Finite Group Databasebeta

← #33091649

Subgroups — #33091649

Order 18688.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988726C211yesmaximal, characteristic
2C2×#17988685C212yesmaximal, complemented
3C4×#14556565C211yesmaximal, characteristic
4#22610286C212yesmaximal, complemented
5#22610248C211yesmaximal, characteristic
6#22610185C212yesmaximal, complemented
7#22610119C211yesmaximal, characteristic, complemented
8#22609866C211yesmaximal, characteristic, complemented
9#22609761C212yesmaximal, complemented
10#22609693C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C73C211yesmaximal, characteristic, complemented
12C22×#11924015C2212yes—
13C2×#14556565C2211yescharacteristic
14C2×C4×D292C2211yescharacteristic
15C2×C4×C22⋊D73—21no—
16#17988726C418yescomplemented
17#17988685—28no—
18C2×#14556524C2212yes—
19C2×#14556515C2212yes—
20C2×#14556515C2212yes—
21C2×#14556515C2212yes—
22C2×#14556511C2212yes—
23C2×C22⋊#10596831C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25C2×#14556502C2212yes—
26#17988489C2211yescharacteristic
27#17988485C2211yescharacteristic
28#17988478—21no—
29#17988466—21no—
30#17988447C2211yescharacteristic
31C4×C4⋊Dic73C2211yescharacteristic
32C4×#11924005—21no—
33#17988349C2212yes—
34#17988349C2212yes—
35#17988342C2212yes—
36C23×Q8×C73C2211yescharacteristic
37C2×C4×Q8×C73—28no—
38C2×C4×Q8×C73—21no—
39C2×C4×Q8×C73—21no—
40C2×C4×Q8×C73C2212yes—
41C2×C4×Q8×C73C2212yes—
42C22×C4⋊C4×C73—21no—
43C22×C4⋊C4×C73C2211yescharacteristic
44C22×C42×C73—21no—
45C22×C42×C73C2211yescharacteristic
46C22×D292C2311yescharacteristic
47C22×C22⋊D73—21no—
48C22×C4×D73—21no—
49C2×#11924015—24no—
50C2×#11924015—24no—
51C2×#11924015C2×C418yescomplemented
52C22×#10596831C2312yes—
53C22×#10596831C2312yes—
54#14556565—22no—
55#14556565C2×C414yes—
56C4×D292—24no—
57C4×C22⋊D73—44no—
58C2×C4⋊Dic73C2311yescharacteristic
59C2×C4⋊Dic73C2311yescharacteristic
60C2×C4⋊Dic73C2311yescharacteristic
61C2×#11924007—21no—
62C2×C22⋊Dic73—21no—
63C2×#11924005—21no—
64C2×#11924005—21no—
65C2×#11924005—21no—
66C2×#11924005—21no—
67C2×C4×Dic73—21no—
68#14556524—24no—
69#14556515—24no—
70#14556515—24no—
71#14556515C2×C418yescomplemented
72#14556511—24no—
73C22⋊#10596831—22no—
74C22⋊#10596831C2×C414yescomplemented
75#14556502—24no—
76#14556498—21no—
77#14556498—21no—
78#14556498—21no—
79C22×Q8×C73—24no—
80C22×Q8×C73—24no—
81C22×Q8×C73—21no—
82C22×Q8×C73—22no—
83C22×Q8×C73C2×C414yescomplemented
84C22×Q8×C73C2312yes—
85C23×C4×C73—21no—
86C23×C4×C73—21no—
87C23×C4×C73D411yescharacteristic
88C23×C4×C73D411yescharacteristic
89C23×C4×C73C2311yescharacteristic
90C4×Q8×C73—28no—
91C4×Q8×C73—28no—
92C4×Q8×C73—24no—
93C4×Q8×C73—24no—
94C4×Q8×C73—24no—
95C4×Q8×C73D418yescomplemented
96C2×C4⋊C4×C73—28no—
97C2×C4⋊C4×C73—22no—
98C2×C4⋊C4×C73—22no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—24no—
101C2×C4⋊C4×C73C2312yes—
102C2×C42×C73—28no—
103C2×C42×C73—22no—
104C2×C42×C73—22no—
105C2×C42×C73—24no—
106C2×C42×C73D411yescharacteristic
107C2×C42×C73D411yescharacteristic
108C2×C42×C73C2311yescharacteristic
109C2×C42×C73C2311yescharacteristic
110C23×D73—21no—
111C2×D292—22no—
112C2×D292—22no—
113C2×D292—21no—
114C2×D292C22×C412yes—
