Finite Group Databasebeta

← #30613429

Subgroups — #30613429

Order 15616.

654 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#21462665C211yesmaximal, characteristic, complemented
2#21462656C211yesmaximal, characteristic, complemented
3#21461455C212yesmaximal, complemented
4#21461432C212yesmaximal, complemented
5#21461420C212yesmaximal, complemented
6SD16×#92420C212yesmaximal, complemented
7#21461379C211yesmaximal, characteristic, complemented
8#21461150C211yesmaximal, characteristic, complemented
9#21461150C211yesmaximal, characteristic, complemented
10C8×#11758217C211yesmaximal, characteristic, complemented
11#135×Dic61C211yesmaximal, characteristic
12#16840813C2212yes—
13Q8×#92420C2212yes—
14#16840797—22no—
15#16840797C2212yes—
16#16840793—22no—
17D4×#92420C2212yes—
18#16840783—21no—
19#16840783C2211yescharacteristic
20#16840783C2211yescharacteristic
21#16840782—21no—
22#16840779C2211yescharacteristic
23C4×#11758217—21no—
24C4×#11758217C2211yescharacteristic
25C2×#14519832—21no—
26#16840626C2212yes—
27#16840623C2212yes—
28Q8⋊#92420C2212yes—
29Q8⋊#92420C2212yes—
30D4⋊#92420C2212yes—
31D4⋊#92420C2212yes—
32#92420⋊C8C2212yes—
33C8×#92420C2212yes—
34#41×Dic61C2211yescharacteristic
35#41×Dic61C2211yescharacteristic
36Q16×Dic61C411yescharacteristic
37Q16×Dic61C2211yescharacteristic
38SD16×Dic61C411yescharacteristic
39SD16×Dic61C411yescharacteristic
40SD16×Dic61C2211yescharacteristic
41SD16×Dic61C2211yescharacteristic
42D8×Dic61C411yescharacteristic
43D8×Dic61C2211yescharacteristic
44#16839800C411yescharacteristic
45#16839758C411yescharacteristic
46#16839758C411yescharacteristic
47C2×C8×Dic61C2211yescharacteristic
48C2×#135×C61C411yescharacteristic
49C2×#11758217—22no—
50C2×#11758217D412yes—
51C22×#92420—22no—
52#14519844—22no—
53#14519844—21no—
54C4⋊#92420—22no—
55C4⋊#92420C2312yes—
56C4⋊#92420C2312yes—
57C22⋊#92420—21no—
58C22⋊#92420—21no—
59C22⋊#92420—21no—
60C22⋊#92420—22no—
61C22⋊#92420—22no—
62#14519832—22no—
63#14519832—21no—
64#14519832—21no—
65#14519832D411yescharacteristic
66#14519832D411yescharacteristic
67C4×#92420—22no—
68C4×#92420C2312yes—
69C4×#92420C2312yes—
70C22×#92418C2×C411yescharacteristic
71Q8×Dic61C2×C411yescharacteristic
72Q8×Dic61C2×C411yescharacteristic
73Q8×Dic61C2311yescharacteristic
74Q8×Dic61C2311yescharacteristic
75D4×Dic61—21no—
76D4×Dic61—21no—
77D4×Dic61—21no—
78D4×Dic61—21no—
79D4×Dic61C2×C411yescharacteristic
80D4×Dic61C2×C411yescharacteristic
81D4×Dic61C2311yescharacteristic
82D4×Dic61C2311yescharacteristic
83#14519736—21no—
84#14519736—21no—
85#14519736C2×C411yescharacteristic
86#14519736C2×C411yescharacteristic
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90#14519711C2×C411yescharacteristic
91#14519711C2×C411yescharacteristic
92#14519709C2×C411yescharacteristic
93#14519709C2×C411yescharacteristic
94#14519701C2×C411yescharacteristic
95#14519701C2×C411yescharacteristic
96#14519701C2×C411yescharacteristic
97#14519701C2×C411yescharacteristic
98#14519698C2×C411yescharacteristic
99#14519698C2×C411yescharacteristic
100#14519698C2×C411yescharacteristic
101#14519698C2×C411yescharacteristic
102C8×Dic61C2×C411yescharacteristic
103C8×Dic61C2×C411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C8×Dic61C2311yescharacteristic
106C2×#41×C61C2×C411yescharacteristic
107C2×#41×C61C2×C411yescharacteristic
108#135×C61—24no—
109C2×Q16×C61C2×C411yescharacteristic
110C2×SD16×C61C2×C411yescharacteristic
111C2×SD16×C61C2×C411yescharacteristic
112C2×D8×C61C2×C411yescharacteristic
113C22×C8×C61C2×C411yescharacteristic
114#41×C61—22no—
115#41×C61—22no—
116#41×C61—41no—
117#41×C61—41no—
118C2×Q8×C61C22×C411yescharacteristic
119C2×Q8×C61C22×C411yescharacteristic
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61C22×C411yescharacteristic
123C2×D4×C61C22×C411yescharacteristic
