Finite Group Databasebeta

← #33120329

Subgroups — #33120329

Order 18688.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988869C211yesmaximal, characteristic, complemented
2C2×#17988784C211yesmaximal, characteristic, complemented
3C4×#14556570C211yesmaximal, characteristic, complemented
4#22611117C212yesmaximal, complemented
5#22610934C212yesmaximal, complemented
6#22610884C211yesmaximal, characteristic, complemented
7#22610859C211yesmaximal, characteristic, complemented
8#22610265C212yesmaximal, complemented
9#22610192C212yesmaximal, complemented
10#22609911C211yesmaximal, characteristic, complemented
11C2×#532×C73C211yesmaximal, characteristic
12C22×#11924016C2212yes—
13C2×#14556570C2211yescharacteristic
14C2×C4×Dic146C2211yescharacteristic
15#17988869—24no—
16#17988784—24no—
17C2×#14556522C2212yes—
18C2×C4⋊#10596831C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×#14556514C2212yes—
21C2×#14556512C2212yes—
22C2×#14556512C2212yes—
23C2×#14556512C2212yes—
24C2×C22⋊#10596831C2212yes—
25#17988493C2211yescharacteristic
26#17988485C2211yescharacteristic
27#17988471—21no—
28#17988460—21no—
29#17988447C2211yescharacteristic
30C4×#11924007—21no—
31C4×C4⋊Dic73C2211yescharacteristic
32C4×C22⋊Dic73—21no—
33#17988349C2212yes—
34#17988349C2212yes—
35#17988342C2212yes—
36C22×#41×C73—21no—
37#532×C73—24no—
38#532×C73—24no—
39C2×C4⋊Q8×C73C2211yescharacteristic
40C2×#127×C73C2211yescharacteristic
41C2×#124×C73—21no—
42C2×#121×C73—21no—
43C2×#121×C73C2212yes—
44C2×C4×D4×C73—21no—
45C2×C4×D4×C73C2212yes—
46C22×C42×C73C2211yescharacteristic
47C22×Dic146C2311yescharacteristic
48C2×#11924016—24no—
49C2×#11924016—24no—
50C2×#11924016—42no—
51C22×#10596831C2312yes—
52C22×#10596831C2312yes—
53#14556570—22no—
54#14556570—22no—
55C4×Dic146—22no—
56C4×Dic146—22no—
57C2×C4⋊Dic73C2311yescharacteristic
58C2×C4⋊Dic73C2311yescharacteristic
59C2×C4⋊Dic73C2311yescharacteristic
60C2×#11924007—21no—
61C2×#11924007—21no—
62C2×#11924007—21no—
63C2×C22⋊Dic73—21no—
64C2×C22⋊Dic73—21no—
65C2×C4×Dic73—21no—
66C2×C4×Dic73—21no—
67#14556522—24no—
68C4⋊#10596831—22no—
69C4⋊#10596831—22no—
70#14556514—24no—
71#14556512—24no—
72#14556512—24no—
73#14556512—42no—
74C22⋊#10596831—42no—
75#14556498—21no—
76#14556498—21no—
77#14556498—21no—
78#14556498—21no—
79#14556498—21no—
80C2×#41×C73—28no—
81C2×#41×C73—44no—
82C2×#41×C73—44no—
83C22×Q8×C73—21no—
84C22×D4×C73—22no—
85C22×D4×C73—21no—
86C22×D4×C73—21no—
87C22×D4×C73C2312yes—
88C23×C4×C73—22no—
89C23×C4×C73D411yescharacteristic
90C23×C4×C73D411yescharacteristic
91C23×C4×C73C2311yescharacteristic
92C4⋊Q8×C73—22no—
93C4⋊Q8×C73D414yes—
94#127×C73—22no—
95#127×C73D414yes—
96#124×C73—28no—
97#121×C73—28no—
98#121×C73—24no—
99#121×C73—24no—
100C4×D4×C73—28no—
101C4×D4×C73—24no—
102C4×D4×C73—24no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73C2312yes—
105C2×C22⋊C4×C73—22no—
106C2×C22⋊C4×C73—22no—
107C2×C42×C73—22no—
108C2×C42×C73—22no—
109C2×C42×C73D411yescharacteristic
110C2×C42×C73D411yescharacteristic
111C2×C42×C73C2311yescharacteristic
112C2×C42×C73C2311yescharacteristic
113C2×Dic146—21no—
114C2×Dic146—21no—
