Finite Group Databasebeta

← #29657367

Subgroups — #29657367

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459631C211yesmaximal, characteristic
2C2×#16840112C211yesmaximal, characteristic, complemented
3#21455628C211yesmaximal, characteristic
4#21444982C214yesmaximal, complemented
5#21444975C214yesmaximal, complemented
6#21444503C211yesmaximal, characteristic, complemented
7#21444274C211yesmaximal, characteristic, complemented
8#21444199C211yesmaximal, characteristic
9C61×#3230C211yesmaximal, characteristic, complemented
10Q8×Dic122—21no—
11#16840490C2212yes—
12#16840472—21no—
13#16840470—22no—
14Q8×D244C2212yes—
15#16840425—22no—
16#16840392—22no—
17#16840390—21no—
18#16840317—21no—
19C2×#14519777C2211yescharacteristic
20C4⋊Q8×D61—21no—
21#16840242—21no—
22C4×#11758222C2211yescharacteristic
23#16840112—22no—
24#16840112C2214yes—
25#16840054C2212yes—
26#16840049C2212yes—
27#16840044—21no—
28#16840039—21no—
29#16839802—21no—
30#16839786C2211yescharacteristic
31#16839761—21no—
32C2×#14519701—22no—
33C2×C4×#92418C2211yescharacteristic
34#16839509C2214yes—
35Q8⋊#92418C2214yes—
36#16839492C2212yes—
37#16839489C2212yes—
38C61×#533C2211yescharacteristic
39C2×C4⋊Q8×C61C2211yescharacteristic
40C61×#500—21no—
41C61×#487—21no—
42C61×#484—22no—
43C61×#477C2214yes—
44C61×#431C2211yescharacteristic
45C2×#11758222D411yescharacteristic
46C22×Dic122—21no—
47C2×Q8×D61—22no—
48C2×#11758194—21no—
49C22×#92418—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777—22no—
53#14519777D412yes—
54#14519777C2312yes—
55#14519776—22no—
56#14519776—24no—
57#14519776—22no—
58#14519775—22no—
59#14519769—22no—
60#14519768—22no—
61#14519767—24no—
62#14519757—22no—
63#14519757—24no—
64#14519755—24no—
65#14519754—24no—
66#14519753—22no—
67Q8×Dic61—22no—
68C4×Dic122—22no—
69C4×Dic122D411yescharacteristic
70C4×Dic122C2311yescharacteristic
71#14519742—22no—
72C4×D244—22no—
73C4×D244D411yescharacteristic
74C4×D244C2311yescharacteristic
75C4×C22⋊D61—21no—
76C4×C22⋊D61—21no—
77#14519737—24no—
78#14519736D411yescharacteristic
79#14519735—22no—
80#14519734—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61D412yes—
83C4⋊C4×D61—22no—
84C4⋊C4×D61—22no—
85C42×D61—21no—
86#14519710—22no—
87#14519708—22no—
88C4⋊#92418—21no—
89C4⋊#92418C2312yes—
90#14519701—28no—
91#14519700—22no—
92#14519697—22no—
93C4×#92418—21no—
94C4×#92418C2312yes—
95C22×Q8×C61—21no—
96C4⋊Q8×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61D411yescharacteristic
100C4⋊Q8×C61D411yescharacteristic
101C4⋊Q8×C61C2312yes—
102#125×C61—21no—
103C22⋊Q8×C61—21no—
104C22⋊Q8×C61D412yes—
105C4×Q8×C61—21no—
106C4×Q8×C61C2312yes—
107#117×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C42×C61C2311yescharacteristic
111#107×C61—24no—
112C4⋊C8×C61C2312yes—
113#103×C61—24no—
114C22⋊C8×C61—21no—
115C4×C8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—22no—
118C2×Q8×C61—44no—
119C2×Q8×C61—41no—
120C2×Q8×C61C2×D412yes—
121C22×C4×C61—41no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C2×C22⋊D61—21no—
126C22×Dic61—22no—
127Q8×D61—84no—
128#11758222—22no—
129#11758222Q1614yes—
130C2×D244C2×D411yescharacteristic
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—24no—
135C2×Dic122—22no—
136C2×Dic122—24no—
137C2×Dic122—22no—
138C2×Dic122—82no—
139C2×Dic122C2×D411yescharacteristic
140C2×C8×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61C2×D412yes—
151C22⋊C4×C61—22no—
152C42×C61—22no—
