Finite Group Databasebeta

← #33239762

Subgroups — #33239762

Order 18688.

1150 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989017C212yesmaximal, complemented
2#22619512C211yesmaximal, characteristic, complemented
3#22611871C212yesmaximal, complemented
4#22611865C211yesmaximal, characteristic, complemented
5#22611860C211yesmaximal, characteristic, complemented
6#22611854C212yesmaximal, complemented
7#22611848C211yesmaximal, characteristic, complemented
8#22611843C211yesmaximal, characteristic, complemented
9#22611638C212yesmaximal, complemented
10C2×C22⋊#11924003C211yesmaximal, characteristic, complemented
11C22×#135×C73C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic, complemented
13#17989098—21no—
14#17989017—28no—
15C2×#14556537C2212yescomplemented
16C2×#14556535C2212yescomplemented
17C2×#14556534C2212yescomplemented
18C2×#14556532C2212yescomplemented
19C22×#11924003C2212yescomplemented
20#17988828C2211yescharacteristic
21#17988828C2211yescharacteristic, complemented
22#17988823C2211yescharacteristic
23#17988823C2211yescharacteristic, complemented
24#17988816—21no—
25#17988816—21no—
26#17988625C2211yescharacteristic, complemented
27#17988584—21no—
28#17988391C2212yescomplemented
29#17988391C2212yescomplemented
30#17988388C2212yescomplemented
31#17988388C2212yescomplemented
32C22⋊#11924003C2211yescharacteristic
33C22⋊#11924003C2211yescharacteristic
34C22⋊#11924003C2212yescomplemented
35C22×#41×C73—21no—
36C2×#135×C73—28no—
37C2×#135×C73—21no—
38C2×#135×C73—21no—
39C2×#135×C73C2212yes—
40C2×#135×C73C2212yes—
41C22×Q16×C73C2211yescharacteristic
42C22×SD16×C73—21no—
43C22×D8×C73C2211yescharacteristic
44C23×C8×C73C2211yescharacteristic
45C2×#11924028—21no—
46#14556613—24no—
47C2×Dic292C2311yescharacteristic
48C2×#11924017—21no—
49C2×D584C2311yescharacteristic
50#14556574—41no—
51#14556570—21no—
52#14556570—21no—
53#14556565—21no—
54#14556565—21no—
55#14556560—41no—
56C4×C22⋊D73—41no—
57C4×C22⋊D73—41no—
58#14556550—21no—
59#14556537—24no—
60#14556535—24no—
61#14556534—24no—
62#14556532—24no—
63C2×#11924003—24no—
64C2×#11924003C2312yes—
65C2×#11924003C2312yes—
66#14556525C2311yescharacteristic
67#14556525C2311yescharacteristic
68#14556523—21no—
69#14556516—21no—
70#14556516—21no—
71#14556513—21no—
72#14556513—21no—
73C2×#41×C73—28no—
74C2×#41×C73—22no—
75C2×#41×C73—22no—
76C2×#41×C73—44no—
77C2×#41×C73—41no—
78C2×#41×C73—41no—
79C22×Q8×C73—21no—
80C22×D4×C73—21no—
81C22×D4×C73—41no—
82C23×C4×C73—41no—
83C23×C4×C73D411yescharacteristic
84#135×C73—28no—
85#135×C73—28no—
86#135×C73—24no—
87#135×C73—24no—
88#135×C73—24no—
89#135×C73D418yescomplemented
90C2×Q16×C73—24no—
91C2×Q16×C73—21no—
92C2×Q16×C73C2312yes—
93C2×SD16×C73—28no—
94C2×SD16×C73—22no—
95C2×SD16×C73—22no—
96C2×D8×C73—24no—
97C2×D8×C73—21no—
98C2×D8×C73C2312yes—
99C22×C8×C73—24no—
100C22×C8×C73D411yescharacteristic
101C22×C8×C73D411yescharacteristic
102C22×C8×C73D411yescharacteristic
103C22×C8×C73C2311yescharacteristic
104C22×C8×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106#11924028—42no—
107#11924028—42no—
108C2×Dic146—21no—
109C2×D292—21no—
110C2×C22⋊D73—41no—