115C2×Dic146—21no—
116C2×Dic146—21no—
117C2×Dic146—21no—
118C2×Dic146#4112yescomplemented
119C22×Dic73—21no—
120C22×Dic73—21no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123#11924016—48no—
124C2×#10596831—22no—
125C2×#10596831—22no—
126C2×#10596831—22no—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C4⋊Dic73—21no—
131C4⋊Dic73—21no—
132C4⋊Dic73—21no—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73#4112yescomplemented
136#11924007—24no—
137#11924007—42no—
138#11924007—42no—
139C22⋊Dic73—24no—
140C22⋊Dic73—42no—
141C4×Dic73—42no—
142C4×Dic73—42no—
143C24×C73—22no—
144C24×C73C2×D411yescharacteristic
145#41×C73—432no—
146C2×Q8×C73—24no—
147C2×Q8×C73—42no—
148C2×Q8×C73—42no—
149C2×D4×C73—28no—
150C2×D4×C73—24no—
151C2×D4×C73—24no—
152C2×D4×C73—22no—
153C2×D4×C73—22no—
154C2×D4×C73—22no—
155C2×D4×C73—22no—
156C2×D4×C73—22no—
157C2×D4×C73—42no—
158C2×D4×C73—42no—
159C2×D4×C73—44no—
160C2×D4×C73—44no—
161C2×D4×C73—42no—
162C2×D4×C73—42no—
163C2×D4×C73C2×D414yes—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—28no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—41no—
176C22×C4×C73—41no—
177C22×C4×C73—44no—
178C22×C4×C73—44no—
179C22×C4×C73SD1614yescomplemented
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2411yescharacteristic
186C4⋊C4×C73—24no—
187C4⋊C4×C73—22no—
188C4⋊C4×C73C2×D414yes—
189C22⋊C4×C73—28no—
190C22⋊C4×C73—28no—
191C42×C73—22no—
192C42×C73—22no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73C2×D412yes—
196C42×C73C2×D412yes—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—24no—
204C23×C73—42no—
205C23×C73—42no—
206C23×C73—42no—
207C23×C73—42no—
208C23×C73—42no—
209C23×C73—42no—
210C23×C73C2×SD1612yes—
211C23×C73C22×D411yescharacteristic
212Q8×C73—48no—
213D4×C73—24no—
214D4×C73—24no—
215D4×C73—48no—
216D4×C73—416no—
217D4×C73—42no—
218D4×C73—42no—
219D4×C73—48no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—28no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—24no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—44no—
233C2×C4×C73—416no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73#12012yes—
241C2×C4×C73#12012yes—
242C2×C4×C73C2×SD1612yes—
243C2×C4×C73C2×SD1612yes—
244C2×C4×C73AGL(1,ℤ/8)12yes—
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C73C22×D411yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73H(5,𝔽2)11yescharacteristic
250C2×C4×C732−1+411yescharacteristic
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263Dic146—22no—
264Dic146—22no—
265Dic146—41no—
266Dic146—41no—
267#10596831—42no—
268#10596831—42no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—44no—
277C22×C73—44no—
278C22×C73—44no—
279C22×C73—82no—
280C22×C73—82no—
281C22×C73—82no—
282C22×C73—82no—
283C22×C73C2×#12011yescharacteristic
284C22×C73#52511yescharacteristic
285C22×C73#53711yescharacteristic
286C22×C73C22×SD1611yescharacteristic
287C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C73#57511yescharacteristic
289C22×C73#57511yescharacteristic
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—41no—
293C4×C73—44no—
294C4×C73—41no—