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C22×C411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61#4111yescharacteristic
133#11758217—24no—
134#11758217—44no—
135C2×#92420—22no—
136C2×#92420—22no—
137C2×#92420—22no—
138C2×#92420—44no—
139C2×#92420C2×D412yes—
140C2×#92420C2×D412yes—
141Q16×C61—22no—
142SD16×C61—22no—
143SD16×C61M1614yescomplemented
144D8×C61—22no—
145C2×C8×C61—22no—
146C2×C8×C61C22×C411yescharacteristic
147C2×C8×C61C22×C411yescharacteristic
148C22⋊Dic61—21no—
149C22⋊Dic61—21no—
150C22⋊Dic61—21no—
151C22⋊Dic61—21no—
152C22⋊Dic61—21no—
153C22⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61C22×C411yescharacteristic
159C4⋊Dic61C22×C411yescharacteristic
160C4⋊Dic61C22×C411yescharacteristic
161C4⋊Dic61C22×C411yescharacteristic
162C4×Dic61—21no—
163C4×Dic61—21no—
164C4×Dic61—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—21no—
168C4×Dic61C22×C411yescharacteristic
169C4×Dic61C2×D411yescharacteristic
170C4×Dic61#4111yescharacteristic
171C4×Dic61C2411yescharacteristic
172C2×#92418—41no—
173C2×#92418C22×C411yescharacteristic
174C2×#92418C22×C411yescharacteristic
175C23×C61—21no—
176C23×C61—21no—
177C23×C61C4×D411yescharacteristic
178Q8×C61—21no—
179Q8×C61—21no—
180Q8×C61—21no—
181Q8×C61C2×M1612yes—
182D4×C61—21no—
183D4×C61—21no—
184D4×C61—21no—
185D4×C61—42no—
186D4×C61—42no—
187D4×C61C2×M1612yes—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61C4×D411yescharacteristic
195C2×C4×C61C23×C411yescharacteristic
196C2×Dic61—22no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—41no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—81no—
205C2×Dic61—81no—
206C2×Dic61C4×D411yescharacteristic
207C2×Dic61C4×D411yescharacteristic
208C2×Dic61C22×D411yescharacteristic
209C2×Dic61C2×#4111yescharacteristic
210#92420—22no—
211#92420—42no—
212#92420—42no—
213#92420—42no—
214#92420#13514yescomplemented
215C8×C61—21no—
216C8×C61C2×M1612yes—
217#92418—21no—
218#92418—21no—
219#18037—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—22no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61C2×C4×D411yescharacteristic
230C4×C61—21no—
231C4×C61—22no—
232C4×C61—41no—
233C4×C61C22×M1611yescharacteristic
234C4×C61#56811yescharacteristic
235Dic61—21no—
236Dic61—21no—
237Dic61—21no—
238Dic61—41no—
239Dic61—41no—
240Dic61—41no—
241Dic61—41no—
242Dic61C2×#13512yes—
243#2971—611no—
244#2970—611no—
245C4×#135—611no—
246#1590—612no—
247#1584—612no—
248#1581—612no—
249C8×SD16—612no—
250#1568—611no—
251#1488—611no—
252#1488—611no—
253C8×M16—611no—
254C2×C61—41no—
255C2×C61—42no—
256C2×C61—81no—
257C2×C61D4×M1611yescharacteristic
258C2×C61C4×#13511yescharacteristic
259C2×C61#295711yescharacteristic
260C2×#135#692311yescharacteristic
261C4×#41—611no—
262C4×#41—611no—
263#446—612no—
264C8×Q8—612no—
265C4×Q16—611no—
266C4×Q16—611no—
267C4×SD16—611no—
268C4×SD16—611no—
269C4×SD16—611no—
270C4×SD16—611no—
271C4×D8—611no—
272C4×D8—611no—
273#436—612no—
274#436—1222no—
275#435—1222no—
276C8×D4—612no—
277#432—611no—
278#432—611no—
279#432—1221no—
280#432—1221no—
281#431—611no—
282#427—611no—
283#416—611no—
284#416—611no—
285C4×M16—611no—
286C4×M16—1221no—
287C2×#97—1221no—
288C2×C4×C8—611no—
289#335—612no—
290#334—612no—
291Q8⋊C8—612no—
292Q8⋊C8—612no—
293D4⋊C8—612no—
294D4⋊C8—612no—
295#322—612no—
296C82—612no—
297C61#1803711yescharacteristic, complemented, Sylow
298C2×#41C2×#692311yescharacteristic
299C2×#41C2×#692311yescharacteristic
300#135—24no—
301C2×Q16C2×#692311yescharacteristic
302C2×SD16C2×#692311yescharacteristic
303C2×SD16C2×#692311yescharacteristic
304C2×D8C2×#692311yescharacteristic
305C2×M16—612no—
306C2×M16—1222no—
307C22×C8—611no—
308C22×C8—1222no—
309C22×C8C2×#692311yescharacteristic
310C4×Q8—611no—
311C4×Q8—611no—
312C4×Q8—611no—
313C4×Q8—611no—
314C4×D4—611no—
315C4×D4—611no—
