Finite Group Databasebeta

← C4×#16863396 #30853624

Subgroups — C4×#16863396 #30853624

Order 17152.

1706 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11761629C211yesmaximal, characteristic
2C2×#16863396C212yesmaximal
3C2×C4×#11761611C211yesmaximal, characteristic, complemented
4C4×C4⋊C4×D67C211yesmaximal, characteristic, complemented
5#22560041C212yesmaximal, complemented
6#22560033C212yesmaximal, complemented
7C4×#14523967C211yesmaximal, characteristic, complemented
8C4×#14523963C211yesmaximal, characteristic, complemented
9C4×#14523965C211yesmaximal, characteristic, complemented
10#22559927C212yesmaximal, complemented
11C4×#103×C67C211yesmaximal, characteristic, complemented
12C22×#11761629C2211yescharacteristic
13C22×#11761611C2212yes—
14C4×#11761629—24no—
15C4×#11761629C418yescomplemented
16C2×Q8×Dic67C2211yescharacteristic
17C2×C4×D268—21no—
18C2×C4×D268C2211yescharacteristic
19C2×#14524005—21no—
20C2×#14524005C2211yescharacteristic
21C2×#14524001—21no—
22C2×#14524001C2211yescharacteristic
23C2×C4⋊C4×D67C2212yes—
24C2×C42×D67—21no—
25C2×C42×D67—21no—
26C2×C42×D67C2211yescharacteristic
27#16863396C4116yesdirect product factor
28C2×#14523967C2212yes—
29C2×#14523965C2212yes—
30C2×#14523963C2212yes—
31C4×#11761611—24no—
32C2×#14523953C2212yes—
33C2×C4×C67⋊C8C2211yescharacteristic
34C4⋊C4×Dic67C2212yes—
35C4×Dic67⋊C4—21no—
36C4×C4⋊Dic67C2211yescharacteristic
37#16863247—21no—
38#95×D67—21no—
39#16863159C2212yes—
40#16863158C2212yes—
41#16863158C2212yes—
42C2×C4×Q8×C67C2211yescharacteristic
43C2×#103×C67C2212yes—
44C2×C4×C8×C67C2211yescharacteristic
45C4×C4⋊C4×C67C2211yescharacteristic
46#340×C67C2212yes—
47C2×#11761629—22no—
48C2×#11761629—24no—
49C2×#11761629C2×C414yes—
50C2×#11761629C2×C418yes—
51C22×D268—21no—
52C22×D268C2311yescharacteristic
53C22×C4×D67—21no—
54C22×C4×D67—22no—
55C22×C4×D67—21no—
56C22×C4×D67—21no—
57C22×C4×D67D411yescharacteristic
58C22×C4×D67D411yescharacteristic
59C22×C4×D67C2311yescharacteristic
60C2×#11761611—24no—
61C2×#11761611—24no—
62C2×#11761611C2×C418yescomplemented
63C22×C67⋊C8C2312yes—
64Q8×Dic67—22no—
65Q8×Dic67C2×C414yescomplemented
66C4×D268—28no—
67C4×D268—22no—
68C4×D268C2×C414yescomplemented
69#14524005—28no—
70#14524005—22no—
71#14524005C2×C414yescomplemented
72#14524001—22no—
73#14524001—28no—
74#14524001C2×C414yes—
75C2×C4⋊Dic67—21no—
76C2×C4⋊Dic67C2312yes—
77C2×C4⋊Dic67C2311yescharacteristic
78C2×Dic67⋊C4—22no—
79C4⋊C4×D67—24no—
80C4⋊C4×D67C2×C418yescomplemented
81C2×(C2×D67)⋊C4—21no—
82C2×(C2×D67)⋊C4—21no—
83C2×(C2×D67)⋊C4—21no—
84C2×C4×Dic67—21no—
85C2×C4×Dic67—21no—
86C2×C4×Dic67—22no—
87C2×C4×Dic67—21no—
88C2×C4×Dic67D411yescharacteristic
89C2×C4×Dic67D411yescharacteristic
90C2×C4×Dic67C2311yescharacteristic
91C42×D67—28no—
92C42×D67—44no—
93C42×D67C2×C414yes—