295C4×C73#45014yes—
296C4×C73#54611yescharacteristic
297C4×C73#54711yescharacteristic
298C4×C73#54811yescharacteristic
299C4×C73#55011yescharacteristic
300Dic73—42no—
301Dic73—42no—
302Dic73—81no—
303Dic73—81no—
304#24154—731nomaximal, Sylow
305C2×C73—42no—
306C2×C73—84no—
307C2×C73#240711yescharacteristic
308C2×C73C2×#45011yescharacteristic
309C2×C73#300511yescharacteristic
310C2×C73#310311yescharacteristic
311C2×C73#311511yescharacteristic
312C2×C73#324611yescharacteristic
313C2×C73#325111yescharacteristic
314C2×#532D7311yescharacteristic
315C2×#475—731no—
316C2×#460—731no—
317C4×C22⋊Q8—731no—
318#2034—732no—
319#2000—732no—
320#1988—731no—
321#1982—731no—
322#1877—732no—
323#1864—732no—
324#1797—731no—
325C73#2415411yescharacteristic, complemented, Sylow
326C22×#41—21no—
327C22×SD16—732no—
328#532—24no—
329#532—24no—
330C2×C4⋊Q8C2×D7311yescharacteristic
331C2×#127C2×D7311yescharacteristic
332C2×#124—21no—
333C2×C22⋊Q8—731no—
334C2×#121—21no—
335C2×#121C2×D7312yes—
336C2×C4×Q8—731no—
337C2×C4×D4—21no—
338C2×C4×D4C2×D7312yes—
339C22×C42C2×D7311yescharacteristic
340#475—1464no—
341#460—1464no—
342C2×#106—732no—
343C2×C4⋊C8—731no—
344C2×C4⋊C8—731no—
345C2×#103—732no—
346C2×#102—732no—
347C2×#102—732no—
348C2×#102—732no—
349C2×C22⋊C8—732no—
350#391—731no—
351#389—731no—
352#387—1461no—
353#385—1461no—
354#380—731no—
355C4×C4⋊C4—731no—
356C4×C4⋊C4—1461no—
357C4×C22⋊C4—1461no—
358#340—732no—
359#340—732no—
360#336—732no—
361C2×#41—28no—
362C2×#41—44no—
363C2×#41—44no—
364C22×Q8—21no—
365C22×Q8—731no—
366C22×D4—22no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4C22×D7312yes—
370C23×C4—22no—
371C23×C4C22⋊D7311yescharacteristic
372C23×C4C22⋊D7311yescharacteristic
373C23×C4C22×D7311yescharacteristic
374C2×SD16—2922no—
375C2×SD16—1464no—
376C2×SD16—1464no—
377C22×C8—732no—
378C22×C8—732no—
379C4⋊Q8—22no—
380C4⋊Q8C22⋊D7314yes—
381#127—22no—
382#127C22⋊D7314yes—
383#124—28no—
384C22⋊Q8—1462no—
385C22⋊Q8—1462no—
386#121—28no—
387#121—24no—
388#121—24no—
389C4×Q8—1462no—
390C4×Q8—1462no—
391C4×D4—28no—
392C4×D4—24no—
393C4×D4—24no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4C22×D7312yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—1461no—
405C2×C22⋊C4—1461no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—1461no—
409C2×C42—1461no—
410C2×C42C22⋊D7311yescharacteristic
411C2×C42C22⋊D7311yescharacteristic
412C2×C42C22×D7311yescharacteristic
413C2×C42C22×D7311yescharacteristic
414#106—1464no—
415C4⋊C8—1462no—
416C4⋊C8—1462no—
417#103—1464no—
418#102—2922no—
419#102—1464no—
420#102—1464no—
421C22⋊C8—2922no—
422#95—1461no—
423#95—1461no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427C24—22no—
428C24C2×C22⋊D7311yescharacteristic
429#41—432no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—732no—
434C2×Q8—1461no—
435C2×Q8—1461no—
436C2×Q8—1461no—
437C2×Q8—1461no—
438C2×Q8—1461no—
439C2×D4—28no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4C2×C22⋊D7314yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4#1192401614yescomplemented
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C2×C22⋊D7311yescharacteristic