316C4×D4—611no—
317C4×D4—611no—
318C4×D4—1221no—
319C4×D4—1221no—
320C4×D4—1221no—
321C4×D4—1221no—
322#117—611no—
323#117—611no—
324#117—1221no—
325#117—1221no—
326C2×C42—611no—
327C2×C42—1221no—
328C2×C42—1221no—
329#107—611no—
330#107—611no—
331#106—611no—
332#106—611no—
333C4⋊C8—612no—
334C4⋊C8—612no—
335C4⋊C8—1221no—
336C4⋊C8—1222no—
337C4⋊C8—1222no—
338#103—611no—
339#103—611no—
340#103—611no—
341#103—611no—
342#102—611no—
343#102—611no—
344#102—611no—
345#102—611no—
346C22⋊C8—1222no—
347C22⋊C8—1222no—
348C22⋊C8—1221no—
349C22⋊C8—1221no—
350C22⋊C8—1221no—
351#97—611no—
352#97—611no—
353#97—1221no—
354#97—1221no—
355#97—1222no—
356C4×C8—612no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8—611no—
360C4×C8—611no—
361C4×C8—612no—
362C4×C8—1222no—
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367C2×Q8C22×#692311yescharacteristic
368C2×Q8C22×#692311yescharacteristic
369C2×D4—21no—
370C2×D4—21no—
371C2×D4C22×#692311yescharacteristic
372C2×D4C22×#692311yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—611no—
376C22×C4—611no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4C22×#692311yescharacteristic
382Q16—22no—
383SD16—22no—
384SD16#1175821714yescomplemented
385D8—22no—
386M16—1224no—
387M16—2444no—
388C2×C8—22no—
389C2×C8—612no—
390C2×C8—612no—
391C2×C8—611no—
392C2×C8—611no—
393C2×C8—1222no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—2444no—
397C2×C8—2441no—
398C2×C8C22×#692311yescharacteristic
399C2×C8C22×#692311yescharacteristic
400C4⋊C4—611no—
401C4⋊C4—611no—
402C4⋊C4—611no—
403C4⋊C4—611no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C22⋊C4—1221no—
409C22⋊C4—1221no—
410C22⋊C4—1221no—
411C22⋊C4—1221no—
412C22⋊C4—1221no—
413C22⋊C4—1221no—
414C42—611no—
415C42—611no—
416C42—611no—
417C42—611no—
418C42—1221no—
419C42—1221no—
420C42—1221no—
421C42—1221no—
422C42—1221no—
423C42—1221no—
424C23—21no—
425C23—21no—
426C23D4×#692311yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—21no—
430Q8C2×#1175821712yes—
431D4—21no—
432D4—21no—
433D4—21no—
434D4—42no—
435D4—42no—
436D4C2×#1175821712yes—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—611no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—1222no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—4881no—
453C2×C4—4881no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2441no—
457C2×C4D4×#692311yescharacteristic
458C2×C4C23×#692311yescharacteristic
459C8—21no—
460C8—614no—
461C8—1222no—
462C8—1221no—
463C8—1221no—
464C8—2442no—
465C8—2442no—
466C8—2442no—
467C8C2×#1175821712yes—
468C22—21no—
469C22—22no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22C2×D4×#692311yescharacteristic
478C4—21no—
479C4—22no—
480C4—41no—
481C4—612no—
482C4—1221no—
483C4—1221no—
484C4—1221no—
485C4—2441no—
486C4—2441no—
487C4—2441no—
488C4—2441no—
489C4C22×#1175821711yescharacteristic
490C4#1684084011yescharacteristic
491C2—41no—
492C2—42no—
493C2—81no—
494C2D4×#1175821711yescharacteristic
495C2#135×#692311yescharacteristic
496C2#2146259811yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
Fitting, prime radicalC2×#135×C61C4
socleC22×C61C2×C4×D4
cosocleC4×Dic61C24
hypercenterC2×#135#6923
hypocenterC61#18037
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #21462665C2semidirect
2. #21462656C2semidirect
3. #21461455C2semidirect
4. #21461432C2semidirect
5. #21461420C2semidirect
6. SD16×#92420C2semidirect
7. #21461379C2semidirect
8. #21461150C2semidirect
9. #21461150C2semidirect
10. C8×#11758217C2semidirect
143. SD16×C61M16semidirect
214. #92420#135semidirect
297. C61#18037semidirect
384. SD16#11758217semidirect

Maximal subgroups

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

subgroupindex
#214626652
#214626562
#214614552
#214614322
#214614202
SD16×#924202
#214613792
#214611502
C8×#117582172
#135×Dic612
#1803761