Finite Group Databasebeta

← #29657265

Subgroups — #29657265

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459631C211yesmaximal, characteristic
2C2×#16840114C211yesmaximal, characteristic, complemented
3#21455628C211yesmaximal, characteristic
4#21445038C214yesmaximal, complemented
5#21445031C214yesmaximal, complemented
6#21444499C211yesmaximal, characteristic, complemented
7#21444274C211yesmaximal, characteristic, complemented
8#21444199C211yesmaximal, characteristic
9C61×#3209C211yesmaximal, characteristic, complemented
10Q8×Dic122C2212yes—
11#16840490—21no—
12#16840472C2212yes—
13#16840470—22no—
14Q8×D244—21no—
15#16840425—22no—
16#16840392—22no—
17#16840390—21no—
18#16840317—21no—
19C2×#14519777C2211yescharacteristic
20C4⋊Q8×D61—21no—
21#16840242—21no—
22C4×#11758222C2211yescharacteristic
23#16840114—22no—
24#16840114C2214yes—
25#16840054—21no—
26#16840049—21no—
27#16840044C2212yes—
28#16840039C2212yes—
29#16839802—21no—
30#16839786C2211yescharacteristic
31#16839761—21no—
32C2×#14519700—22no—
33C2×C4×#92418C2211yescharacteristic
34#16839510C2214yes—
35Q8⋊#92418C2214yes—
36#16839492C2212yes—
37#16839489C2212yes—
38C61×#533C2211yescharacteristic
39C2×C4⋊Q8×C61C2211yescharacteristic
40C61×#494—21no—
41C61×#487—21no—
42C61×#462C2214yes—
43C61×#451—22no—
44C61×#431C2211yescharacteristic
45C2×#11758222D411yescharacteristic
46C22×Dic122D412yes—
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—24no—
63#14519757—22no—
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⋊Dic61—21no—
83C4⋊C4×D61—22no—
84C4⋊C4×D61—22no—
85C42×D61—21no—
86#14519710—22no—
87#14519708—22no—
88C4⋊#92418—21no—
89C4⋊#92418C2312yes—
90#14519700—28no—
91#14519700—22no—
92#14519697—22no—
93C4×#92418—21no—
94C4×#92418C2312yes—
95C22×Q8×C61—21no—
96C2×Q16×C61—24no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61D411yescharacteristic
101C4⋊Q8×C61D411yescharacteristic
102C4⋊Q8×C61C2312yes—
103#125×C61—21no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61D412yes—
106C4×Q8×C61—21no—
107C4×Q8×C61C2312yes—
108#117×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C42×C61C2311yescharacteristic
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—41no—
119C2×Q8×C61—82no—
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#11758222Q1614yescomplemented
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—44no—
138C2×Dic122C2×D411yescharacteristic
139C2×Dic122C2×D414yes—
140Q16×C61—84no—
141C2×C8×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—22no—
153C42×C61—22no—
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2411yescharacteristic
158C22⋊Dic61—21no—
159#11758198—24no—
160#11758198—21no—
161#11758198—21no—
162C4⋊Dic61—24no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61C2×D411yescharacteristic
168C4⋊Dic61C2×D411yescharacteristic
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172#11758196—21no—
173#11758196—21no—
174C4×Dic61—24no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178#11758194—42no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—42no—
182C23×C61#12011yescharacteristic
183C22×D61—21no—
184Q8×C61—24no—