94C42×D67D414yes—
95#14523967C2×C418yescomplemented
96#14523965C2×C418yescomplemented
97#14523963C2×C418yescomplemented
98#14523953—24no—
99C4×C67⋊C8—22no—
100C67⋊#95—22no—
101C67⋊#95—21no—
102C22×Q8×C67C2311yescharacteristic
103C22×C8×C67C2312yes—
104C4×Q8×C67—22no—
105C4×Q8×C67C2×C414yescomplemented
106C2×C4⋊C4×C67—21no—
107C2×C4⋊C4×C67C2312yes—
108C2×C4⋊C4×C67C2311yescharacteristic
109C2×C42×C67—21no—
110C2×C42×C67—21no—
111C2×C42×C67C2311yescharacteristic
112#103×C67C2×C418yescomplemented
113C4×C8×C67—22no—
114#95×C67—21no—
115C23×D67—21no—
116C23×D67C2×D411yescharacteristic
117#11761629—28no—
118#11761629—216no—
119#11761629C42116yescomplemented
120C2×D268—24no—
121C2×D268—28no—
122C2×D268—21no—
123C2×D268—22no—
124C2×D268C22×C414yes—
125C2×D268C22×C412yes—
126C22×Dic67—22no—
127C22×Dic67—21no—
128C22×Dic67C2×D411yescharacteristic
129C22×Dic67C2×D411yescharacteristic
130C22×Dic67C2×D411yescharacteristic
131C2×C4×D67—22no—
132C2×C4×D67—22no—
133C2×C4×D67—24no—
134C2×C4×D67—28no—
135C2×C4×D67—28no—
136C2×C4×D67—24no—
137C2×C4×D67—24no—
138C2×C4×D67—42no—
139C2×C4×D67—44no—
140C2×C4×D67—44no—
141C2×C4×D67—44no—
142C2×C4×D67—42no—
143C2×C4×D67—42no—
144C2×C4×D67C22⋊C414yes—
145C2×C4×D67C22⋊C414yes—
146C2×C4×D67C22×C412yes—
147C2×C4×D67C22×C414yes—
148C2×C4×D67C2×D412yes—
149C2×C4×D67#4114yes—
150#11761611—216no—
151C2×C67⋊C8—22no—
152C2×C67⋊C8—22no—
153C2×C67⋊C8C22×C414yes—
154C4⋊Dic67—21no—
155C4⋊Dic67—22no—
156C4⋊Dic67—24no—
157C4⋊Dic67C22×C414yes—
158C4⋊Dic67C22×C412yes—
159Dic67⋊C4—28no—
160(C2×D67)⋊C4—24no—
161(C2×D67)⋊C4—24no—
162(C2×D67)⋊C4—24no—
163C4×Dic67—24no—
164C4×Dic67—24no—
165C4×Dic67—44no—
166C4×Dic67—42no—
167C4×Dic67C22⋊C414yes—
168C4×Dic67C22⋊C414yes—
169C4×Dic67C22×C412yes—
170C4×Dic67C2×D412yes—
171C2×Q8×C67—21no—
172C2×Q8×C67—22no—
173C2×Q8×C67C22×C414yes—
174C2×Q8×C67C22×C412yes—
175C22×C4×C67—21no—
176C22×C4×C67—21no—
177C22×C4×C67—22no—
178C22×C4×C67C2×D411yescharacteristic
179C22×C4×C67C2×D411yescharacteristic
180C22×C4×C67C2411yescharacteristic
181C2×C8×C67—22no—
182C2×C8×C67—22no—
183C2×C8×C67C22×C414yes—
184C4⋊C4×C67—24no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—22no—
187C4⋊C4×C67C22×C414yes—
188C4⋊C4×C67C22×C412yes—
189C42×C67—24no—
190C42×C67—42no—
191C42×C67C22×C412yes—
192C42×C67C2×D412yes—
193C23×C67C22×D411yescharacteristic
194C22×D67—21no—
195C22×D67—21no—
196C22×D67—22no—
197C22×D67—22no—
198C22×D67—24no—
199C22×D67—41no—
200C22×D67—41no—
201C22×D67—42no—
202C22×D67C2×C22⋊C412yes—
203C22×D67C4×D414yes—
204Q8×C67—22no—
205Q8×C67—24no—
206Q8×C67C2×C4214yescomplemented
207C2×C4×C67—22no—
208C2×C4×C67—24no—
209C2×C4×C67—22no—
210C2×C4×C67—22no—
211C2×C4×C67—24no—
212C2×C4×C67—42no—
213C2×C4×C67—42no—
214C2×C4×C67—42no—
215C2×C4×C67C2×C22⋊C412yes—
