Finite Group Databasebeta

← #29707385

Subgroups — #29707385

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459224C211yesmaximal, characteristic, complemented
2#21454339C211yesmaximal, characteristic
3#21454336C211yesmaximal, characteristic
4#21448309C211yesmaximal, characteristic
5#21446979C214yesmaximal, complemented
6#21446974C214yesmaximal, complemented
7#21445557C211yesmaximal, characteristic, complemented
8C2×#16839522C211yesmaximal, characteristic, complemented
9C2×C61×#339C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840393—22no—
13#16840390—22no—
14#16840390C2214yes—
15#16840388—22no—
16#16840319—22no—
17#16840315—22no—
18#16840315—22no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839772C2212yes—
22#16839770C2212yes—
23#16839768C2212yes—
24#16839766C2212yes—
25#16839764—22no—
26#16839761—22no—
27#16839759—22no—
28#16839756—22no—
29#16839668C2212yes—
30#16839664C2212yes—
31#16839641—22no—
32C4×#11758198—22no—
33#16839530C2214yes—
34#16839522C2214yes—
35C22×M16×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C22×C42×C61C2211yescharacteristic
38C61×#339C2214yes—
39C2×#11758222—22no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222—21no—
43C2×#11758222—21no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222Q814yescomplemented
47C22×Dic122C2311yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—21no—
50C22×C4×D61—21no—
51C2×#11758194—21no—
52C2×#11758194—22no—
53C2×#11758194—22no—
54C2×#11758194C2312yes—
55C22×#92418—21no—
56#14519777—24no—
57#14519776—24no—
58#14519776—24no—
59#14519769—24no—
60#14519768—24no—
61#14519768—24no—
62#14519756—28no—
63#14519754—28no—
64#14519754—28no—
65C4×Dic122—28no—
66#14519736—21no—
67#14519736—21no—
68#14519736—21no—
69#14519736—21no—
70#14519736—21no—
71#14519736—21no—
72#14519736D412yes—
73#14519736C2312yes—
74C2×C4⋊Dic61D412yes—
75C2×C4⋊Dic61C2312yes—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61D412yes—
78C2×#11758198—24no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61D412yes—
81#14519702—24no—
82#14519700—24no—
83#14519699—24no—
84#14519697—24no—
85#14519684—24no—
86C23×C4×C61—21no—
87C23×C4×C61C2311yescharacteristic
88C2×M16×C61—21no—
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C2×M16×C61C2312yes—
92C22×C8×C61—21no—
93#117×C61—21no—
94#117×C61—21no—
95#117×C61D412yes—
96#117×C61C2312yes—
97C2×C4⋊C4×C61D412yes—
98C2×C22⋊C4×C61—21no—
99C2×C42×C61—21no—
100C2×C42×C61—22no—
101C2×C42×C61—22no—
102C2×C42×C61—21no—
103C2×C42×C61C2312yes—
104C24×C61C2×D411yescharacteristic
105C23×D61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—44no—
109C22×C4×C61—42no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×Q812yes—
116C22×C4×C61#4112yes—
117C22×C4×C61#4112yes—
118C22×C4×C61#4112yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—24no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C2×C22⋊D61—42no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61C2×D414yes—
127#11758222—24no—
128#11758222—24no—
129#11758222—24no—
130#11758222—24no—
131#11758222—42no—
132#11758222—42no—
133#11758222—42no—
134#11758222—42no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244C2×Q812yes—
141C2×C4×D61—24no—
142C2×C4×D61—41no—
143C2×C4×D61—41no—
144C2×C4×D61—42no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—28no—
151C2×Dic122C2×Q812yes—
152M16×C61—22no—
153M16×C61—22no—
154M16×C61—42no—
155C2×C8×C61—22no—
156C2×C8×C61—42no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61C2×D414yes—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C42×C61—22no—
162C42×C61—22no—
163C42×C61—22no—
164C42×C61—22no—
165C42×C61—42no—
166C22⋊Dic61—22no—
167C22⋊Dic61—22no—
168C22⋊Dic61—22no—
169C22⋊Dic61—22no—
170#11758198—48no—
171C4⋊Dic61—28no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61C2×D414yes—
176#11758196—28no—
177#11758196—28no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182#11758194—22no—