115C2×C22⋊D73—24no—
116C2×C22⋊D73—42no—
117C2×C22⋊D73—42no—
118C22×Dic73—21no—
119C22×Dic73—21no—
120C22×Dic73—21no—
121C2×C4×D73—42no—
122C2×C4×D73—42no—
123C2×C4×D73—42no—
124#11924015—216no—
125C2×#10596831—22no—
126C2×#10596831—22no—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C2×#10596831C22×C414yes—
131C4⋊Dic73—21no—
132C4⋊Dic73—22no—
133C4⋊Dic73—22no—
134C4⋊Dic73C22×C412yes—
135#11924007—42no—
136C22⋊Dic73—42no—
137#11924005—24no—
138#11924005—42no—
139#11924005—42no—
140#11924005—42no—
141C4×Dic73—42no—
142C24×C73C2×D411yescharacteristic
143C2×Q8×C73—24no—
144C2×Q8×C73—216no—
145C2×Q8×C73—24no—
146C2×Q8×C73—216no—
147C2×Q8×C73—24no—
148C2×Q8×C73—22no—
149C2×Q8×C73—22no—
150C2×Q8×C73—22no—
151C2×Q8×C73C22×C414yes—
152C2×Q8×C73C2×D414yes—
153C2×Q8×C73#4114yes—
154C22×C4×C73—24no—
155C22×C4×C73—24no—
156C22×C4×C73—24no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—24no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—41no—
168C22×C4×C73—42no—
169C22×C4×C73—42no—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73#4112yes—
177C22×C4×C73C2411yescharacteristic
178C4⋊C4×C73—28no—
179C4⋊C4×C73—24no—
180C4⋊C4×C73—28no—
181C4⋊C4×C73—22no—
182C4⋊C4×C73C2×D414yes—
183C42×C73—28no—
184C42×C73—28no—
185C42×C73—22no—
186C42×C73—22no—
187C42×C73—22no—
188C42×C73C2×D414yes—
189C23×C73—21no—
190C23×C73—21no—
191C23×C73—21no—
192C23×C73—21no—
193C23×C73C4×D412yes—
194C23×C73AGL(1,ℤ/8)12yes—
195C23×C73#13712yes—
196C23×C73C22×D411yescharacteristic
197C22×D73—22no—
198C22×D73—41no—
199C22×D73—41no—
200C22×D73—41no—
201C22×D73—41no—
202C22×D73—41no—
203C22×D73—41no—
204Q8×C73—216no—
205Q8×C73—24no—
206Q8×C73—24no—
207Q8×C73—24no—
208Q8×C73C4×D418yescomplemented
209C2×C4×C73—28no—
210C2×C4×C73—216no—
211C2×C4×C73—24no—
212C2×C4×C73—21no—
213C2×C4×C73—24no—
214C2×C4×C73—21no—
215C2×C4×C73—24no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—24no—
223C2×C4×C73—24no—
224C2×C4×C73—21no—
225C2×C4×C73—24no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—44no—
229C2×C4×C73—48no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73C4×D412yes—
233C2×C4×C73#12012yes—
234C2×C4×C73#12012yes—
235C2×C4×C73#13712yes—
236C2×C4×C73C23×C411yescharacteristic
237C2×C4×C73C22×D412yes—
238C2×C4×C73C2×#4112yes—
239C2×C4×C73C2×#4111yescharacteristic
240C22⋊D73—48no—
241C2×Dic73—22no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250D292—24no—
251D292—42no—
252C4×D73—84no—
253#10596831—24no—
254#10596831—42no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73—22no—
259C22×C73—22no—
260C22×C73—22no—
261C22×C73#44014yescomplemented
262C22×C73C2×C4×D411yescharacteristic
263C22×C73C2×#12011yescharacteristic
264C22×C73#52511yescharacteristic
265C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C73C2×#13711yescharacteristic
267C22×C73C2×#13711yescharacteristic
268C22×C73#57511yescharacteristic
269C2×D73—42no—
270C2×D73—42no—
271C2×D73—42no—
272C2×D73—81no—
273C2×D73—81no—
274C2×D73—81no—
275C2×D73—81no—
276C4×C73—22no—
277C4×C73—22no—
278C4×C73—22no—
279C4×C73—28no—
280C4×C73—22no—
281C4×C73—41no—
282C4×C73—44no—
283C4×C73C2×C4×D412yes—
284C4×C73#52512yes—
285Dic73—42no—
286Dic73—81no—
287Dic73—81no—
