8⋊C8—612no—
301Q8⋊C8—612no—
302D4⋊C8—612no—
303D4⋊C8—612no—
304C61#1860511yescharacteristic, complemented, Sylow
305C2×#41—21no—
306C22×Q8C22⋊D6111yescharacteristic
307C2×Q16—1222no—
308C2×Q16—1222no—
309C2×M16—1221no—
310C4⋊Q8—22no—
311C4⋊Q8—611no—
312C4⋊Q8—611no—
313C4⋊Q8C22⋊D6111yescharacteristic
314C4⋊Q8C22×D6111yescharacteristic
315C4⋊Q8C22×D6111yescharacteristic
316#127C22⋊D6111yescharacteristic
317#126—24no—
318#126—24no—
319#125—24no—
320#125—22no—
321#125—21no—
322#125—1221no—
323#124—22no—
324#124—22no—
325#124—21no—
326#123—24no—
327#123—24no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—24no—
331C22⋊Q8—21no—
332C22⋊Q8—1221no—
333C22⋊Q8—1221no—
334C22⋊Q8C22⋊D6112yes—
335#121—21no—
336#121C22⋊D6112yes—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8—21no—
340C4×Q8—21no—
341C4×Q8—21no—
342C4×Q8—1221no—
343C4×Q8—1221no—
344C4×Q8C22⋊D6112yes—
345C4×Q8C22⋊D6112yes—
346C4×Q8C22×D6112yes—
347C4×D4—24no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—21no—
351C4×D4C22×D6112yes—
352#117—24no—
353#117—21no—
354#117—22no—
355#117—1221no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4C22⋊D6111yescharacteristic
359C2×C42—21no—
360C2×C42C22×D6111yescharacteristic
361#107—1222no—
362#107—1222no—
363#106—1222no—
364#106—1222no—
365C4⋊C8—612no—
366C4⋊C8—612no—
367#103—1222no—
368#103—1222no—
369#103—1222no—
370#103—1222no—
371#102—1222no—
372#102—1222no—
373C22⋊C8—1221no—
374C22⋊C8—1221no—
375#97—1221no—
376#97—1221no—
377C4×C8—611no—
378C4×C8—611no—
379C4×C8—1221no—
380#41—44no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—42no—
385C2×Q8—1221no—
386C2×Q8—1221no—
387C2×Q8C2×C22⋊D6112yes—
388C2×Q8C2×C22⋊D6112yes—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4C2×C22⋊D6112yes—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4C2×C22⋊D6111yescharacteristic
398C22×C4C2×C22⋊D6111yescharacteristic
399C22×C4C2×C22⋊D6111yescharacteristic
400Q16—2444no—
401Q16—2444no—
402M16—2442no—
403C2×C8—1221no—
404C2×C8—1221no—
405C2×C8—1222no—
406C2×C8—1222no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C4⋊C4C2×C22⋊D6112yes—
426C4⋊C4C2×C22⋊D6112yes—
427C4⋊C4C2×C22⋊D6112yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—2441no—
435C42—24no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—2441no—
440C42C2×C22⋊D6111yescharacteristic
441C42C2×C22⋊D6111yescharacteristic
442C42C2×C22⋊D6111yescharacteristic
443C42C23×D6111yescharacteristic
444C23—22no—
445C23#1451974611yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—2442no—
453Q8—2442no—
454D4—24no—
455D4—42no—
456D4—44no—
457D4—81no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—81no—
469C2×C4—84no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4#1451974611yescharacteristic
473C2×C4#1451974611yescharacteristic
474C2×C4#1451974611yescharacteristic
475C2×C4C22×C22⋊D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C22×C22⋊D6111yescharacteristic
478C8—1222no—
479C8—2441no—
480C8—2442no—
481C8—2442no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22C2×#1451974611yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—4881no—
499C4—4881no—
500C4#1684056012yes—
501C4#1684057712yes—
502C2—41no—
503C2—82no—
504C2#2145506111yescharacteristic
505C2#2145511411yescharacteristic
506C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3507C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3507D61
hypocenterC61#18605
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3507

Complements

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

NH ≅ G/Ntype
2. #21456560C2semidirect
3. #21455858C2semidirect
4. #21445089C2semidirect
5. #21445033C2semidirect
6. #21444991C2semidirect
7. #21444953C2semidirect
8. #21444529C2semidirect
9. #21444266C2semidirect
304. C61#18605semidirect

Maximal subgroups

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

subgroupindex
#214570132
#214565602
#214558582
#214450892
#214450332
#214449912
#214449532
#214445292
#214442662
#214442362
C61×#35072
#1860561