185Q8×C61—42no—
186Q8×C61—44no—
187Q8×C61—82no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—42no—
193C2×C4×C61—82no—
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61C2×Q1612yes—
198C2×C4×C61#13712yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C22⋊D61—24no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—24no—
207C2×Dic61—82no—
208D244—21no—
209D244C2×Q1612yes—
210C4×D61—24no—
211C4×D61—42no—
212C4×D61—42no—
213C4×D61—84no—
214Dic122—21no—
215Dic122—44no—
216Dic122—44no—
217Dic122—84no—
218Dic122C2×Q1612yes—
219C8×C61—42no—
220#92418—42no—
221#92418—42no—
222#18633—611nomaximal, Sylow
223C22×C61—21no—
224C22×C61—21no—
225C22×C61#45112yes—
226C22×C61C2×#12011yescharacteristic
227C2×D61—22no—
228C2×D61—41no—
229C2×D61—41no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—42no—
233C4×C61—42no—
234C4×C61—82no—
235C4×C61#45112yes—
236C4×C61#45512yes—
237C4×C61C22×Q1611yescharacteristic
238C4×C61C2×#13711yescharacteristic
239C4×C61#57712yes—
240Dic61—22no—
241Dic61—41no—
242Dic61—44no—
243Dic61—41no—
244#3503—611no—
245#3209D6111yescharacteristic, complemented
246C2×#487—611no—
247#3079—611no—
248#1645—614no—
249#1644—614no—
250#1548—611no—
251#1493—611no—
252#1476—611no—
253C2×C61—22no—
254C2×C61C2×#45111yescharacteristic
255C2×C61#301011yescharacteristic
256C2×C61C2×#45511yescharacteristic
257D61—42no—
258Q82—612no—
259#557—1221no—
260#554—612no—
261#554—1222no—
262D4×Q8—1221no—
263D4×Q8—1222no—
264#544—1221no—
265#544—1222no—
266#533—1221no—
267#533C2×D6111yescharacteristic
268C2×C4⋊Q8—611no—
269C2×C4⋊Q8—1221no—
270C2×C4⋊Q8C2×D6111yescharacteristic
271#520—1221no—
272C4×#41—611no—
273#494—21no—
274#487—21no—
275#487—614no—
276#487—1222no—
277#477—1221no—
278#476—1221no—
279#475—612no—
280#474—612no—
281#462C2×D6114yes—
282#451—22no—
283#431C2×D6111yescharacteristic
284#428—1221no—
285#423—611no—
286#417—1221no—
287C2×#103—1222no—
288C2×C4×C8—611no—
289#333—614no—
290Q8⋊C8—614no—
291Q8⋊C8—612no—
292D4⋊C8—612no—
293C61#1863311yescharacteristic, complemented, Sylow
294C2×#41—611no—
295C22×Q8—21no—
296C22×Q8—612no—
297C22×Q8—1222no—
298C2×Q16—24no—
299C2×M16—1221no—
300C22×C8—1221no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—612no—
305C4⋊Q8—612no—
306C4⋊Q8—1222no—
307C4⋊Q8—1224no—
308C4⋊Q8—1222no—
309C4⋊Q8—1222no—
310C4⋊Q8—1222no—
311C4⋊Q8—1221no—
312C4⋊Q8—1221no—
313C4⋊Q8C22⋊D6111yescharacteristic
314C4⋊Q8C22⋊D6111yescharacteristic
315C4⋊Q8C22×D6112yes—
316#125—21no—
317#125—1222no—
318#125—1222no—
319#125—1224no—
320C22⋊Q8—21no—
321C22⋊Q8—1224no—
322C22⋊Q8—1224no—
323C22⋊Q8—1222no—
324C22⋊Q8—1224no—
325C22⋊Q8—1222no—
326C22⋊Q8C22⋊D6112yes—
327C4×Q8—21no—
328C4×Q8—611no—
329C4×Q8—611no—
330C4×Q8—1222no—
331C4×Q8—1222no—
332C4×Q8—1222no—
333C4×Q8C22×D6112yes—
334C4×D4—611no—
335C4×D4—611no—
336C4×D4—1222no—
337C4×D4—1224no—
338C4×D4—1221no—
339C4×D4—1221no—
340#117—21no—
341#117—611no—
342#117—1222no—
343#117—1221no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1222no—
350C2×C42—1221no—
351C2×C42C22×D6111yescharacteristic
352#107—1222no—
353#106—1222no—
354C4⋊C8—612no—
355C4⋊C8—1221no—
356C4⋊C8C22×D6112yes—