288#21454—731nomaximal, Sylow
289C2×C73—24no—
290C2×C73C4×#12011yescharacteristic
291C2×C73C2×#44011yescharacteristic
292C2×C73C4×#13711yescharacteristic
293C2×C73#300411yescharacteristic
294C2×C73#300611yescharacteristic
295C2×C73#311211yescharacteristic
296C2×C73#311711yescharacteristic
297D73—82no—
298C22×C4×Q8D7311yescharacteristic, complemented
299C2×#448—731no—
300C2×#440—732no—
301C4×#121—731no—
302#1880—732no—
303#1874—731no—
304#1863—732no—
305#1852—731no—
306#1787—731no—
307#1763—732no—
308#1749—731no—
309C73#2145411yescharacteristic, complemented, Sylow
310C23×Q8C2×D7311yescharacteristic
311C22×SD16—732no—
312C2×#121—731no—
313C2×C4×Q8—28no—
314C2×C4×Q8—21no—
315C2×C4×Q8—21no—
316C2×C4×Q8C2×D7312yes—
317C2×C4×Q8C2×D7312yes—
318C2×C4×D4—731no—
319C2×C4×D4—1461no—
320C22×C4⋊C4—21no—
321C22×C4⋊C4C2×D7311yescharacteristic
322C22×C42—21no—
323C22×C42C2×D7311yescharacteristic
324#448—738no—
325#440—1468no—
326C2×#107—732no—
327C2×#103—732no—
328C2×#103—732no—
329C2×#103—732no—
330C2×#102—732no—
331C2×C22⋊C8—731no—
332C2×C22⋊C8—731no—
333C2×#97—732no—
334#390—731no—
335#389—731no—
336#388—1461no—
337#386—1461no—
338#380—731no—
339C4×C4⋊C4—731no—
340C4×C22⋊C4—1461no—
341#340—732no—
342#340—732no—
343#336—732no—
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8C4×D7314yescomplemented
349C22×Q8C22×D7312yes—
350C22×D4—731no—
351C22×D4—1461no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—1461no—
355C23×C4C22⋊D7311yescharacteristic
356C23×C4C22⋊D7311yescharacteristic
357C23×C4C22×D7311yescharacteristic
358C2×SD16—738no—
359C2×SD16—1464no—
360C2×SD16—1464no—
361C22×C8—732no—
362C22×C8—732no—
363#121—734no—
364#121—1462no—
365C4×Q8—28no—
366C4×Q8—28no—
367C4×Q8—24no—
368C4×Q8—24no—
369C4×Q8—24no—
370C4×Q8C22⋊D7318yescomplemented
371C4×D4—2924no—
372C4×D4—1464no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—1461no—
382C2×C4⋊C4C22×D7312yes—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C22⋊C4—1461no—
387C2×C22⋊C4—1461no—
388C2×C42—28no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—24no—
392C2×C42—1461no—
393C2×C42C22⋊D7311yescharacteristic
394C2×C42C22⋊D7311yescharacteristic
395C2×C42C22×D7311yescharacteristic
396C2×C42C22×D7311yescharacteristic
397#107—1464no—
398#103—738no—
399#103—1464no—
400#103—1464no—
401#102—1464no—
402C22⋊C8—734no—
403C22⋊C8—1462no—
404#97—1464no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408C24—1461no—
409C24C2×C22⋊D7311yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8C2×C4×D7314yes—
419C2×Q8C2×C22⋊D7314yes—
420C2×Q8#1192402814yes—
421C2×D4—2922no—
422C2×D4—2922no—
423C2×D4—732no—
424C2×D4—1464no—
425C2×D4—1462no—
426C2×D4—1461no—
427C2×D4—1462no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—2922no—
445C22×C4—2922no—
446C22×C4—2922no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4C2×C4×D7312yes—
451C22×C4C2×C22⋊D7311yescharacteristic
452C22×C4C2×C22⋊D7311yescharacteristic
453C22×C4C2×C22⋊D7311yescharacteristic
454C22×C4C2×C22⋊D7311yescharacteristic
455C22×C4C2×C22⋊D7311yescharacteristic
456C22×C4#1192402812yes—
457C22×C4C23×D7311yescharacteristic
458SD16—14616no—
459C2×C8—734no—
460C2×C8—1462no—
461C2×C8—1462no—
462C2×C8—1462no—