216C2×C4×C67C2×C22⋊C412yes—
217C2×C4×C67AGL(1,ℤ/8)12yes—
218C2×C4×C67#13712yes—
219C2×C4×C67C23×C411yescharacteristic
220C2×C4×C67C23×C412yes—
221C2×C4×C67C22×D411yescharacteristic
222C2×C4×C67C2×#4112yes—
223C2×Dic67—21no—
224C2×Dic67—21no—
225C2×Dic67—22no—
226C2×Dic67—22no—
227C2×Dic67—24no—
228C2×Dic67—42no—
229C2×Dic67—41no—
230C2×Dic67—41no—
231C2×Dic67—42no—
232C2×Dic67C2×C22⋊C412yes—
233C2×Dic67C2×C22⋊C412yes—
234C2×Dic67C2×C22⋊C412yes—
235C2×Dic67C4×D414yes—
236D268—216no—
237D268—22no—
238D268—24no—
239D268C2×C4214yescomplemented
240C4×D67—28no—
241C4×D67—216no—
242C4×D67—44no—
243C4×D67—48no—
244C4×D67—416no—
245C4×D67—48no—
246C4×D67C2×C4214yes—
247C4×D67#11714yes—
248C4×D67C4×D418yes—
249C8×C67—24no—
250C67⋊C8—24no—
251C22×C67C22×C22⋊C411yescharacteristic
252C22×C67C2×C4×D412yes—
253C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C67C2×#13711yescharacteristic
255C22×C67#57512yes—
256C2×D67—22no—
257C2×D67—22no—
258C2×D67—24no—
259C2×D67—24no—
260C2×D67—24no—
261C2×D67—42no—
262C2×D67—42no—
263C2×D67—44no—
264C2×D67—44no—
265C2×D67C4×C22⋊C414yes—
266C4×C67—22no—
267C4×C67—24no—
268C4×C67—42no—
269C4×C67—44no—
270C4×C67#41714yescomplemented
271C4×C67C22×C4211yescharacteristic
272C4×C67C2×#11711yescharacteristic
273C4×C67C2×C4×D412yes—
274Dic67—22no—
275Dic67—22no—
276Dic67—42no—
277Dic67—44no—
278Dic67C4×C22⋊C414yes—
279C4×#417—671nomaximal, Sylow
280C2×C67C2×C4×C22⋊C411yescharacteristic
281C2×C67C2×#41711yescharacteristic
282C2×C67#289611yescharacteristic
283C2×C67C4×AGL(1,ℤ/8)12yes—
284C2×C67C4×#13712yes—
285D67—24no—
286D67—44no—
287C2×C4×#41—671no—
288C2×#417—672no—
289C2×C4×M16—671no—
290C2×C4×C4⋊C4—671no—
291#1763—672no—
292#1762—672no—
293C4×#103—671no—
294C4×#103D6711yescharacteristic, complemented
295C4×#102—671no—
296C4×#102—671no—
297#1736—672no—
298C67C4×#41711yescharacteristic, complemented, Sylow
299C22×#41—671no—
300C22×M16—672no—
301C4×#41—678no—
302C4×#41—1344no—
303C2×C4×Q8—671no—
304C2×C4×Q8C2×D6711yescharacteristic
305C2×C4×D4—671no—
306C2×C4×D4—671no—
307C2×C4×D4—1341no—
308C2×C4×D4—1341no—
309C2×#117—671no—
310C2×#117—1341no—
311C22×C4⋊C4—672no—
312C22×C42—671no—
313C22×C42—1341no—
314C22×C42—1341no—
315#417—6716no—
316C2×#103—672no—
317C2×#103C2×D6712yes—
318C2×#102—672no—
319C2×#102—672no—
320C4×M16—1344no—
321C2×#97—672no—
322C2×C4×C8—671no—
323C2×C4×C8C2×D6711yescharacteristic
324C4×C4⋊C4—672no—
325C4×C4⋊C4—671no—
326C4×C4⋊C4—1341no—
327C4×C4⋊C4—1341no—
328C4×C4⋊C4C2×D6711yescharacteristic
329C2×#95—1341no—
330#340—672no—
331#340—672no—
332#340—672no—
333#340C2×D6712yes—
334C2×#41—674no—
335C2×#41—678no—
336C2×#41—1342no—
337C2×#41—1344no—
338C22×Q8C22×D6711yescharacteristic
339C22×D4—671no—
340C22×D4—1341no—
341C23×C4—671no—
342C23×C4—671no—
343C23×C4—671no—
344C23×C4—1341no—
