Finite Group Databasebeta

← #29686899

Subgroups — #29686899

Order 15616.

882 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840038C211yesmaximal, characteristic
2#21454637C212yesmaximal
3C2×#16839856C211yesmaximal, characteristic, complemented
4#21454241C211yesmaximal, characteristic
5#21446734C211yesmaximal, characteristic, complemented
6#21446473C212yesmaximal, complemented
7#21446335C212yesmaximal
8#21445884C211yesmaximal, characteristic
9#108×Dic61C211yesmaximal, characteristic, complemented
10#21445614C212yesmaximal, complemented
11C61×#1952C211yesmaximal, characteristic, complemented
12C2×#14519810C2212yes—
13C22×Dic244C2211yescharacteristic
14C2×#14519785C2211yescharacteristic
15C2×#14519783C2211yescharacteristic
16#16840230C2211yescharacteristic
17C2×#14519742C2212yes—
18#16840038C418yescomplemented
19#16839868—22no—
20#16839861—22no—
21#16839856—22no—
22#16839856—21no—
23#16839856—21no—
24#16839856—21no—
25#16839856—21no—
26#16839849—22no—
27C2×#14519713C2211yescharacteristic
28#16839770C2212yes—
29#16839761C2212yes—
30C2×#14519700C2212yes—
31C2×#14519695C2212yes—
32C2×#14519695C2212yes—
33#16839719—21no—
34#16839718C2211yescharacteristic
35#16839716—21no—
36#16839714—21no—
37M16×Dic61C2212yes—
38M16×Dic61C2211yescharacteristic
39C2×C8×Dic61C2211yescharacteristic
40C4×C8×D61—21no—
41#16839531C2212yes—
42#16839526C2212yes—
43C2×#108×C61C2211yescharacteristic
44C61×#428C2212yes—
45C2×#107×C61C2211yescharacteristic
46C61×#405C2211yescharacteristic
47C61×#343C2212yes—
48C2×#11758222C2311yescharacteristic
49C22×Dic122C2312yes—
50#14519810—24no—
51#14519810C2×C418yes—
52C2×Dic244—22no—
53C2×Dic244—21no—
54C2×Dic244—21no—
55C2×Dic244—22no—
56C2×Dic244C2×C414yescomplemented
57C2×#11758190—22no—
58#14519785—22no—
59#14519785—22no—
60#14519783—22no—
61#14519783C2×C414yes—
62C2×#11758194C2312yes—
63C2×#11758194C2311yescharacteristic
64C2×#11758189—21no—
65C2×#11758189—21no—
66C22×#92418C2311yescharacteristic
67C2×C8×D61—21no—
68C2×C8×D61—21no—
69#14519742—22no—
70#14519742—22no—
71#14519742—22no—
72#14519742—22no—
73#14519742D411yescharacteristic
74#14519742D411yescharacteristic
75#14519739D411yescharacteristic
76#14519739D411yescharacteristic
77#14519737—21no—
78#14519737—21no—
79#14519734—21no—
80C2×#11758196—22no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61C2311yescharacteristic
83C42×D61—21no—
84#14519713—22no—
85#14519700—22no—
86#14519700—22no—
87#14519700—22no—
88#14519700—22no—
89#14519700—22no—
90#14519697—22no—
91#14519695—24no—
92#14519695C2×C418yescomplemented
93#14519688—21no—
94#14519687—21no—
95#14519687—22no—
96C4×#92418—21no—
97C8×Dic61—21no—
98C8×Dic61—22no—
99C8×Dic61D411yescharacteristic
100C8×Dic61D411yescharacteristic
101C8×Dic61D411yescharacteristic
102C8×Dic61D411yescharacteristic
103C2×M16×C61C2312yes—
104C2×M16×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106#117×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108#108×C61C2×C414yescomplemented
109#107×C61—21no—
110#107×C61—21no—
111#107×C61—22no—
112#106×C61—22no—
113#97×C61—21no—
114C4×C8×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61C2411yescharacteristic
117C2×C22⋊D61—21no—
118C22×Dic61—22no—
119C22×Dic61#4111yescharacteristic
120C22×Dic61#4111yescharacteristic
121#11758222—21no—
122#11758222—21no—
123#11758222C22×C412yes—
124#11758222#4112yes—
125C2×D244C2×D411yescharacteristic
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122C22×C414yes—
135C2×Dic122C2×D411yescharacteristic
136M16×C61—22no—
137M16×C61—21no—
138M16×C61C22×C414yes—
139M16×C61#4112yes—
140C2×C8×C61—21no—
141C2×C8×C61—22no—
142C2×C8×C61C22×C412yes—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61#4112yes—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61C2×D411yescharacteristic
149C4⋊C4×C61C2×D411yescharacteristic