183#11758194—22no—
184#11758194—42no—
185C2×#92418—22no—
186C2×#92418—42no—
187C23×C61—42no—
188C23×C61#12011yescharacteristic
189C23×C61#12011yescharacteristic
190C23×C61C22⋊Q812yes—
191C23×C61#12412yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—44no—
195C22×D61—41no—
196C22×D61—41no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—42no—
200C2×C4×C61—48no—
201C2×C4×C61—42no—
202C2×C4×C61—82no—
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#12011yescharacteristic
205C2×C4×C61C22⋊Q812yes—
206C2×C4×C61C22⋊Q812yes—
207C2×C4×C61C22⋊Q812yes—
208C2×C4×C61C22⋊Q812yes—
209C2×C4×C61C22⋊Q812yes—
210C2×C4×C61#12412yes—
211C2×C4×C61#12412yes—
212C2×C4×C61#12412yes—
213C2×C4×C61#12412yes—
214C2×C4×C61#12412yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×Q811yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C22⋊D61—44no—
222C22⋊D61—44no—
223C2×Dic61—24no—
224C2×Dic61—24no—
225C2×Dic61—22no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—44no—
229C2×Dic61—82no—
230D244—22no—
231D244—22no—
232D244—41no—
233D244—41no—
234C4×D61—44no—
235C4×D61—44no—
236Dic122—22no—
237Dic122—22no—
238Dic122—41no—
239Dic122—41no—
240Dic122—88no—
241C8×C61—42no—
242#92418—42no—
243#23461—611nomaximal, Sylow
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61#39312yes—
250C22×C61#39312yes—
251C22×C61C2×#12011yescharacteristic
252C22×C61C2×C22⋊Q811yescharacteristic
253C22×C61C2×#12411yescharacteristic
254C2×D61—44no—
255C2×D61—42no—
256C2×D61—82no—
257C2×D61—82no—
258C4×C61—44no—
259C4×C61—44no—
260C4×C61#39312yes—
261C4×C61#39312yes—
262C4×C61C2×C22⋊Q811yescharacteristic
263C4×C61C2×C22⋊Q811yescharacteristic
264C4×C61C2×#12411yescharacteristic
265C4×C61C2×#12411yescharacteristic
266Dic61—42no—
267Dic61—44no—
268Dic61—44no—
269#3458—611no—
270#2896—611no—
271#2896—611no—
272#2297—611no—
273#2028—614no—
274#2027—614no—
275#1742—611no—
276C2×#339—611no—
277C2×#339D6111yescharacteristic, complemented
278C2×C61—21no—
279C2×C61—21no—
280C2×C61—22no—
281C2×C61—22no—
282C2×C61C2×#39311yescharacteristic
283C2×C61#301812yes—
284D61—82no—
285C22×#41—611no—
286C22×M16—611no—
287C22×M16C2×D6111yescharacteristic
288#544—614no—
289#544—1222no—
290#544—1222no—
291#543—1222no—
292#533—1222no—
293#533—1222no—
294#533—1222no—
295C2×#117—611no—
296C2×#117—611no—
297C2×#117C2×D6111yescharacteristic
298C22×C42C2×D6111yescharacteristic
299#419—612no—
300#419—612no—
301#418—612no—
302#418—612no—
303#417—1222no—
304#417—1222no—
305#416—1222no—
306#416—1222no—
307#391—612no—
308#390—612no—
309#386—1222no—
310C4×C22⋊C4—1222no—
311#343—614no—
312#339—614no—
313#339C2×D6114yes—
314C61#2346111yescharacteristic, complemented, Sylow
315C2×#41—614no—
316C2×#41—1222no—
317C2×#41—1221no—
318C2×#41—1221no—
319C2×#41—1222no—
320C2×#41—1221no—
321C2×#41—1221no—
322C2×#41—1222no—
323C22×Q8—611no—
324C22×D4—611no—
325C22×D4—1221no—
326C23×C4—21no—
327C23×C4—1221no—
328C23×C4C22×D6111yescharacteristic
329C2×M16—21no—
330C2×M16—22no—
331C2×M16—22no—
332C2×M16—612no—
333C2×M16—1221no—
334C2×M16—1222no—
335C2×M16—1222no—
336C2×M16C22×D6112yes—
337C22×C8—21no—
338C22×C8—1221no—
339C4⋊Q8—1224no—
340C4⋊Q8—1224no—
341C4⋊Q8—1224no—
342#125—1224no—
343#125—1224no—
344#125—1224no—
345C22⋊Q8—1228no—
346C22⋊Q8—1228no—
347C22⋊Q8—1228no—
348C4×Q8—1228no—
349#117—21no—
350#117—21no—
351#117—612no—
352#117—612no—
353#117—1221no—
354#117—1221no—
355#117—1221no—
356#117—1221no—
357#117—1221no—
358#117—1221no—
359#117D24412yes—
360#117C22×D6112yes—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4D24412yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—612no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1224no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—612no—
373C2×C42—1221no—
374C2×C42C22×D6112yes—
375#103—1224no—
376#103—1224no—
377#102—1224no—
378#102—1224no—
379#95—1224no—
380C24—1221no—
381C24D4×D6111yescharacteristic
382#41—1224no—
383#41—1224no—