345C23×C4—1342no—
346C23×C4—1341no—
347C23×C4—1341no—
348C2×M16—678no—
349C2×M16—1344no—
350C2×M16—1344no—
351C22×C8—672no—
352C22×C8C22×D6712yes—
353C4×Q8—22no—
354C4×Q8—674no—
355C4×Q8—1342no—
356C4×Q8C4×D6714yescomplemented
357C4×D4—674no—
358C4×D4—674no—
359C4×D4—1348no—
360C4×D4—1342no—
361C4×D4—1348no—
362C4×D4—1342no—
363#117—674no—
364#117—1348no—
365#117—1342no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—672no—
368C2×C4⋊C4—671no—
369C2×C4⋊C4—678no—
370C2×C4⋊C4—1344no—
371C2×C4⋊C4—1342no—
372C2×C4⋊C4—1341no—
373C2×C4⋊C4C22×D6712yes—
374C2×C4⋊C4C22×D6711yescharacteristic
375C2×C22⋊C4—1341no—
376C2×C22⋊C4—1341no—
377C2×C22⋊C4—1341no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—2684no—
381C2×C42—671no—
382C2×C42—671no—
383C2×C42—671no—
384C2×C42—674no—
385C2×C42—674no—
386C2×C42—1348no—
387C2×C42—1342no—
388C2×C42—1341no—
389C2×C42—1341no—
390C2×C42—1341no—
391C2×C42C22×D6711yescharacteristic
392#103—678no—
393#103C4×D6718yescomplemented
394#102—678no—
395#102—678no—
396#97—1344no—
397C4×C8—22no—
398C4×C8—1342no—
399#95—21no—
400#95—1342no—
401#95—1341no—
402C24—671no—
403C24—1341no—
404#41—6716no—
405#41—1348no—
406#41—13416no—
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8C2×C4×D6714yes—
410C2×Q8C2×C4×D6712yes—
411C2×D4—674no—
412C2×D4—672no—
413C2×D4—1344no—
414C2×D4—1348no—
415C2×D4—1341no—
416C2×D4—1342no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—2682no—
421C22×C4—2684no—
422C22×C4—2684no—
423C22×C4—2684no—
424C22×C4—2682no—
425C22×C4—2682no—
426C22×C4—674no—
427C22×C4—671no—
428C22×C4—672no—
429C22×C4—674no—
430C22×C4—672no—
431C22×C4—674no—
432C22×C4—671no—
433C22×C4—671no—
434C22×C4—674no—
435C22×C4—1344no—
436C22×C4—1348no—
437C22×C4—1344no—
438C22×C4—1348no—
439C22×C4—1342no—
440C22×C4—1341no—
441C22×C4—1344no—
442C22×C4—1342no—
443C22×C4—1342no—
444C22×C4D4×D6711yescharacteristic
445C22×C4D4×D6711yescharacteristic
446C22×C4C23×D6711yescharacteristic
447M16—13416no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—674no—
451C2×C8—1342no—
452C2×C8—1342no—
453C2×C8C2×C4×D6714yes—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—674no—
458C4⋊C4—672no—
459C4⋊C4—1348no—
460C4⋊C4—1344no—
461C4⋊C4—1341no—
462C4⋊C4—1342no—
463C4⋊C4C2×C4×D6714yes—
464C4⋊C4C2×C4×D6712yes—
465C22⋊C4—1344no—
466C22⋊C4—1344no—
467C22⋊C4—1344no—
468C42—24no—
469C42—42no—
470C42—2684no—
471C42—2682no—
472C42—674no—
473C42—674no—
474C42—672no—
475C42—672no—
476C42—1344no—
477C42—1344no—
478C42C2×C4×D6712yes—
479C42D4×D6712yes—
480C23—2681no—
481C23—2681no—
482C23—2682no—
483C23—672no—
484C23—674no—
485C23—1342no—
486C23—1344no—
487C23—1341no—
488C23—1341no—
489C23—1342no—
490C23C2×D4×D6711yescharacteristic
491Q8—22no—
492Q8—24no—
493Q8C42×D6714yescomplemented
494D4—674no—
495D4—13416no—
