Finite Group Databasebeta

← #33127282

Subgroups — #33127282

Order 18688.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988868C211yesmaximal, characteristic, complemented
2#22618283C211yesmaximal, characteristic, complemented
3C4×#14556565C211yesmaximal, characteristic
4#22611359C212yesmaximal, complemented
5#22611121C212yesmaximal
6#22610882C211yesmaximal, characteristic, complemented
7#22610695C211yesmaximal, characteristic, complemented
8#22610674C212yesmaximal, complemented
9#22610628C212yesmaximal, complemented
10#22609932C211yesmaximal, characteristic, complemented
11C2×#432×C73C211yesmaximal, characteristic, complemented
12C2×#14556591C2211yescharacteristic
13C2×#14556570—21no—
14C2×#14556565C2211yescharacteristic
15C2×C4×D292C2211yescharacteristic
16C2×C4×C22⋊D73—21no—
17#17988868—24no—
18C2×#14556525C2212yes—
19C2×#14556523C2212yes—
20C2×#14556513C2212yes—
21C2×#14556513C2212yes—
22#17988493—21no—
23#17988489C2211yescharacteristic
24#17988485—21no—
25#17988485C2212yes—
26#17988485C2211yescharacteristic
27#17988478—21no—
28#17988466—21no—
29#17988457C2211yescharacteristic
30#17988447—21no—
31#17988447C2211yescharacteristic
32#17988447C2212yes—
33C4×C4⋊Dic73C2211yescharacteristic
34C4×#11924005—21no—
35#17988350C2212yes—
36#17988350C2212yes—
37#17988350C2212yes—
38#17988350C2212yes—
39C22×C42×C73C2211yescharacteristic
40#432×C73—24no—
41C2×C4⋊C8×C73C2211yescharacteristic
42C2×C4⋊C8×C73C2211yescharacteristic
43C2×C22⋊C8×C73C2212yes—
44C2×#97×C73C2212yes—
45C22×Dic146—21no—
46C22×D292C2311yescharacteristic
47C22×C22⋊D73—21no—
48C22×C4×D73—21no—
49#14556591—22no—
50#14556591—22no—
51#14556570—44no—
52#14556565—22no—
53#14556565—22no—
54C4×D292—22no—
55C4×D292—22no—
56C4×C22⋊D73—44no—
57C2×C4⋊Dic73—21no—
58C2×C4⋊Dic73—21no—
59C2×C4⋊Dic73—21no—
60C2×C4⋊Dic73C2312yes—
61C2×C4⋊Dic73C2312yes—
62C2×C4⋊Dic73C2311yescharacteristic
63C2×C4⋊Dic73C2311yescharacteristic
64C2×C4⋊Dic73C2311yescharacteristic
65C2×#11924007—21no—
66C2×#11924007—22no—
67C2×C22⋊Dic73—22no—
68C2×C22⋊Dic73—21no—
69C2×#11924005—21no—
70C2×#11924005—21no—
71C2×#11924005—21no—
72C2×#11924005—21no—
73C2×C4×Dic73—21no—
74#14556525—24no—
75#14556523—24no—
76#14556513—24no—
77#14556513—24no—
78#14556498—21no—
79#14556498—21no—
80#14556498—21no—
81#14556498—22no—
82#14556498—22no—
83C23×C4×C73D411yescharacteristic
84C23×C4×C73D411yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C22×C8×C73C2312yes—
87C22×C8×C73C2312yes—
88C2×C42×C73—22no—
89C2×C42×C73—22no—
90C2×C42×C73D411yescharacteristic
91C2×C42×C73D411yescharacteristic
92C2×C42×C73C2311yescharacteristic
93C2×C42×C73C2311yescharacteristic
94C4⋊C8×C73—22no—
95C4⋊C8×C73D414yescomplemented
96C22⋊C8×C73—24no—
97#97×C73—24no—
98C23×D73—21no—
99C2×Dic146—42no—
100C2×Dic146—42no—
101C2×Dic146—42no—
102C2×D292—21no—
103C2×D292—21no—
104C2×D292—21no—
105C2×D292—21no—
106C2×D292—21no—
107C2×D292#4112yes—
108C2×C22⋊D73—24no—
109C2×C22⋊D73—42no—
110C2×C22⋊D73—42no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—21no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C2×C4×D73—42no—
