Finite Group Databasebeta

← #29686907

Subgroups — #29686907

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×#16840034C211yesmaximal, characteristic
2#21454634C212yesmaximal
3C2×Q16×Dic61C211yesmaximal, characteristic, complemented
4#21454241C211yesmaximal, characteristic
5#21446604C211yesmaximal, characteristic, complemented
6#21446477C212yesmaximal, complemented
7#21446331C212yesmaximal
8#21445880C211yesmaximal, characteristic
9#108×Dic61C211yesmaximal, characteristic, complemented
10#101×Dic61C212yesmaximal, complemented
11C61×#1979C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519785C2211yescharacteristic
14C2×#14519783C2211yescharacteristic
15#16840230C2211yescharacteristic
16C2×Q8×Dic61C2212yes—
17#16840034C418yescomplemented
18#16839866—22no—
19#16839865—22no—
20Q16×Dic61—21no—
21Q16×Dic61—21no—
22Q16×Dic61—21no—
23Q16×Dic61—21no—
24#16839854—22no—
25#16839847—22no—
26C2×#14519713C2211yescharacteristic
27#16839805C2212yes—
28C2×#14519711C2211yescharacteristic
29#16839764C2212yes—
30C2×#14519701C2212yes—
31C2×#14519696C2212yes—
32#16839719—21no—
33#16839718C2211yescharacteristic
34#16839716—21no—
35#16839714—21no—
36M16×Dic61C2212yes—
37M16×Dic61C2211yescharacteristic
38C2×C8×Dic61C2211yescharacteristic
39C4×C8×D61—21no—
40#16839530C2212yes—
41#16839530C2212yes—
42#16839526C2212yes—
43C22×Q16×C61C2211yescharacteristic
44C2×#108×C61C2211yescharacteristic
45C61×#419C2212yes—
46C2×#101×C61C2212yes—
47C61×#405C2211yescharacteristic
48C2×#11758222C2311yescharacteristic
49#14519808—24no—
50#14519808C2×C418yes—
51C2×#11758202—22no—
52C2×#11758201—22no—
53#14519785—22no—
54#14519785—22no—
55#14519783—22no—
56#14519783C2×C414yes—
57C2×#11758194C2311yescharacteristic
58C2×#11758194C2312yes—
59C2×#11758189—21no—
60C2×#11758189—21no—
61C22×#92418C2311yescharacteristic
62C2×C8×D61—21no—
63C2×C8×D61—21no—
64Q8×Dic61—22no—
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67Q8×Dic61—22no—
68#14519742D411yescharacteristic
69#14519742D411yescharacteristic
70#14519739D411yescharacteristic
71#14519739D411yescharacteristic
72#14519737—21no—
73#14519737—21no—
74#14519734—21no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61C2312yes—
77C2×C4×Dic61—22no—
78C2×C4×Dic61C2311yescharacteristic
79C42×D61—21no—
80#14519713C2×C414yescomplemented
81#14519711—21no—
82#14519711—21no—
83#14519710—22no—
84#14519708—22no—
85#14519702—22no—
86#14519701—22no—
87#14519701—22no—
88#14519699—22no—
89#14519696C2×C418yescomplemented
90#14519688—21no—
91#14519687—21no—
92#14519687—22no—
93C4×#92418—21no—
94C8×Dic61—21no—
95C8×Dic61—22no—
96C8×Dic61D411yescharacteristic
97C8×Dic61D411yescharacteristic
98C8×Dic61D411yescharacteristic
99C8×Dic61D411yescharacteristic
100C22×Q8×C61C2312yes—
101C2×Q16×C61—21no—
102C2×Q16×C61—21no—
103C2×Q16×C61—22no—
104C2×Q16×C61C2×C414yescomplemented
105C2×M16×C61C2312yes—
106C2×M16×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108#117×C61C2311yescharacteristic
109#108×C61—22no—
110#103×C61—22no—
111#103×C61—22no—
112#101×C61—24no—
113#97×C61—21no—
114C4×C8×C61—21no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61C22×C414yes—
121C22×C4×C61—22no—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—21no—
124C22×Dic61—22no—