496D4—1342no—
497D4—1344no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—2684no—
507C2×C4—26816no—
508C2×C4—2688no—
509C2×C4—2682no—
510C2×C4—2681no—
511C2×C4—2681no—
512C2×C4—2682no—
513C2×C4—2688no—
514C2×C4—672no—
515C2×C4—674no—
516C2×C4—674no—
517C2×C4—674no—
518C2×C4—678no—
519C2×C4—672no—
520C2×C4—672no—
521C2×C4—13416no—
522C2×C4—1342no—
523C2×C4—1344no—
524C2×C4—1341no—
525C2×C4—1341no—
526C2×C4—1342no—
527C2×C4—1348no—
528C2×C4C22⋊C4×D6712yes—
529C2×C4C22⋊C4×D6712yes—
530C2×C4#1452407012yes—
531C2×C4#1452407512yes—
532C2×C4C22×C4×D6711yescharacteristic
533C2×C4C22×C4×D6712yes—
534C2×C4C2×D4×D6711yescharacteristic
535C2×C4#41×D6712yes—
536C8—24no—
537C8—1344no—
538C22—2682no—
539C22—2682no—
540C22—2684no—
541C22—2684no—
542C22—674no—
543C22—1344no—
544C22—1342no—
545C22—1342no—
546C22—1344no—
547C22—1344no—
548C22C2×C22⋊C4×D6711yescharacteristic
549C22C4×D4×D6712yes—
550C22C2×#1452407011yescharacteristic
551C22C2×#1452407511yescharacteristic
552C22#1686419312yes—
553C4—22no—
554C4—24no—
555C4—42no—
556C4—44no—
557C4—2682no—
558C4—2684no—
559C4—674no—
560C4—1342no—
561C4—1342no—
562C4#1686339614yesdirect product factor
563C4C2×C42×D6711yescharacteristic
564C4#117×D6711yescharacteristic
565C4C4×D4×D6712yes—
566C2—2684no—
567C2—1344no—
568C2C4×C22⋊C4×D6711yescharacteristic
569C2C2×#1686339611yescharacteristic
570C2#2256872411yescharacteristic
571C2C4×#1452407012yes—
572C2C4×#1452407512yes—
subgroupG/H
centerC22×C4D4×D67
commutatorC4×C67C22×C42
FrattiniC22×C4C23×D67
FittingC4×#103×C67C2
socleC23×C67C22×D4
prime radicalC22×#11761629C22
cosocleC22×C4×C67C24
hypercenterC4×#103D67
hypocenterC67C4×#417
LCSG ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
3. C2×C4×#11761611C2semidirect
4. C4×C4⋊C4×D67C2semidirect
5. #22560041C2semidirect
6. #22560033C2semidirect
7. C4×#14523967C2semidirect
8. C4×#14523963C2semidirect
9. C4×#14523965C2semidirect
10. #22559927C2semidirect
11. C4×#103×C67C2semidirect
15. C4×#11761629C4semidirect
27. #16863396C4direct (N × H)
62. C2×#11761611C2×C4semidirect
65. Q8×Dic67C2×C4semidirect
68. C4×D268C2×C4semidirect
71. #14524005C2×C4semidirect
80. C4⋊C4×D67C2×C4semidirect
95. #14523967C2×C4semidirect
96. #14523965C2×C4semidirect
97. #14523963C2×C4semidirect
105. C4×Q8×C67C2×C4semidirect
112. #103×C67C2×C4semidirect
119. #11761629C42semidirect
206. Q8×C67C2×C42semidirect
239. D268C2×C42semidirect
270. C4×C67#417semidirect
294. C4×#103D67semidirect
298. C67C4×#417semidirect
356. C4×Q8C4×D67semidirect
393. #103C4×D67semidirect
493. Q8C42×D67semidirect
562. C4#16863396direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#117616292
C2×#168633962
C2×C4×#117616112
C4×C4⋊C4×D672
#225600412
#225600332
C4×#145239672
C4×#145239632
C4×#145239652
#225599272
C4×#103×C672
C4×#41767