117C2×C4×D73—42no—
118C2×C4×D73—42no—
119C4⋊Dic73—21no—
120C4⋊Dic73—22no—
121C4⋊Dic73—21no—
122C4⋊Dic73—22no—
123C4⋊Dic73—22no—
124C4⋊Dic73—21no—
125C4⋊Dic73—22no—
126C4⋊Dic73—21no—
127C4⋊Dic73—21no—
128C4⋊Dic73—42no—
129C4⋊Dic73—42no—
130C4⋊Dic73—42no—
131C4⋊Dic73#4112yes—
132#11924007—42no—
133#11924007—44no—
134C22⋊Dic73—42no—
135C22⋊Dic73—44no—
136#11924005—24no—
137#11924005—42no—
138#11924005—42no—
139#11924005—42no—
140C4×Dic73—42no—
141C24×C73C2×D411yescharacteristic
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73—41no—
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158C22×C4×C73C2×D411yescharacteristic
159C22×C4×C73C2411yescharacteristic
160C2×C8×C73—22no—
161C2×C8×C73—22no—
162C2×C8×C73—22no—
163C2×C8×C73—22no—
164C2×C8×C73—22no—
165C2×C8×C73C2×D414yes—
166C42×C73—22no—
167C42×C73—22no—
168C42×C73—21no—
169C42×C73—21no—
170C42×C73C2×D412yes—
171C42×C73#4112yes—
172C23×C73—21no—
173C23×C73—21no—
174C23×C73—21no—
175C23×C73—21no—
176C23×C73—21no—
177C23×C73—21no—
178C23×C73#13712yes—
179C23×C73C22×D411yescharacteristic
180C22×D73—22no—
181C22×D73—41no—
182C22×D73—41no—
183C22×D73—41no—
184C22×D73—41no—
185C22×D73—41no—
186C22×D73—41no—
187C2×C4×C73—21no—
188C2×C4×C73—21no—
189C2×C4×C73—21no—
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73—21no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—21no—
196C2×C4×C73—21no—
197C2×C4×C73—21no—
198C2×C4×C73—44no—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—41no—
202C2×C4×C73—41no—
203C2×C4×C73—44no—
204C2×C4×C73#12112yes—
205C2×C4×C73#12112yes—
206C2×C4×C73AGL(1,ℤ/8)12yes—
207C2×C4×C73AGL(1,ℤ/8)12yes—
208C2×C4×C73C22×D411yescharacteristic
209C2×C4×C73C2×#4111yescharacteristic
210C2×C4×C73C2×#4111yescharacteristic
211C2×C4×C73C2×#4111yescharacteristic
212C2×C4×C732−1+411yescharacteristic
213C2×C4×C732−1+411yescharacteristic
214C22⋊D73—48no—
215C2×Dic73—22no—
216C2×Dic73—41no—
217C2×Dic73—41no—
218C2×Dic73—41no—
219C2×Dic73—42no—
220C2×Dic73—42no—
221C2×Dic73—41no—
222C2×Dic73—41no—
223C2×Dic73—42no—
224C2×Dic73—41no—
225C2×Dic73—42no—
226C2×Dic73—41no—
227C2×Dic73—42no—
228C2×Dic73—41no—
229C2×Dic73—42no—
230D292—22no—
231D292—22no—
232D292—41no—
233D292—41no—
234C4×D73—84no—
235Dic146—84no—
236C8×C73—24no—
237C8×C73—42no—
238C22×C73—22no—
239C22×C73—22no—
240C22×C73—22no—
241C22×C73—41no—
242C22×C73—41no—
243C22×C73—41no—
244C22×C73—41no—
245C22×C73C2×#12111yescharacteristic
246C22×C73#52511yescharacteristic
247C22×C73#53611yescharacteristic
248C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
249C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C73C2×#13711yescharacteristic
251C22×C73#57511yescharacteristic
252C2×D73—42no—
253C2×D73—42no—
254C2×D73—42no—
255C2×D73—81no—
256C2×D73—81no—
257C2×D73—81no—
258C2×D73—81no—
259C4×C73—22no—
260C4×C73—41no—
261C4×C73—41no—
262C4×C73—41no—
263C4×C73#46914yes—
264C4×C73#54711yescharacteristic
265C4×C73#54811yescharacteristic
266C4×C73#55211yescharacteristic
267C4×C73#55611yescharacteristic
268Dic73—42no—
269Dic73—82no—
270Dic73—81no—