463C2×C8—1462no—
464C2×C8—1462no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—2922no—
470C4⋊C4—732no—
471C4⋊C4—1462no—
472C4⋊C4—1462no—
473C4⋊C4—1461no—
474C4⋊C4C2×C22⋊D7314yes—
475C22⋊C4—2922no—
476C22⋊C4—2922no—
477C22⋊C4—2922no—
478C22⋊C4—2922no—
479C22⋊C4—1464no—
480C42—28no—
481C42—28no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—2922no—
486C42C2×C22⋊D7314yes—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—2921no—
492C23—2921no—
493C23—2921no—
494C23—2921no—
495C23—2921no—
496C23—2921no—
497C23—1462no—
498C23C4×C22⋊D7312yes—
499C23#1455661612yes—
500C23#1455662212yes—
501C23C22×C22⋊D7311yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8C4×C22⋊D7318yescomplemented
507D4—2928no—
508D4—2922no—
509D4—1464no—
510C2×C4—216no—
511C2×C4—28no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2921no—
536C2×C4—2921no—
537C2×C4—2921no—
538C2×C4—2921no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—5844no—
542C2×C4—1462no—
543C2×C4C4×C22⋊D7312yes—
544C2×C4#1455656012yes—
545C2×C4#1455656012yes—
546C2×C4#1455662212yes—
547C2×C4C22×C4×D7311yescharacteristic
548C2×C4C22×C22⋊D7312yes—
549C2×C4C2×#1192402812yes—
550C2×C4C2×#1192402811yescharacteristic
551C8—2922no—
552C8—1464no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—2922no—
560C22—2922no—
561C22—2922no—
562C22—5841no—
563C22—5841no—
564C22—5841no—
565C22—5841no—
566C22#1798868514yescomplemented
567C22C2×C4×C22⋊D7311yescharacteristic
568C22C2×#1455656011yescharacteristic
569C22#1798909811yescharacteristic
570C22C2×#1455661611yescharacteristic
571C22C2×#1455662211yescharacteristic
572C22C2×#1455662211yescharacteristic
573C22#1798938511yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—28no—
578C4—22no—
579C4—41no—
580C4—44no—
581C4—2922no—
582C4—5841no—
583C4—5841no—
584C4C2×C4×C22⋊D7312yes—
585C4#1798909812yes—
586C2—24no—
587C2—5842no—
588C2C4×#1455656011yescharacteristic
589C2C2×#1798868511yescharacteristic
590C2C4×#1455662211yescharacteristic
591C2#2261916511yescharacteristic
592C2#2261918311yescharacteristic
593C2#2262005311yescharacteristic
594C2#2262010211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC22×C4×Q8×C73C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
cosocleC22×C4×C73C24
hypercenterC22×C4×Q8D73
hypocenterC73#21454
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#17988685C2semidirect
4. #22610286C2semidirect
6. #22610185C2semidirect
7. #22610119C2semidirect
8. #22609866C2semidirect
9. #22609761C2semidirect
10. #22609693C2semidirect
11. C22×C4×Q8×C73C2semidirect
16. #17988726C4semidirect
51. C2×#11924015C2×C4semidirect
71. #14556515C2×C4semidirect
74. C22⋊#10596831C2×C4semidirect
83. C22×Q8×C73C2×C4semidirect
95. C4×Q8×C73D4semidirect
208. Q8×C73C4×D4semidirect
261. C22×C73#440semidirect
298. C22×C4×Q8D73semidirect
309. C73#21454semidirect
348. C22×Q8C4×D73semidirect
370. C4×Q8C22⋊D73semidirect
506. Q8C4×C22⋊D73semidirect
566. C22#17988685semidirect

Maximal subgroups

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

subgroupindex
C2×#179887262
C2×#179886852
C4×#145565652
#226102862
#226102482
#226101852
#226101192
#226098662
#226097612
#226096932
C22×C4×Q8×C732
#2145473