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2411yescharacteristic
157C22⋊Dic61—21no—
158#11758198—24no—
159#11758198—21no—
160#11758198—21no—
161C4⋊Dic61—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61C2×D414yes—
166C4⋊Dic61C2×D411yescharacteristic
167C4⋊Dic61C2×D411yescharacteristic
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—21no—
172#11758196—21no—
173C4×Dic61—24no—
174C4×Dic61—21no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177#11758194—42no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418—42no—
181C23×C61#12011yescharacteristic
182C22×D61—21no—
183Q8×C61—24no—
184Q8×C61—44no—
185Q8×C61—42no—
186Q8×C61—82no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—42no—
192C2×C4×C61—82no—
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61C2×Q1612yes—
197C2×C4×C61#13712yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C22⋊D61—24no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—24no—
206C2×Dic61—82no—
207D244—21no—
208D244C2×Q1612yes—
209C4×D61—24no—
210C4×D61—42no—
211C4×D61—42no—
212C4×D61—84no—
213Dic122—21no—
214Dic122—44no—
215Dic122—82no—
216Dic122—84no—
217Dic122C2×Q1612yes—
218C8×C61—42no—
219#92418—42no—
220#92418—42no—
221#18643—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61#45112yes—
225C22×C61C2×#12011yescharacteristic
226C2×D61—22no—
227C2×D61—41no—
228C2×D61—41no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—82no—
234C4×C61#45112yes—
235C4×C61#45412yes—
236C4×C61C22×Q1611yescharacteristic
237C4×C61C2×#13711yescharacteristic
238C4×C61#57712yes—
239Dic61—22no—
240Dic61—44no—
241Dic61—41no—
242Dic61—41no—
243#3503—611no—
244#3230D6111yescharacteristic, complemented
245C2×#487—611no—
246#3079—611no—
247#1637—614no—
248#1636—614no—
249#1549—611no—
250#1493—611no—
251#1476—611no—
252C2×C61—22no—
253C2×C61C2×#45111yescharacteristic
254C2×C61#300911yescharacteristic
255C2×C61C2×#45411yescharacteristic
256D61—42no—
257Q82—1221no—
258#557—612no—
259#554—1222no—
260#554—1221no—
261D4×Q8—612no—
262D4×Q8—1222no—
263#544—1221no—
264#544—1222no—
265#533—1221no—
266#533C2×D6111yescharacteristic
267C2×C4⋊Q8—611no—
268C2×C4⋊Q8—1221no—
269C2×C4⋊Q8C2×D6111yescharacteristic
270#520—1221no—
271C4×#41—611no—
272#500—21no—
273#487—21no—
274#487—614no—
275#487—1222no—
276#484—22no—
277#477—612no—
278#477C2×D6114yes—
279#476—612no—
280#475—1221no—
281#474—1221no—
282#431C2×D6111yescharacteristic
283#428—1221no—
284#423—611no—
285#417—1221no—
286C2×#103—1222no—
287C2×C4×C8—611no—
288#332—614no—
289Q8⋊C8—614no—
290Q8⋊C8—612no—
291D4⋊C8—612no—
292C61#1864311yescharacteristic, complemented, Sylow
293C2×#41—611no—
294C22×Q8—21no—
295C22×Q8—1222no—
296C22×Q8—1221no—
297C2×M16—1221no—
298C22×C8—1221no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—612no—
303C4⋊Q8—612no—
304C4⋊Q8—1222no—
305C4⋊Q8—1222no—
306C4⋊Q8—1222no—
307C4⋊Q8—1224no—
308C4⋊Q8—1222no—
309C4⋊Q8—1221no—
310C4⋊Q8—1221no—
311C4⋊Q8C22⋊D6111yescharacteristic
312C4⋊Q8C22⋊D6111yescharacteristic
313C4⋊Q8C22×D6112yes—
314#125—21no—
315#125—1222no—
316#125—1224no—
317#125—1222no—
318C22⋊Q8—21no—
319C22⋊Q8—1224no—
320C22⋊Q8—1224no—
321C22⋊Q8—1224no—
322C22⋊Q8—1222no—
323C22⋊Q8—1222no—
324C22⋊Q8C22⋊D6112yes—
325C4×Q8—21no—
326C4×Q8—611no—
327C4×Q8—611no—
328C4×Q8—1222no—
329C4×Q8—1222no—
330C4×Q8—1222no—
331C4×Q8C22×D6112yes—
332C4×D4—611no—
333C4×D4—611no—
334C4×D4—1222no—
335C4×D4—1224no—
336C4×D4—1221no—
337C4×D4—1221no—
338#117—21no—
339#117—611no—
340#117—1222no—