271Dic73—82no—
272Dic73—81no—
273#24709—731nomaximal, Sylow
274C2×C73—42no—
275C2×C73#241511yescharacteristic
276C2×C73C2×#46911yescharacteristic
277C2×C73#305611yescharacteristic
278C2×C73#311011yescharacteristic
279C2×C73#311511yescharacteristic
280C2×C73#323511yescharacteristic
281C2×C73#323911yescharacteristic
282D73—82no—
283C2×#474—731no—
284C2×#432D7311yescharacteristic, complemented
285#2857—731no—
286C4×#121—731no—
287#2076—732no—
288#2034—732no—
289#1988—731no—
290#1948—731no—
291#1944—732no—
292#1936—732no—
293#1802—731no—
294C73#2470911yescharacteristic, complemented, Sylow
295C2×C4⋊Q8—731no—
296C2×C22⋊Q8—1461no—
297C2×#121—731no—
298C2×C4×D4—731no—
299C2×C4×D4—1461no—
300C22×C42C2×D7311yescharacteristic
301#474—1464no—
302#432—24no—
303C2×#107—732no—
304C2×#106—732no—
305C2×C4⋊C8C2×D7311yescharacteristic
306C2×C4⋊C8C2×D7311yescharacteristic
307C2×#102—732no—
308C2×#102—732no—
309C2×C22⋊C8C2×D7312yes—
310C2×#97C2×D7312yes—
311#391—1461no—
312#390—731no—
313#389—732no—
314#389—731no—
315#389—1461no—
316#388—1461no—
317#386—1461no—
318#384—731no—
319#380—731no—
320#380—732no—
321#380—1461no—
322C4×C4⋊C4—731no—
323C4×C22⋊C4—1461no—
324#340—732no—
325#340—732no—
326#340—732no—
327#340—732no—
328C22×Q8—1461no—
329C22×D4—731no—
330C22×D4—1461no—
331C23×C4—1461no—
332C23×C4D29211yescharacteristic
333C23×C4D29211yescharacteristic
334C23×C4C22×D7311yescharacteristic
335C22×C8C22×D7312yes—
336C22×C8C22×D7312yes—
337C4⋊Q8—1462no—
338C4⋊Q8—1462no—
339C22⋊Q8—2924no—
340#121—1462no—
341#121—1462no—
342C4×D4—2924no—
343C4×D4—1462no—
344C4×D4—1462no—
345C2×C4⋊C4—731no—
346C2×C4⋊C4—731no—
347C2×C4⋊C4—732no—
348C2×C4⋊C4—731no—
349C2×C4⋊C4—732no—
350C2×C4⋊C4—1461no—
351C2×C4⋊C4—1461no—
352C2×C4⋊C4—1462no—
353C2×C4⋊C4—1461no—
354C2×C4⋊C4—1461no—
355C2×C22⋊C4—1461no—
356C2×C22⋊C4—1462no—
357C2×C22⋊C4—1461no—
358C2×C22⋊C4—1461no—
359C2×C22⋊C4—1461no—
360C2×C22⋊C4—1461no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—1461no—
364C2×C42D29211yescharacteristic
365C2×C42D29211yescharacteristic
366C2×C42C22×D7311yescharacteristic
367C2×C42C22×D7311yescharacteristic
368#107—1464no—
369#106—1464no—
370C4⋊C8—22no—
371C4⋊C8C22⋊D7314yescomplemented
372#102—1464no—
373#102—1464no—
374C22⋊C8—24no—
375#97—24no—
376#95—1461no—
377#95—1461no—
378#95—1461no—
379#95—1462no—
380#95—1462no—
381C24—1461no—
382C24C2×D29211yescharacteristic
383C2×Q8—2922no—
384C2×Q8—2922no—
385C2×Q8—2922no—
386C2×D4—2922no—
387C2×D4—2922no—
388C2×D4—732no—
389C2×D4—1464no—
390C2×D4—1461no—
391C2×D4—1461no—
392C2×D4—1461no—
393C2×D4—1461no—
394C2×D4—1461no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—2922no—
408C22×C4—2922no—
409C22×C4—2922no—
410C22×C4—1461no—
411C22×C4—1461no—
412C22×C4—1462no—
413C22×C4—1462no—
414C22×C4—1461no—
415C22×C4C2×D29211yescharacteristic
416C22×C4C2×D29211yescharacteristic
417C22×C4C2×D29211yescharacteristic
418C22×C4C2×D29211yescharacteristic
419C22×C4C2×D29211yescharacteristic
420C22×C4C23×D7311yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8C2×C22⋊D7314yes—
427C4⋊C4—2922no—