150C22⋊C4×C61—21no—
151C42×C61—21no—
152#11758198—21no—
153#11758198—21no—
154#11758196—21no—
155#11758196—21no—
156#11758196—42no—
157#11758196—42no—
158C4×Dic61—21no—
159C4×Dic61—21no—
160C4×Dic61—42no—
161C4×Dic61—42no—
162C4×Dic61C2×D411yescharacteristic
163C4×Dic61C2×D411yescharacteristic
164C4×Dic61C2×D411yescharacteristic
165C4×Dic61C2×D411yescharacteristic
166#11758194—21no—
167#11758194—22no—
168#11758194—22no—
169#11758194C22×C412yes—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418C2×D411yescharacteristic
175C2×#92418C2×D411yescharacteristic
176Dic244—28no—
177Dic244—24no—
178Dic244—24no—
179#11758190—28no—
180#11758189—24no—
181#11758189—24no—
182C8×D61—24no—
183C8×D61—24no—
184C23×C61C2×#4111yescharacteristic
185C22×D61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—42no—
189C2×C4×C61—82no—
190C2×C4×C61C23×C411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C22⋊D61—22no—
197C22⋊D61—22no—
198C2×Dic61—21no—
199C2×Dic61—22no—
200C2×Dic61—24no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61#12111yescharacteristic
205C2×Dic61#12111yescharacteristic
206C2×Dic61#12111yescharacteristic
207C2×Dic61#12111yescharacteristic
208D244—21no—
209D244C4×D412yes—
210C4×D61—22no—
211C4×D61—22no—
212C4×D61—42no—
213C4×D61—42no—
214Dic122—21no—
215Dic122—24no—
216Dic122—24no—
217Dic122—42no—
218Dic122—42no—
219Dic122C4×D412yes—
220C8×C61—24no—
221C8×C61—22no—
222C8×C61C4×D412yes—
223C8×C61C4×D412yes—
224#92418—22no—
225#92418—21no—
226#92418—21no—
227#92418—42no—
228C4×#473—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61C4×#4111yescharacteristic
232C22×C61C2×#12111yescharacteristic
233C22×C61#52911yescharacteristic
234C2×D61—22no—
235C2×D61—41no—
236C2×D61—41no—
237C4×C61—41no—
238C4×C61—41no—
239C4×C61—82no—
240C4×C61C2×C4×D411yescharacteristic
241C4×C61C2×C4×D411yescharacteristic
242C4×C61#52511yescharacteristic
243C4×C61#53211yescharacteristic
244Dic61—21no—
245Dic61—21no—
246Dic61—21no—
247Dic61—22no—
248Dic61—44no—
249Dic61#47312yescomplemented
250C2×#473—611no—
251C4×#137—612no—
252C2×C4×Q16—611no—
253C4×#131—611no—
254#1979—611no—
255#1952D6111yescharacteristic, complemented
256#1930—612no—
257#1908—612no—
258#1816—611no—
259C4×#108—611no—
260C4×#101—612no—
261C2×C61—21no—
262C2×C61—21no—
263C2×C61C4×#12111yescharacteristic
264C2×C61C2×#47311yescharacteristic
265C2×C61#306711yescharacteristic
266D61—42no—
267C2×#137—612no—
268C22×Q16—611no—
269C2×#131—611no—
270C2×#131—611no—
271C4×#41—611no—
272C2×C4×Q8—612no—
273#473—618no—
274#441—1222no—
275#440—1222no—
276C4×Q16—1221no—
277C4×Q16—1221no—
278C4×Q16—1221no—
279C4×Q16—1221no—
280C4×Q16—1222no—
281C4×SD16—1222no—
282C2×#108—611no—
283C2×#108C2×D6111yescharacteristic
284#428C2×D6112yes—
285C2×#107C2×D6111yescharacteristic
286#419—612no—
287#417—612no—
288C2×#103—612no—
289C2×#101—612no—
290C2×#101—612no—
291#405—611no—
292#405—1221no—
293#405—1221no—
294#405C2×D6111yescharacteristic
295C4×M16—611no—
296C4×M16—612no—
297C4×M16—1221no—
298C2×C4×C8—611no—
299C2×C4×C8—1221no—
300#343—612no—
301#343C2×D6112yes—
302#341—612no—
303C61C4×#47311yescharacteristic, complemented, Sylow
304C2×#41—611no—
305C22×Q8—612no—
306#137—618no—
307#137—1224no—
308C2×Q16—614no—
309C2×Q16—1222no—
310C2×Q16—1221no—
311C2×Q16—1221no—
312C2×Q16—1222no—
313C2×SD16—1222no—
314#131—614no—
315#131—1222no—
316#131—1222no—
317#131—1222no—
318C2×M16—611no—
319C2×M16—612no—
320C2×M16—1221no—
321C2×M16—1221no—
322C2×M16C22×D6112yes—
323C2×M16C22×D6111yescharacteristic
324C22×C8—611no—
325C22×C8—1221no—
326C22×C8—1221no—
327C22×C8C22×D6111yescharacteristic
328C4×Q8—611no—
329C4×Q8—611no—
330C4×Q8—1222no—
331C4×Q8—1222no—
332C4×Q8—1222no—
333C4×Q8—1222no—
334C4×D4—611no—
335C4×D4—611no—
336C4×D4—1221no—
337C4×D4—1221no—
338#117—1221no—
339#117C22×D6111yescharacteristic
340C2×C4⋊C4—1222no—