125C22×Dic61#4111yescharacteristic
126C22×Dic61#4111yescharacteristic
127#11758222—21no—
128#11758222—21no—
129#11758222C22×C412yes—
130#11758222#4112yes—
131C2×D244C2×D411yescharacteristic
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×Dic122C2×D411yescharacteristic
136Q16×C61—24no—
137Q16×C61—24no—
138M16×C61—21no—
139M16×C61—22no—
140M16×C61—22no—
141M16×C61#4112yes—
142C2×C8×C61—21no—
143C2×C8×C61—22no—
144C2×C8×C61C22×C412yes—
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61#4112yes—
148C4⋊C4×C61C2×D411yescharacteristic
149C4⋊C4×C61C2×D411yescharacteristic
150C22⋊C4×C61—21no—
151C42×C61—21no—
152#11758202—28no—
153#11758201—28no—
154#11758198—21no—
155#11758198—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—42no—
159C4⋊Dic61—42no—
160#11758196—21no—
161#11758196—21no—
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—42no—
165C4×Dic61—42no—
166C4×Dic61C2×D411yescharacteristic
167C4×Dic61C2×D411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169C4×Dic61C2×D411yescharacteristic
170#11758194—22no—
171#11758194—21no—
172#11758194C22×C414yes—
173#11758194C22×C412yes—
174C2×#92418—21no—
175C2×#92418—22no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418C2×D411yescharacteristic
179C2×#92418C2×D411yescharacteristic
180#11758189—24no—
181#11758189—24no—
182C8×D61—24no—
183C8×D61—24no—
184C23×C61C2×#4111yescharacteristic
185C22×D61—21no—
186Q8×C61—24no—
187Q8×C61—24no—
188Q8×C61—42no—
189Q8×C61—42no—
190C2×C4×C61—24no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61C23×C411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C22⋊D61—22no—
202C22⋊D61—22no—
203C2×Dic61—21no—
204C2×Dic61—22no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207C2×Dic61—82no—
208C2×Dic61#12111yescharacteristic
209C2×Dic61#12111yescharacteristic
210C2×Dic61#12111yescharacteristic
211C2×Dic61#12111yescharacteristic
212D244—21no—
213D244C4×D412yes—
214C4×D61—22no—
215C4×D61—22no—
216C4×D61—42no—
217C4×D61—42no—
218Dic122—21no—
219Dic122C4×D412yes—
220C8×C61—22no—
221C8×C61—42no—
222C8×C61C4×D412yes—
223C8×C61C4×D412yes—
224#92418—22no—
225#92418—24no—
226#92418—21no—
227#92418—21no—
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—44no—
240C4×C61C2×C4×D411yescharacteristic
241C4×C61C2×C4×D411yescharacteristic
242C4×C61#52511yescharacteristic
243C4×C61#53211yescharacteristic
244Dic61—21no—
245Dic61—21no—
246Dic61—21no—
247Dic61—22no—
248Dic61—82no—
249Dic61#47312yescomplemented
250C2×#473—611no—
251C4×#137—612no—
252C2×C4×Q16—611no—
253C4×#131—611no—
254#1979D6111yescharacteristic, complemented
255#1952—611no—
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×Q16C2×D6111yescharacteristic
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#428—612no—
285C2×#107—611no—
286#419C2×D6112yes—
287#417—612no—
288C2×#103—612no—
289C2×#101—612no—
290C2×#101C2×D6112yes—
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#343—612no—
302#341—612no—
303C61C4×#47311yescharacteristic, complemented, Sylow
304C2×#41—611no—
305C22×Q8C22×D6112yes—
306#137—618no—
307#137—1224no—
308C2×Q16—21no—
309C2×Q16—21no—
310C2×Q16—22no—
311C2×Q16—1222no—
312C2×Q16C4×D6114yescomplemented
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—612no—
341C2×C4⋊C4—1222no—
