Finite Group Databasebeta

← #29708920

Subgroups — #29708920

Order 15616.

906 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
1C2×#16839798C211yesmaximal, characteristic, complemented
2#21454339C211yesmaximal, characteristic
3#21454336C211yesmaximal, characteristic
4C2×#16839730C211yesmaximal, characteristic
5#21447268C214yesmaximal, complemented
6#21447263C214yesmaximal, complemented
7#21445557C211yesmaximal, characteristic, complemented
8C2×#16839526C211yesmaximal, characteristic, complemented
9C61×#1742C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C23×#92418C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14#16839798—22no—
15#16839798—22no—
16C2×#14519710C2212yescomplemented
17C2×#14519708C2212yescomplemented
18#16839772—21no—
19#16839770—21no—
20#16839768—21no—
21#16839766—21no—
22#16839764C2214yes—
23#16839761C2214yes—
24#16839759—22no—
25#16839756—22no—
26#16839730—22no—
27#16839730C2212yes—
28#16839730C2212yes—
29C2×#14519689—22no—
30#16839530C2214yescomplemented
31#16839526C2212yescomplemented
32#16839526C2212yescomplemented
33C22×M16×C61C2211yescharacteristic
34C22×M16×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#343C2214yescomplemented
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—22no—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222D412yes—
44C2×#11758222C2312yes—
45C22×Dic122D411yescharacteristic
46C22×D244D411yescharacteristic
47C22×C22⋊D61—21no—
48C22×C4×D61—21no—
49C2×#11758194—21no—
50C2×#11758194—22no—
51C2×#11758194—22no—
52C2×#11758194C2312yes—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418—21no—
56C22×#92418D412yes—
57C22×#92418D412yes—
58C22×#92418C2311yescharacteristic
59C22×#92418C2311yescharacteristic
60#14519736—21no—
61#14519736—21no—
62#14519736—21no—
63#14519736—21no—
64C2×C4⋊Dic61C2312yes—
65C2×C22⋊Dic61—21no—
66C2×C4×Dic61—21no—
67#14519710—24no—
68#14519708—24no—
69#14519702—24no—
70#14519700—24no—
71#14519699—24no—
72#14519697—24no—
73#14519689—24no—
74#14519689—24no—
75C23×C4×C61C2311yescharacteristic
76C2×M16×C61—21no—
77C2×M16×C61—21no—
78C2×M16×C61—22no—
79C2×M16×C61—22no—
80C2×M16×C61—22no—
81C2×M16×C61—22no—
82C2×M16×C61C2312yes—
83C2×M16×C61C2312yes—
84C22×C8×C61—21no—
85C22×C8×C61—21no—
86#117×C61—21no—
87#117×C61—21no—
88#117×C61—21no—
89#117×C61—21no—
90C2×C4⋊C4×C61C2312yes—
91C2×C22⋊C4×C61—21no—
92C2×C42×C61—21no—
93C24×C61#4111yescharacteristic
94C23×D61—21no—
95C22×C4×C61—22no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61#4111yescharacteristic
101C22×C4×C61#4112yes—
102C22×C4×C61#4112yes—
103C22×C4×C61#4112yes—
104C22×C4×C61#4112yes—
105C22×C4×C61C2411yescharacteristic
106C2×C22⋊D61—22no—
107C2×C22⋊D61—22no—
108C2×C22⋊D61—42no—
109C2×C22⋊D61—42no—
110C22×Dic61—21no—
111C22×Dic61—22no—
112#11758222—24no—
113#11758222—24no—
114#11758222—22no—
115#11758222—22no—
116#11758222—22no—
117#11758222—22no—
118#11758222—42no—
119#11758222—42no—
120#11758222—42no—
121#11758222—42no—
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—21no—
125C2×D244C2×D412yes—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—21no—
133C2×Dic122C2×D412yes—
134M16×C61—22no—
135M16×C61—22no—
136M16×C61—22no—
137M16×C61—22no—
138M16×C61—42no—
139M16×C61—42no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—42no—
143C2×C8×C61—42no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C42×C61—41no—
149C42×C61—41no—
150C22⋊Dic61—41no—
151C22⋊Dic61—41no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4×Dic61—41no—
155C4×Dic61—41no—
156#11758194—22no—
157#11758194—22no—
158#11758194—42no—
159C2×#92418—24no—
160C2×#92418—24no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—22no—