111C2×C22⋊D73—41no—
112C2×C22⋊D73—41no—
113C2×C4×D73—41no—
114Dic292—22no—
115#11924017—42no—
116D584—22no—
117C4⋊Dic73—21no—
118C4⋊Dic73—21no—
119#11924007—41no—
120#11924007—41no—
121C22⋊Dic73—41no—
122C22⋊Dic73—41no—
123C22⋊Dic73—41no—
124#11924005—41no—
125#11924005—41no—
126C4×Dic73—41no—
127#11924003—22no—
128#11924003—22no—
129C24×C73—41no—
130C24×C73D811yescharacteristic
131#41×C73—28no—
132#41×C73—28no—
133#41×C73—48no—
134#41×C73—44no—
135#41×C73—48no—
136#41×C73—44no—
137C2×Q8×C73—24no—
138C2×Q8×C73—22no—
139C2×Q8×C73—42no—
140C2×Q8×C73—41no—
141C2×D4×C73—24no—
142C2×D4×C73—22no—
143C2×D4×C73—42no—
144C2×D4×C73—48no—
145C2×D4×C73—42no—
146C2×D4×C73—42no—
147C2×D4×C73—41no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—48no—
151C22×C4×C73—41no—
152C22×C4×C73—42no—
153C22×C4×C73—42no—
154C22×C4×C73D811yescharacteristic
155C22×C4×C73D811yescharacteristic
156C22×C4×C73D811yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158C22×C4×C73C2×D411yescharacteristic
159C22×C4×C73C2×D411yescharacteristic
160Q16×C73—24no—
161Q16×C73—22no—
162Q16×C73C2×D414yes—
163SD16×C73—28no—
164SD16×C73—28no—
165D8×C73—24no—
166D8×C73—22no—
167D8×C73C2×D414yes—
168C2×C8×C73—22no—
169C2×C8×C73—22no—
170C2×C8×C73—22no—
171C2×C8×C73—22no—
172C2×C8×C73—22no—
173C2×C8×C73C2×D411yescharacteristic
174C2×C8×C73C2×D411yescharacteristic
175C2×C8×C73C2×D411yescharacteristic
176C2×C8×C73C2×D414yes—
177C2×C8×C73C2411yescharacteristic
178C23×C73—22no—
179C23×C73—22no—
180C23×C73—41no—
181C23×C73—44no—
182C23×C73—42no—
183C23×C73—82no—
184C23×C73—82no—
185C23×C73C2×D811yescharacteristic
186C23×C73C2×D811yescharacteristic
187C23×C73C2×D811yescharacteristic
188C22×D73—41no—
189Q8×C73—24no—
190Q8×C73—44no—
191Q8×C73—42no—
192D4×C73—24no—
193D4×C73—44no—
194D4×C73—48no—
195D4×C73—48no—
196D4×C73—42no—
197C2×C4×C73—22no—
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—21no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—48no—
205C2×C4×C73—44no—
206C2×C4×C73—42no—
207C2×C4×C73—48no—
208C2×C4×C73#12012yes—
209C2×C4×C73C2×D811yescharacteristic
210C2×C4×C73C2×D811yescharacteristic
211C2×C4×C73C2×D811yescharacteristic
212C2×C4×C73AGL(1,ℤ/8)12yes—
213C2×C4×C73C22×D411yescharacteristic
214C22⋊D73—82no—
215C22⋊D73—82no—
216C22⋊D73—82no—
217C2×Dic73—41no—
218C2×Dic73—41no—
219C2×Dic73—41no—
220D292—41no—
221D292—41no—
222C4×D73—82no—
223Dic146—41no—
224Dic146—41no—
225C8×C73—21no—
226C8×C73—21no—
227C8×C73#12012yes—
228C8×C73C22×D412yes—
229C22×C73—21no—
230C22×C73—21no—
231C22×C73—21no—
232C22×C73—21no—
233C22×C73—21no—
234C22×C73—21no—
235C22×C73—21no—
236C22×C73—44no—
237C22×C73—44no—
238C22×C73—88no—
239C22×C73—82no—
240C22×C73—82no—
241C22×C73#44712yes—
242C22×C73#50812yescomplemented
243C22×C73C22×D811yescharacteristic
244C2×D73—81no—
245C2×D73—81no—
246C2×D73—81no—
247C4×C73—21no—
248C4×C73—21no—
249C4×C73—44no—
250C4×C73#44712yes—
251C4×C73C2×#12011yescharacteristic
252C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
253Dic73—81no—
254Dic73—81no—
255Dic73—81no—
256#33516—731nomaximal, Sylow
257C2×C73—22no—
258C2×C73—22no—
259C2×C73—41no—
260C2×C73—84no—