384#41—1224no—
385#41—1224no—
386#41—2442no—
387#41—2442no—
388#41—2442no—
389#41—2442no—
390C2×Q8—612no—
391C2×Q8—1228no—
392C2×Q8—1221no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×Q8—1221no—
397C2×D4—612no—
398C2×D4—1224no—
399C2×D4—1221no—
400C2×D4—1221no—
401C2×D4—1221no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—2441no—
405C2×D4—2441no—
406C2×D4—2442no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—614no—
412C22×C4—1222no—
413C22×C4—1221no—
414C22×C4—1224no—
415C22×C4—2441no—
416C22×C4—2441no—
417C22×C4—2442no—
418C22×C4C2×D24411yescharacteristic
419C22×C4C2×D24411yescharacteristic
420C22×C4#1175822212yes—
421C22×C4#1175822212yes—
422C22×C4D4×D6111yescharacteristic
423C22×C4D4×D6111yescharacteristic
424C22×C4D4×D6111yescharacteristic
425C22×C4#1175822412yes—
426C22×C4Q8×D6112yes—
427C22×C4C23×D6111yescharacteristic
428M16—22no—
429M16—22no—
430M16—42no—
431M16—1222no—
432M16—1222no—
433M16—2442no—
434C2×C8—22no—
435C2×C8—42no—
436C2×C8—1222no—
437C2×C8—2442no—
438C4⋊C4—22no—
439C4⋊C4—614no—
440C4⋊C4—1228no—
441C4⋊C4—1228no—
442C4⋊C4—1228no—
443C4⋊C4—1222no—
444C4⋊C4—1222no—
445C4⋊C4—1222no—
446C4⋊C4C2×D24414yes—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—1222no—
450C22⋊C4—1222no—
451C22⋊C4—1222no—
452C22⋊C4—1222no—
453C22⋊C4—2448no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—42no—
459C42—1222no—
460C42—1222no—
461C42—1222no—
462C42—1222no—
463C23—42no—
464C23—1222no—
465C23—2444no—
466C23—2441no—
467C23—2441no—
468C23#1451974711yescharacteristic
469C23#1451974711yescharacteristic
470C23#1451975512yes—
471C23#1451976312yes—
472C23C2×D4×D6111yescharacteristic
473Q8—1222no—
474Q8—1222no—
475Q8—4888no—
476Q8—2441no—
477Q8—2441no—
478D4—1222no—
479D4—1222no—
480D4—2444no—
481D4—2444no—
482D4—2441no—
483D4—2441no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—82no—
490C2×C4—1224no—
491C2×C4—1224no—
492C2×C4—1222no—
493C2×C4—4882no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2444no—
497C2×C4—2444no—
498C2×C4—2444no—
499C2×C4#1451974711yescharacteristic
500C2×C4#1451974711yescharacteristic
501C2×C4#1451975512yes—
502C2×C4#1451975512yes—
503C2×C4#1451975512yes—
504C2×C4#1451975712yes—
505C2×C4#1451975712yes—
506C2×C4#1451976312yes—
507C2×C4#1451976312yes—
508C2×C4#1451976312yes—
509C2×C4#1451976512yes—
510C2×C4#1451976512yes—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C22×D24411yescharacteristic
513C2×C4C2×Q8×D6111yescharacteristic
514C2×C4C2×#1175822411yescharacteristic
515C2×C4C2×#1175822211yescharacteristic
516C2×C4C2×#1175822211yescharacteristic
517C8—42no—
518C8—2442no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—4882no—
525C22—4882no—
526C22—2444no—
527C22—2442no—
528C22#1683967512yes—
529C22#1683967512yes—
530C22C2×#1451974711yescharacteristic
531C22C2×#1451975511yescharacteristic
532C22C2×#1451976311yescharacteristic
533C4—44no—
534C4—44no—
535C4—2442no—
536C4—2444no—
537C4—2444no—
538C4#1683967512yes—
539C4#1683967512yes—
540C4C2×#1451975511yescharacteristic
541C4C2×#1451975711yescharacteristic
542C4C2×#1451976311yescharacteristic
543C4C2×#1451976511yescharacteristic
544C2—21no—
545C2—21no—
546C2—22no—
547C2—22no—
548C2—4882no—
549C2C2×#1683967511yescharacteristic
550C2#2145517012yes—
subgroupG/H
centerC2×C4#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#339C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
hypercenterC2×#339D61
hypocenterC61#23461
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
1. #21459224C2semidirect
5. #21446979C2semidirect
6. #21446974C2semidirect
7. #21445557C2semidirect
8. C2×#16839522C2semidirect
9. C2×C61×#339C2semidirect
46. C2×#11758222Q8semidirect
277. C2×#339D61semidirect
314. C61#23461semidirect

Maximal subgroups

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

subgroupindex
#214592242
#214543392
#214543362
#214483092
#214469792
#214469742
#214455572
C2×#168395222
C2×C61×#3392
#2346161