341C2×C4⋊C4C22×D6112yes—
342C2×C42—611no—
343C2×C42—1221no—
344C2×C42—1222no—
345#108—1222no—
346#108C4×D6114yescomplemented
347#107—21no—
348#107—21no—
349#107—22no—
350#106—22no—
351#103—1222no—
352#103—1222no—
353#103—1222no—
354#103—1222no—
355#103—1222no—
356#102—1222no—
357#101—618no—
358#101—1224no—
359#97—21no—
360#97—1221no—
361#97—1221no—
362#97—1222no—
363C4×C8—21no—
364C4×C8—611no—
365C4×C8—611no—
366C4×C8—611no—
367C4×C8—611no—
368C4×C8—1221no—
369C4×C8—1221no—
370C4×C8—1222no—
371#41—612no—
372#41—612no—
373#41—1221no—
374#41—1221no—
375C2×Q8—614no—
376C2×Q8—611no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—1222no—
380C2×Q8—1222no—
381C2×Q8—1222no—
382C2×D4—611no—
383C2×D4—1221no—
384C22×C4—22no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—1222no—
391C22×C4C23×D6111yescharacteristic
392Q16—1228no—
393Q16—1224no—
394Q16—1224no—
395SD16—1228no—
396M16—21no—
397M16—22no—
398M16—612no—
399M16—1224no—
400M16—1221no—
401M16—1224no—
402M16—1222no—
403M16—1222no—
404M16C2×C4×D6114yes—
405M16#1175822612yes—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—611no—
409C2×C8—611no—
410C2×C8—1224no—
411C2×C8—1221no—
412C2×C8—1221no—
413C2×C8—1221no—
414C2×C8—1224no—
415C2×C8—1222no—
416C2×C8C2×C4×D6112yes—
417C2×C8D4×D6111yescharacteristic
418C2×C8D4×D6111yescharacteristic
419C2×C8#1175822612yes—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—2442no—
425C4⋊C4—2442no—
426C4⋊C4D4×D6111yescharacteristic
427C4⋊C4D4×D6111yescharacteristic
428C22⋊C4—21no—
429C22⋊C4—1221no—
430C22⋊C4—1221no—
431C42—21no—
432C42—611no—
433C42—611no—
434C42—611no—
435C42—611no—
436C42—1221no—
437C42—1221no—
438C42—2442no—
439C42—2442no—
440C23—1221no—
441C23#41×D6111yescharacteristic
442Q8—612no—
443Q8—1221no—
444Q8—1224no—
445Q8—1224no—
446Q8—2442no—
447Q8—2442no—
448D4—612no—
449D4—1222no—
450D4—1222no—
451D4—1221no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—611no—
457C2×C4—611no—
458C2×C4—611no—
459C2×C4—611no—
460C2×C4—1222no—
461C2×C4—1221no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1224no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2441no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4C22×C4×D6111yescharacteristic
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C2×C4#41×D6111yescharacteristic
474C2×C4C2×#1175822611yescharacteristic
475C2×C4C2×#1175822611yescharacteristic
476C8—24no—
477C8—22no—
478C8—1222no—
479C8—1221no—
480C8—1221no—
481C8—2442no—
482C8#1451973912yes—
483C8#1451973912yes—
484C22—21no—
485C22—21no—
486C22—1222no—
487C22—2441no—
488C22—2441no—
489C22C4×#1175822611yescharacteristic
490C22#121×D6111yescharacteristic
491C22#1684029711yescharacteristic
492C4—41no—
493C4—41no—
494C4—82no—
495C4—612no—
496C4—1222no—
497C4—1221no—
498C4—1221no—
499C4—1221no—
500C4—2444no—
501C4C4×D4×D6111yescharacteristic
502C4C2×#1451973911yescharacteristic
503C4#1684027511yescharacteristic
504C4#1684031311yescharacteristic
505C2—21no—
506C2—21no—
507C2—2442no—
508C2#2144832711yescharacteristic
509C2#473×D6111yescharacteristic
510C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1952C2
socleC22×C61C2×#121
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenter#1952D61
hypocenterC61C4×#473
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
3. C2×#16839856C2semidirect
5. #21446734C2semidirect
6. #21446473C2semidirect
9. #108×Dic61C2semidirect
10. #21445614C2semidirect
11. C61×#1952C2semidirect
18. #16840038C4semidirect
56. C2×Dic244C2×C4semidirect
92. #14519695C2×C4semidirect
108. #108×C61C2×C4semidirect
249. Dic61#473semidirect
255. #1952D61semidirect
303. C61C4×#473semidirect
346. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400382
#214546372
C2×#168398562
#214542412
#214467342
#214464732
#214463352
#214458842
#108×Dic612
#214456142
C61×#19522
C4×#47361