357#103—24no—
358#103—1228no—
359#103—1222no—
360#102—1222no—
361C22⋊C8—21no—
362C4×C8—21no—
363C4×C8—612no—
364C4×C8—1221no—
365#41—614no—
366#41—1222no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—41no—
370C2×Q8—82no—
371C2×Q8—611no—
372C2×Q8—614no—
373C2×Q8—1224no—
374C2×Q8—1222no—
375C2×Q8—4884no—
376C2×Q8—2444no—
377C2×Q8C2×C22⋊D6112yes—
378C2×D4—611no—
379C2×D4—1221no—
380C22×C4—41no—
381C22×C4—1224no—
382C22×C4—1221no—
383C22×C4—1222no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4D4×D6111yescharacteristic
387C22×C4D4×D6111yescharacteristic
388C22×C4C2×C22⋊D6111yescharacteristic
389Q16—84no—
390M16—2442no—
391C2×C8—22no—
392C2×C8—1222no—
393C2×C8—1222no—
394C2×C8—2442no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—611no—
405C4⋊C4—611no—
406C4⋊C4—1224no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1222no—
411C4⋊C4—1221no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—1221no—
416C4⋊C4C2×C22⋊D6112yes—
417C22⋊C4—22no—
418C22⋊C4—1224no—
419C22⋊C4—1221no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C42—22no—
423C42—1224no—
424C42—1221no—
425C42—1221no—
426C42—1221no—
427C42D4×D6111yescharacteristic
428C42D4×D6111yescharacteristic
429C42C2×C22⋊D6111yescharacteristic
430C42C23×D6111yescharacteristic
431C23—1221no—
432C23#1451974511yescharacteristic
433Q8—24no—
434Q8—42no—
435Q8—44no—
436Q8—82no—
437Q8—612no—
438Q8—1221no—
439Q8—4884no—
440Q8—2444no—
441Q8—2444no—
442D4—612no—
443D4—1224no—
444D4—1221no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4—1221no—
452C2×C4—1224no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—1224no—
456C2×C4—4884no—
457C2×C4—4882no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4#1451974511yescharacteristic
461C2×C4#1451974511yescharacteristic
462C2×C4#1451974511yescharacteristic
463C2×C4Q16×D6112yes—
464C2×C4#1451980912yes—
465C2×C4C2×D4×D6111yescharacteristic
466C2×C4C2×D4×D6111yescharacteristic
467C2×C4C22×C22⋊D6111yescharacteristic
468C8—42no—
469C8—2442no—
470C8—2442no—
471C22—21no—
472C22—21no—
473C22—1222no—
474C22—2441no—
475C22—2441no—
476C22#1683991612yes—
477C22C2×#1451974511yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—1222no—
484C4—2444no—
485C4—2441no—
486C4—2441no—
487C4#1683991612yes—
488C4#1683993812yes—
489C4C2×Q16×D6111yescharacteristic
490C4C2×#1451980911yescharacteristic
491C4#1684057312yes—
492C2—22no—
493C2—2442no—
494C2C2×#1683991611yescharacteristic
495C2#2145510311yescharacteristic
496C2C2×#1683993811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3209C2
socleC22×C61C2×#120
prime radicalC2×#11758222D4
hypercenter#3209D61
hypocenterC61#18633
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
2. C2×#16840114C2semidirect
4. #21445038C2semidirect
5. #21445031C2semidirect
6. #21444499C2semidirect
7. #21444274C2semidirect
9. C61×#3209C2semidirect
129. #11758222Q16semidirect
245. #3209D61semidirect
293. C61#18633semidirect

Maximal subgroups

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

subgroupindex
#214596312
C2×#168401142
#214556282
#214450382
#214450312
#214444992
#214442742
#214441992
C61×#32092
#1863361