476C22×C4C2×C22⋊D7311yescharacteristic
477C22×C4C2×C22⋊D7311yescharacteristic
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C23×D7311yescharacteristic
480SD16—2928no—
481C2×C8—1462no—
482C2×C8—1462no—
483C2×C8—1462no—
484C2×C8—1462no—
485C2×C8—1462no—
486C2×C8—1462no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—2922no—
490C4⋊C4—2922no—
491C4⋊C4—732no—
492C4⋊C4—1464no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4C2×C22⋊D7314yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—2922no—
502C22⋊C4—1464no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—2922no—
508C42—2922no—
509C42C2×C22⋊D7312yes—
510C42C2×C22⋊D7312yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C2×#1192401612yes—
525C23C22×C22⋊D7311yescharacteristic
526Q8—48no—
527Q8—2921no—
528Q8—2921no—
529Q8—1462no—
530Q8—1462no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—416no—
535D4—42no—
536D4—48no—
537D4—42no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—416no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—1462no—
569C2×C4—1462no—
570C2×C4#1455656012yes—
571C2×C4#1455656012yes—
572C2×C4C2×#1192401612yes—
573C2×C4C2×#1192401612yes—
574C2×C4#1455661712yes—
575C2×C4C22×C22⋊D7311yescharacteristic
576C2×C4C22×C22⋊D7311yescharacteristic
577C2×C4#41×D7311yescharacteristic
578C2×C4#41×D7311yescharacteristic
579C2×C4#1455663711yescharacteristic
580C2×C4#1455663811yescharacteristic
581C8—2922no—
582C8—2922no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×#1455656011yescharacteristic
598C22#1798909411yescharacteristic
599C22#1798916511yescharacteristic
600C22C22×#1192401611yescharacteristic
601C22C2×#1455661711yescharacteristic
602C22#1798938511yescharacteristic
603C22#1798938511yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—44no—
609C4—2922no—
610C4—2922no—
611C4—5841no—
612C4—5841no—
613C4#1798873814yes—
614C4#1798922511yescharacteristic
615C4#1798923611yescharacteristic
616C4#1798924411yescharacteristic
617C4#1798925611yescharacteristic
618C2—42no—
619C2—84no—
620C2#2261327011yescharacteristic
621C2C2×#1798873811yescharacteristic
622C2#2261917311yescharacteristic
623C2#2261995811yescharacteristic
624C2#2262007611yescharacteristic
625C2#2262124711yescharacteristic
626C2#2262128711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
Fitting, prime radicalC2×#532×C73C2
socleC23×C73C22×D4
hypercenterC2×#532D73
hypocenterC73#24154
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#17988869C2semidirect
2. C2×#17988784C2semidirect
3. C4×#14556570C2semidirect
4. #22611117C2semidirect
5. #22610934C2semidirect
6. #22610884C2semidirect
7. #22610859C2semidirect
8. #22610265C2semidirect
9. #22610192C2semidirect
10. #22609911C2semidirect
118. C2×Dic146#41semidirect
135. C4⋊Dic73#41semidirect
179. C22×C4×C73SD16semidirect
325. C73#24154semidirect
473. C22×C4#11924016semidirect

Maximal subgroups

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

subgroupindex
C2×#179888692
C2×#179887842
C4×#145565702
#226111172
#226109342
#226108842
#226108592
#226102652
#226101922
#226099112
C2×#532×C732
#2415473