342C2×C42—611no—
343C2×C42—1221no—
344C2×C42—1222no—
345#108—22no—
346#108—614no—
347#107—1221no—
348#107—1221no—
349#107—1222no—
350#106—1222no—
351#103—22no—
352#103—22no—
353#103—1222no—
354#103—1222no—
355#103—1222no—
356#102—1222no—
357#101—24no—
358#101—618no—
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—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—611no—
381C2×Q8C2×C4×D6114yes—
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—24no—
393Q16—24no—
394Q16—1228no—
395SD16—1228no—
396M16—21no—
397M16—22no—
398M16—22no—
399M16—612no—
400M16—614no—
401M16—1224no—
402M16—1222no—
403M16—1221no—
404M16—1224no—
405M16#1175822412yes—
406C2×C8—21no—
407C2×C8—22no—
408C2×C8—611no—
409C2×C8—611no—
410C2×C8—1224no—
411C2×C8—1221no—
412C2×C8—1222no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8—1224no—
416C2×C8C2×C4×D6112yes—
417C2×C8D4×D6111yescharacteristic
418C2×C8D4×D6111yescharacteristic
419C2×C8#1175822412yes—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
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—24no—
443Q8—24no—
444Q8—42no—
445Q8—42no—
446Q8—612no—
447Q8—1221no—
448D4—612no—
449D4—1222no—
450D4—1222no—
451D4—1221no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—611no—
458C2×C4—611no—
459C2×C4—611no—
460C2×C4—611no—
461C2×C4—1222no—
462C2×C4—1221no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—4882no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2441no—
469C2×C4—2442no—
470C2×C4C22×C4×D6111yescharacteristic
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C2×D4×D6111yescharacteristic
473C2×C4#41×D6111yescharacteristic
474C2×C4C2×#1175822411yescharacteristic
475C2×C4C2×#1175822411yescharacteristic
476C8—22no—
477C8—42no—
478C8—1224no—
479C8—1222no—
480C8—1221no—
481C8—1221no—
482C8#1451973712yes—
483C8#1451973712yes—
484C22—21no—
485C22—21no—
486C22—1222no—
487C22—2441no—
488C22—2441no—
489C22C4×#1175822411yescharacteristic
490C22#121×D6111yescharacteristic
491C22#1684029111yescharacteristic
492C4—44no—
493C4—41no—
494C4—41no—
495C4—612no—
496C4—1222no—
497C4—1221no—
498C4—1221no—
499C4—1221no—
500C4—4882no—
501C4C4×D4×D6111yescharacteristic
502C4C2×#1451973711yescharacteristic
503C4#1684027211yescharacteristic
504C4#1684031011yescharacteristic
505C2—21no—
506C2—21no—
507C2—2442no—
508C2#2144833511yescharacteristic
509C2#473×D6111yescharacteristic
510C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1979C2
socleC22×C61C2×#121
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenter#1979D61
hypocenterC61C4×#473
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1979

Complements

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

NH ≅ G/Ntype
3. C2×Q16×Dic61C2semidirect
5. #21446604C2semidirect
6. #21446477C2semidirect
9. #108×Dic61C2semidirect
10. #101×Dic61C2semidirect
11. C61×#1979C2semidirect
17. #16840034C4semidirect
80. #14519713C2×C4semidirect
89. #14519696C2×C4semidirect
104. C2×Q16×C61C2×C4semidirect
249. Dic61#473semidirect
254. #1979D61semidirect
303. C61C4×#473semidirect
312. C2×Q16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
#214546342
C2×Q16×Dic612
#214542412
#214466042
#214464772
#214463312
#214458802
#108×Dic612
#101×Dic612
C61×#19792
C4×#47361