341#117—1221no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—612no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1221no—
348C2×C42—1221no—
349C2×C42C22×D6111yescharacteristic
350#107—24no—
351#107—1222no—
352#106—1222no—
353C4⋊C8—612no—
354C4⋊C8—1221no—
355C4⋊C8C22×D6112yes—
356#103—24no—
357#103—1222no—
358#103—1228no—
359#102—1222no—
360C22⋊C8—21no—
361C4×C8—21no—
362C4×C8—612no—
363C4×C8—1221no—
364#41—614no—
365#41—1222no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—44no—
369C2×Q8—41no—
370C2×Q8—611no—
371C2×Q8—1224no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—4882no—
375C2×Q8—4884no—
376C2×Q8C2×C22⋊D6112yes—
377C2×D4—611no—
378C2×D4—1221no—
379C22×C4—41no—
380C22×C4—1224no—
381C22×C4—1221no—
382C22×C4—1222no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4D4×D6111yescharacteristic
386C22×C4D4×D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388M16—2442no—
389C2×C8—22no—
390C2×C8—1222no—
391C2×C8—1222no—
392C2×C8—2442no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—614no—
403C4⋊C4—611no—
404C4⋊C4—611no—
405C4⋊C4—1224no—
406C4⋊C4—1224no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1222no—
410C4⋊C4—1221no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—1221no—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—22no—
416C22⋊C4—1224no—
417C22⋊C4—1221no—
418C22⋊C4—1221no—
419C22⋊C4—1221no—
420C42—22no—
421C42—1224no—
422C42—1221no—
423C42—1221no—
424C42—1221no—
425C42D4×D6111yescharacteristic
426C42D4×D6111yescharacteristic
427C42C2×C22⋊D6111yescharacteristic
428C42C23×D6111yescharacteristic
429C23—1221no—
430C23#1451974511yescharacteristic
431Q8—24no—
432Q8—44no—
433Q8—42no—
434Q8—82no—
435Q8—612no—
436Q8—1221no—
437Q8—4882no—
438Q8—4884no—
439Q8—2444no—
440D4—612no—
441D4—1224no—
442D4—1221no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—82no—
449C2×C4—1221no—
450C2×C4—1224no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—1224no—
454C2×C4—4884no—
455C2×C4—4882no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4#1451974511yescharacteristic
459C2×C4#1451974511yescharacteristic
460C2×C4#1451974511yescharacteristic
461C2×C4Q16×D6112yes—
462C2×C4#1451980912yes—
463C2×C4C2×D4×D6111yescharacteristic
464C2×C4C2×D4×D6111yescharacteristic
465C2×C4C22×C22⋊D6111yescharacteristic
466C8—42no—
467C8—2442no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—1222no—
472C22—2441no—
473C22—2441no—
474C22#1683991612yes—
475C22C2×#1451974511yescharacteristic
476C4—21no—
477C4—21no—
478C4—42no—
479C4—42no—
480C4—82no—
481C4—1222no—
482C4—2441no—
483C4—2444no—
484C4—2441no—
485C4#1683991612yes—
486C4#1683993312yes—
487C4C2×Q16×D6111yescharacteristic
488C4C2×#1451980911yescharacteristic
489C4#1684057012yes—
490C2—22no—
491C2—2442no—
492C2C2×#1683991611yescharacteristic
493C2#2145509411yescharacteristic
494C2C2×#1683993311yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3230C2
socleC22×C61C2×#120
prime radicalC2×#11758222D4
hypercenter#3230D61
hypocenterC61#18643
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
2. C2×#16840112C2semidirect
4. #21444982C2semidirect
5. #21444975C2semidirect
6. #21444503C2semidirect
7. #21444274C2semidirect
9. C61×#3230C2semidirect
244. #3230D61semidirect
292. C61#18643semidirect

Maximal subgroups

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

subgroupindex
#214596312
C2×#168401122
#214556282
#214449822
#214449752
#214445032
#214442742
#214441992
C61×#32302
#1864361