261C2×C73C2×#44711yescharacteristic
262C2×C73C2×#50811yescharacteristic
263C2×C73#341411yescharacteristic
264D73—161no—
265C22×#135D7311yescharacteristic, complemented
266C2×#508—732no—
267#3050—731no—
268#2189—732no—
269#2188—731no—
270#2187—731no—
271#2186—732no—
272#2185—731no—
273#2184—731no—
274#2138—732no—
275C2×C22⋊C16—731no—
276C73#3351611yescharacteristic, complemented, Sylow
277C22×#41—21no—
278C2×#135—28no—
279C2×#135—21no—
280C2×#135—21no—
281C2×#135—731no—
282C2×#135C2×D7312yes—
283C2×#135C2×D7312yes—
284C22×Q16C2×D7311yescharacteristic
285C22×SD16—21no—
286C22×D8C2×D7311yescharacteristic
287C23×C8C2×D7311yescharacteristic
288#525—1461no—
289#508—1468no—
290C2×Q32—732no—
291C2×SD32—732no—
292C2×SD32—732no—
293C2×D16—732no—
294C22×C16—732no—
295#467—731no—
296#467—731no—
297#466—731no—
298#466—731no—
299#465—1461no—
300#465—1461no—
301#427—731no—
302#416—1461no—
303#358—732no—
304#358—732no—
305#357—732no—
306#357—732no—
307C22⋊C16—731no—
308C22⋊C16—731no—
309C22⋊C16—732no—
310C2×#41—28no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—44no—
314C2×#41—41no—
315C2×#41—41no—
316C2×#41—1461no—
317C22×Q8—21no—
318C22×D4—21no—
319C22×D4—41no—
320C23×C4—41no—
321C23×C4C22⋊D7311yescharacteristic
322#135—28no—
323#135—28no—
324#135—24no—
325#135—24no—
326#135—24no—
327#135—1464no—
328#135C22⋊D7318yescomplemented
329C2×Q16—24no—
330C2×Q16—21no—
331C2×Q16—731no—
332C2×Q16C22×D7312yes—
333C2×SD16—28no—
334C2×SD16—22no—
335C2×SD16—22no—
336C2×SD16—1461no—
337C2×D8—24no—
338C2×D8—21no—
339C2×D8—731no—
340C2×D8C22×D7312yes—
341C22×C8—24no—
342C22×C8C22⋊D7311yescharacteristic
343C22×C8C22⋊D7311yescharacteristic
344C22×C8C22⋊D7311yescharacteristic
345C22×C8C22×D7311yescharacteristic
346C22×C8C22×D7311yescharacteristic
347C22×C8C22×D7311yescharacteristic
348#123—2921no—
349C22⋊Q8—1461no—
350C22⋊Q8—1461no—
351#121—1461no—
352#121—1461no—
353#120—2921no—
354C4×D4—2921no—
355C4×D4—2921no—
356#117—1461no—
357Q32—1464no—
358SD32—1464no—
359SD32—1464no—
360D16—1464no—
361C2×C16—732no—
362C2×C16—732no—
363C2×C16—1464no—
364#107—731no—
365#107—731no—
366#106—1461no—
367#103—1461no—
368#103—1461no—
369#102—1461no—
370#102—1461no—
371C24—41no—
372C24#1192401311yescharacteristic
373#41—28no—
374#41—28no—
375#41—48no—
376#41—48no—
377#41—44no—
378#41—44no—
379#41—2922no—
380#41—2922no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—42no—
384C2×Q8—41no—
385C2×Q8—1461no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—42no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—41no—
393C2×D4—2921no—
394C2×D4—2921no—
395C2×D4—2921no—
396C2×D4—1461no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—48no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—2921no—
404C22×C4#1192401311yescharacteristic
405C22×C4#1192401311yescharacteristic
406C22×C4#1192401311yescharacteristic
407C22×C4C2×C22⋊D7311yescharacteristic
408C22×C4C2×C22⋊D7311yescharacteristic
409C22×C4C2×C22⋊D7311yescharacteristic
410Q16—24no—
411Q16—22no—
412Q16—1462no—
413Q16C2×C22⋊D7314yes—
414SD16—28no—
415SD16—28no—
416SD16—2922no—
417D8—24no—
418D8—22no—
419D8—1462no—