166C2×#92418—42no—
167C2×#92418C2×D412yes—
168C2×#92418C2×D412yes—
169C23×C61—42no—
170C23×C61#12111yescharacteristic
171C23×C61#12111yescharacteristic
172C23×C61#12612yes—
173C23×C61#12612yes—
174C23×C61C2×#4111yescharacteristic
175C22×D61—22no—
176C22×D61—44no—
177C22×D61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—82no—
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12312yes—
183C2×C4×C61#12312yes—
184C2×C4×C61#12312yes—
185C2×C4×C61#12312yes—
186C2×C4×C61#12312yes—
187C2×C4×C61#12312yes—
188C2×C4×C61#12412yes—
189C2×C4×C61#12412yes—
190C2×C4×C61#12612yes—
191C2×C4×C61#12612yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C22⋊D61—42no—
199C22⋊D61—42no—
200C22⋊D61—44no—
201C2×Dic61—22no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—82no—
205D244—22no—
206D244—22no—
207D244—42no—
208C4×D61—42no—
209C4×D61—42no—
210C4×D61—44no—
211Dic122—22no—
212Dic122—22no—
213Dic122—42no—
214C8×C61—42no—
215C8×C61—42no—
216#92418—22no—
217#92418—22no—
218#92418—42no—
219#23646—611nomaximal, Sylow
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61#39712yes—
226C22×C61#39712yes—
227C22×C61C2×#12111yescharacteristic
228C22×C61C2×#12611yescharacteristic
229C22×C61C2×#12611yescharacteristic
230C2×D61—42no—
231C2×D61—44no—
232C2×D61—84no—
233C4×C61—82no—
234C4×C61#39712yes—
235C4×C61#39712yes—
236C4×C61C2×#12311yescharacteristic
237C4×C61C2×#12311yescharacteristic
238C4×C61C2×#12311yescharacteristic
239C4×C61C2×#12411yescharacteristic
240Dic61—42no—
241Dic61—82no—
242C2×#427—611no—
243#2896—611no—
244#2896—611no—
245C2×#408—611no—
246#2079—614no—
247#2078—614no—
248#1742—611no—
249#1742D6111yescharacteristic, complemented
250C2×#341—611no—
251C2×C61—22no—
252C2×C61—21no—
253C2×C61—21no—
254C2×C61—22no—
255C2×C61C2×#39711yescharacteristic
256C2×C61#306712yes—
257D61—82no—
258C22×#41—611no—
259C22×M16—611no—
260C22×M16C2×D6111yescharacteristic
261C22×M16C2×D6111yescharacteristic
262C23×C8—611no—
263C2×#117—611no—
264C2×#117C2×D6111yescharacteristic
265#427—1222no—
266#427—1222no—
267C2×#107—612no—
268C2×#106—612no—
269#419—1221no—
270#419—1221no—
271#418—1221no—
272#418—1221no—
273#417—614no—
274#417—614no—
275#416—1222no—
276#416—1222no—
277#408—612no—
278#408—612no—
279#408—1222no—
280C2×C22⋊C8—1222no—
281#343—614no—
282#343C2×D6114yescomplemented
283#341—612no—
284#341—612no—
285C61#2364611yescharacteristic, complemented, Sylow
286C2×#41—612no—
287C2×#41—612no—
288C2×#41—1221no—
289C2×#41—1221no—
290C2×#41—1222no—
291C2×#41—1222no—
292C2×#41—1222no—
293C2×#41—1222no—
294C22×Q8—611no—
295C22×D4—611no—
296C22×D4—1221no—
297C23×C4—1221no—
298C23×C4C22×D6111yescharacteristic
299C2×M16—21no—
300C2×M16—21no—
301C2×M16—22no—
302C2×M16—22no—
303C2×M16—22no—
304C2×M16—22no—
305C2×M16—612no—
306C2×M16—1221no—
307C2×M16—1222no—
308C2×M16—1222no—
309C2×M16C22×D6112yes—
310C2×M16C22×D6112yes—
311C22×C8—21no—
312C22×C8—21no—
313C22×C8—612no—
314C22×C8—612no—
315C22×C8—611no—
316C22×C8—611no—
317C22×C8—1222no—
318C22×C8—1222no—
319C22×C8—1221no—
320#117—21no—
321#117—21no—
322#117—21no—
323#117—21no—
324#117—1221no—
325#117—1221no—
326#117—1221no—
327#117—1221no—
328C2×C4⋊C4—612no—
329C2×C4⋊C4C22×D6112yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—1221no—
332C2×C42—21no—
333C2×C42—1221no—
334#107—1224no—
335#106—1224no—
336#103—1224no—
337#103—1224no—
338#102—1224no—
339#102—1224no—
340C22⋊C8—1224no—
341C22⋊C8—1224no—
342C24—1221no—
343C24#1175822211yescharacteristic
344#41—1222no—
345#41—1222no—
346#41—1222no—
347#41—1222no—
348#41—1224no—
349#41—1224no—
350#41—2442no—
351#41—2442no—
352#41—2442no—
353#41—2442no—
354C2×Q8—612no—