428C4⋊C4—2924no—
429C4⋊C4—2922no—
430C4⋊C4—2922no—
431C4⋊C4—2922no—
432C4⋊C4—732no—
433C4⋊C4—1462no—
434C4⋊C4—1461no—
435C4⋊C4—1462no—
436C4⋊C4—1462no—
437C4⋊C4—1461no—
438C4⋊C4—1462no—
439C4⋊C4—1461no—
440C4⋊C4—1461no—
441C4⋊C4—1461no—
442C22⋊C4—2922no—
443C22⋊C4—2922no—
444C22⋊C4—2922no—
445C22⋊C4—2924no—
446C22⋊C4—2922no—
447C22⋊C4—1464no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—2922no—
453C42C2×C22⋊D7312yes—
454C42#1192402812yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—2921no—
462C23—2921no—
463C23—2921no—
464C23—2921no—
465C23—2921no—
466C23—2921no—
467C23—1462no—
468C23#1455662412yes—
469C23C22×D29211yescharacteristic
470Q8—5844no—
471D4—2928no—
472D4—2921no—
473D4—2921no—
474D4—1462no—
475D4—1462no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—2921no—
494C2×C4—2921no—
495C2×C4—2921no—
496C2×C4—2922no—
497C2×C4—2921no—
498C2×C4—2922no—
499C2×C4—2921no—
500C2×C4—2922no—
501C2×C4—2921no—
502C2×C4—2922no—
503C2×C4—2921no—
504C2×C4—2921no—
505C2×C4—2922no—
506C2×C4—2922no—
507C2×C4—5844no—
508C2×C4—1462no—
509C2×C4#1455656512yes—
510C2×C4#1455656512yes—
511C2×C4#1455661912yes—
512C2×C4#1455661912yes—
513C2×C4C22×C22⋊D7311yescharacteristic
514C2×C4#41×D7311yescharacteristic
515C2×C4#41×D7311yescharacteristic
516C2×C4C2×#1192402811yescharacteristic
517C2×C4#1455663811yescharacteristic
518C2×C4#1455663811yescharacteristic
519C8—24no—
520C8—42no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—2922no—
529C22—2922no—
530C22—2922no—
531C22—5841no—
532C22—5841no—
533C22—5841no—
534C22—5841no—
535C22C2×#1455656511yescharacteristic
536C22#1798909911yescharacteristic
537C22#1798916311yescharacteristic
538C22C2×#1455661911yescharacteristic
539C22C2×#1455661911yescharacteristic
540C22C2×#1455662411yescharacteristic
541C22#1798938711yescharacteristic
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—2922no—
547C4—5841no—
548C4—5842no—
549C4—5842no—
550C4—5841no—
551C4#1798884014yes—
552C4#1798923611yescharacteristic
553C4#1798924411yescharacteristic
554C4#1798927611yescharacteristic
555C4#1798930511yescharacteristic
556C2—42no—
557C2—5842no—
558C2#2261335411yescharacteristic
559C2C2×#1798884011yescharacteristic
560C2#2261956111yescharacteristic
561C2#2262003111yescharacteristic
562C2#2262008311yescharacteristic
563C2#2262113111yescharacteristic
564C2#2262116311yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#432×C73C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
hypercenterC2×#432D73
hypocenterC73#24709
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
1. C2×#17988868C2semidirect
2. #22618283C2semidirect
4. #22611359C2semidirect
6. #22610882C2semidirect
7. #22610695C2semidirect
8. #22610674C2semidirect
9. #22610628C2semidirect
10. #22609932C2semidirect
11. C2×#432×C73C2semidirect
95. C4⋊C8×C73D4semidirect
284. C2×#432D73semidirect
294. C73#24709semidirect
371. C4⋊C8C22⋊D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888682
#226182832
C4×#145565652
#226113592
#226111212
#226108822
#226106952
#226106742
#226106282
#226099322
C2×#432×C732
#2470973