420D8C2×C22⋊D7314yes—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8C2×C22⋊D7311yescharacteristic
427C2×C8C2×C22⋊D7311yescharacteristic
428C2×C8C2×C22⋊D7311yescharacteristic
429C2×C8C2×C22⋊D7314yes—
430C2×C8C23×D7311yescharacteristic
431C4⋊C4—2921no—
432C4⋊C4—2921no—
433C4⋊C4—1461no—
434C4⋊C4—1461no—
435C22⋊C4—2921no—
436C22⋊C4—2921no—
437C22⋊C4—2921no—
438C22⋊C4—2921no—
439C22⋊C4—2921no—
440C42—2921no—
441C16—1462no—
442C16—1462no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—44no—
447C23—42no—
448C23—82no—
449C23—82no—
450C23—2921no—
451C23C2×#1192401311yescharacteristic
452C23C2×#1192401311yescharacteristic
453C23C2×#1192401311yescharacteristic
454Q8—24no—
455Q8—44no—
456Q8—42no—
457Q8—2921no—
458Q8—2921no—
459D4—24no—
460D4—44no—
461D4—48no—
462D4—48no—
463D4—42no—
464D4—2921no—
465D4—2921no—
466D4—5842no—
467D4—5842no—
468D4—5842no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—48no—
480C2×C4—2921no—
481C2×C4—2921no—
482C2×C4—2921no—
483C2×C4—5842no—
484C2×C4#1455656012yes—
485C2×C4C2×#1192401311yescharacteristic
486C2×C4C2×#1192401311yescharacteristic
487C2×C4C2×#1192401311yescharacteristic
488C2×C4#1455661712yes—
489C2×C4C22×C22⋊D7311yescharacteristic
490C8—21no—
491C8—21no—
492C8#1455656012yes—
493C8C22×C22⋊D7312yes—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—44no—
502C22—44no—
503C22—88no—
504C22—82no—
505C22—82no—
506C22—5841no—
507C22—5841no—
508C22—5841no—
509C22#1798872112yes—
510C22#1798901712yescomplemented
511C22C22×#1192401311yescharacteristic
512C4—21no—
513C4—21no—
514C4—44no—
515C4—5841no—
516C4—5841no—
517C4—5841no—
518C4#1798872112yes—
519C4C2×#1455656011yescharacteristic
520C4C2×#1455661711yescharacteristic
521C2—22no—
522C2—22no—
523C2—41no—
524C2—84no—
525C2—11681no—
526C2C2×#1798872111yescharacteristic
527C2C2×#1798901711yescharacteristic
528C2#2262306511yescharacteristic
subgroupG/H
centerC2×C4C2×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC22×#135×C73C2
socleC22×C73C22×D8
hypercenterC22×#135D73
hypocenterC73#33516
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
1. C2×#17989017C2semidirect
2. #22619512C2semidirect
3. #22611871C2semidirect
4. #22611865C2semidirect
5. #22611860C2semidirect
6. #22611854C2semidirect
7. #22611848C2semidirect
8. #22611843C2semidirect
9. #22611638C2semidirect
10. C2×C22⋊#11924003C2semidirect
11. C22×#135×C73C2semidirect
12. C2×#14556613C22semidirect
15. C2×#14556537C22semidirect
16. C2×#14556535C22semidirect
17. C2×#14556534C22semidirect
18. C2×#14556532C22semidirect
19. C22×#11924003C22semidirect
21. #17988828C22semidirect
23. #17988823C22semidirect
26. #17988625C22semidirect
28. #17988391C22semidirect
29. #17988391C22semidirect
30. #17988388C22semidirect
31. #17988388C22semidirect
34. C22⋊#11924003C22semidirect
89. #135×C73D4semidirect
242. C22×C73#508semidirect
265. C22×#135D73semidirect
276. C73#33516semidirect
328. #135C22⋊D73semidirect
510. C22#17989017semidirect

Maximal subgroups

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

subgroupindex
C2×#179890172
#226195122
#226118712
#226118652
#226118602
#226118542
#226118482
#226118432
#226116382
C2×C22⋊#119240032
C22×#135×C732
#3351673