355C2×Q8—1221no—
356C2×Q8—1222no—
357C2×Q8—1222no—
358C2×D4—612no—
359C2×D4—1222no—
360C2×D4—1222no—
361C2×D4—1221no—
362C2×D4—1222no—
363C2×D4—1222no—
364C2×D4—2442no—
365C2×D4—2442no—
366C22×C4—22no—
367C22×C4—1221no—
368C22×C4—1222no—
369C22×C4—1222no—
370C22×C4—1222no—
371C22×C4—2442no—
372C22×C4—2442no—
373C22×C4#1175822212yes—
374C22×C4#1175822212yes—
375C22×C4#1175822211yescharacteristic
376C22×C4D4×D6111yescharacteristic
377C22×C4D4×D6111yescharacteristic
378C22×C4D4×D6111yescharacteristic
379C22×C4D4×D6111yescharacteristic
380C22×C4#1175822412yes—
381C22×C4#1175822612yes—
382C22×C4C23×D6111yescharacteristic
383M16—22no—
384M16—22no—
385M16—22no—
386M16—22no—
387M16—42no—
388M16—42no—
389M16—1222no—
390M16—1222no—
391M16—2442no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—612no—
397C2×C8—612no—
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—1224no—
403C2×C8—1224no—
404C2×C8—1222no—
405C2×C8—2442no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C42—41no—
415C42—41no—
416C42—2441no—
417C42—2441no—
418C23—42no—
419C23—1222no—
420C23—2444no—
421C23—2442no—
422C23#1451975011yescharacteristic
423C23#1451975011yescharacteristic
424C23#1451977112yes—
425C23#1451977212yes—
426C23C2×#1175822211yescharacteristic
427Q8—1222no—
428Q8—1222no—
429Q8—2442no—
430D4—1222no—
431D4—1222no—
432D4—2444no—
433D4—2442no—
434D4—2442no—
435D4—2442no—
436C2×C4—42no—
437C2×C4—82no—
438C2×C4—1222no—
439C2×C4—4882no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2444no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4#1451975011yescharacteristic
446C2×C4#1451975011yescharacteristic
447C2×C4#1451975912yes—
448C2×C4#1451975912yes—
449C2×C4#1451976212yes—
450C2×C4#1451976212yes—
451C2×C4#1451976212yes—
452C2×C4#1451976212yes—
453C2×C4#1451976312yes—
454C2×C4#1451976312yes—
455C2×C4#1451977112yes—
456C2×C4#1451977212yes—
457C2×C4C2×D4×D6111yescharacteristic
458C2×C4C2×D4×D6111yescharacteristic
459C2×C4C2×#1175822411yescharacteristic
460C2×C4C2×#1175822211yescharacteristic
461C2×C4C2×#1175822211yescharacteristic
462C2×C4C2×#1175822611yescharacteristic
463C8—42no—
464C8—42no—
465C8—1222no—
466C8—1222no—
467C8—2442no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—42no—
472C22—42no—
473C22—4884no—
474C22—2442no—
475C22—2444no—
476C22#1683969112yes—
477C22#1683969112yes—
478C22C2×#1451975011yescharacteristic
479C22C2×#1451977111yescharacteristic
480C22C2×#1451977211yescharacteristic
481C4—82no—
482C4—4882no—
483C4—2442no—
484C4#1683969112yes—
485C4#1683969112yes—
486C4C2×#1451975911yescharacteristic
487C4C2×#1451976211yescharacteristic
488C4C2×#1451976211yescharacteristic
489C4C2×#1451976311yescharacteristic
490C2—22no—
491C2—22no—
492C2—21no—
493C2—21no—
494C2—4882no—
495C2C2×#1683969111yescharacteristic
496C2#2145553812yes—
subgroupG/H
centerC2×C4#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1742C2
socleC22×C61C2×#121
prime radicalC22×#11758222C22
hypercenter#1742D61
hypocenterC61#23646
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
1. C2×#16839798C2semidirect
5. #21447268C2semidirect
6. #21447263C2semidirect
7. #21445557C2semidirect
8. C2×#16839526C2semidirect
9. C61×#1742C2semidirect
16. C2×#14519710C22semidirect
17. C2×#14519708C22semidirect
30. #16839530C22semidirect
31. #16839526C22semidirect
32. #16839526C22semidirect
36. C61×#343C22semidirect
249. #1742D61semidirect
282. #343C2×D61semidirect
285. C61#23646semidirect

Maximal subgroups

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

subgroupindex
C2×#168397982
#214543392
#214543362
C2×#168397302
#214472682
#214472632
#214455572
C2×#168395262
C61×#17422
#2364661