Finite Group Databasebeta

← #33118253

Subgroups — #33118253

Order 18688.

1278 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988742C211yesmaximal, characteristic, complemented
2C22×#14556514C211yesmaximal, characteristic, complemented
3#22613296C211yesmaximal, characteristic
4#22611069C212yesmaximal, complemented
5#22610939C212yesmaximal, complemented
6#22610355C211yesmaximal, characteristic, complemented
7#22610272C212yesmaximal, complemented
8#22610238C212yesmaximal, complemented
9#22610224C211yesmaximal, characteristic
10#22610021C211yesmaximal, characteristic, complemented
11C2×#554×C73C211yesmaximal, characteristic
12C23×Dic146C2211yescharacteristic
13C22×#11924018C2212yes—
14C23×#10596831C2211yescharacteristic
15C2×#14556570C2212yes—
16#17988742—24no—
17C2×#14556515C2212yes—
18C2×#14556514—24no—
19C2×#14556514C2212yes—
20C2×#14556514C2212yes—
21C2×#14556514C2212yes—
22C2×C22⋊#10596831C2211yescharacteristic
23C2×C22⋊#10596831C2212yes—
24#17988507—21no—
25#17988499—21no—
26#17988493C2212yes—
27#17988467—21no—
28#17988459—21no—
29#17988447C2211yescharacteristic
30#17988349C2212yes—
31#17988349C2212yes—
32#17988349C2212yes—
33#17988342C2212yes—
34#554×C73—24no—
35#554×C73—24no—
36C2×C4⋊Q8×C73C2211yescharacteristic
37C2×#125×C73—21no—
38C2×C22⋊Q8×C73—21no—
39C2×C22⋊Q8×C73—21no—
40C2×C22⋊Q8×C73C2211yescharacteristic
41C2×C22⋊Q8×C73C2211yescharacteristic
42C2×C4×Q8×C73C2211yescharacteristic
43C2×C4×D4×C73—21no—
44C2×C4×D4×C73C2211yescharacteristic
45C22×C4⋊C4×C73C2211yescharacteristic
46C22×C4⋊C4×C73C2211yescharacteristic
47C22×Dic146—24no—
48C22×Dic146—21no—
49C22×Dic146—22no—
50C22×Dic146—22no—
51C22×Dic146—42no—
52C22×Dic146C2312yes—
53C23×Dic73—21no—
54C2×#11924018—24no—
55C2×#11924018—24no—
56C2×#11924018—42no—
57C22×#10596831—22no—
58C22×#10596831—42no—
59C22×#10596831C2312yes—
60C22×#10596831C2312yes—
61#14556570—24no—
62#14556570—42no—
63C2×C4⋊Dic73C2312yes—
64C2×#11924007—22no—
65C2×C22⋊Dic73—21no—
66C2×C22⋊Dic73—22no—
67#14556515—42no—
68#14556514—24no—
69#14556514—24no—
70#14556514—24no—
71C22⋊#10596831—22no—
72C22⋊#10596831—42no—
73#14556498—21no—
74#14556498—22no—
75#14556498—22no—
76C22×Q8×C73—21no—
77C22×Q8×C73C2311yescharacteristic
78C22×D4×C73D411yescharacteristic
79C23×C4×C73—21no—
80C23×C4×C73—21no—
81C23×C4×C73C2311yescharacteristic
82C23×C4×C73C2311yescharacteristic
83C4⋊Q8×C73—22no—
84C4⋊Q8×C73—22no—
85#125×C73—28no—
86C22⋊Q8×C73—28no—
87C22⋊Q8×C73—28no—
88C22⋊Q8×C73—22no—
89C22⋊Q8×C73—22no—
90C22⋊Q8×C73—22no—
91C22⋊Q8×C73D414yes—
92C4×Q8×C73—22no—
93C4×Q8×C73—22no—
94C4×D4×C73—28no—
95C4×D4×C73—22no—
96C4×D4×C73—22no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—22no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—22no—
105C2×C4⋊C4×C73—22no—
106C2×C4⋊C4×C73D411yescharacteristic
107C2×C4⋊C4×C73D414yes—
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73D411yescharacteristic
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C42×C73—21no—
116C2×C42×C73C2311yescharacteristic
117C2×Dic146—28no—
118C2×Dic146—22no—
119C2×Dic146—22no—
120C2×Dic146—22no—
121C2×Dic146—22no—
122C2×Dic146—22no—
123C2×Dic146—48no—
124C2×Dic146—44no—
125C2×Dic146—44no—
126C2×Dic146—42no—
127C2×Dic146#4114yes—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—44no—
131C22×Dic73—42no—
132#11924018—48no—
133C2×#10596831—22no—
134C2×#10596831—22no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C2×#10596831—22no—
138C2×#10596831—22no—
139C2×#10596831—44no—
140C4⋊Dic73—22no—
141C4⋊Dic73—22no—
142#11924007—44no—
143C22⋊Dic73—44no—
144C22⋊Dic73—42no—
145C24×C73C2×D411yescharacteristic
146C24×C73C2×D411yescharacteristic
147C2×Q8×C73—24no—
148C2×Q8×C73—21no—
149C2×Q8×C73—22no—
150C2×Q8×C73—22no—
151C2×Q8×C73—82no—
152C2×Q8×C73C2×D412yes—
153C2×D4×C73—22no—
154C2×D4×C73—22no—
155C2×D4×C73Q1614yescomplemented
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—24no—
163C22×C4×C73—24no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—82no—
169C22×C4×C73—82no—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D412yes—
174C22×C4×C73C2×D412yes—
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2411yescharacteristic
177C4⋊C4×C73—24no—
178C4⋊C4×C73—24no—
179C4⋊C4×C73—24no—
180C4⋊C4×C73—24no—
181C4⋊C4×C73—24no—
182C4⋊C4×C73—24no—
183C4⋊C4×C73—22no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73C2×D412yes—
188C4⋊C4×C73C2×D412yes—
189C4⋊C4×C73C2×D412yes—
190C22⋊C4×C73—24no—
191C22⋊C4×C73—24no—
192C22⋊C4×C73—22no—
193C22⋊C4×C73—22no—
194C42×C73—24no—
195C42×C73—21no—
196C42×C73—21no—
197C23×C73—22no—
198C23×C73—22no—
199C23×C73—21no—
200C23×C73—22no—
201C23×C73—21no—
202C23×C73—22no—
203C23×C73#12012yes—
204C23×C73C2×Q1612yes—
205C23×C73C2×Q1612yes—
206C23×C73#13712yes—
207C23×C73C22×D411yescharacteristic
208Q8×C73—24no—
209Q8×C73—41no—
210Q8×C73—41no—
211Q8×C73—84no—
212D4×C73—28no—
213C2×C4×C73—21no—
214C2×C4×C73—22no—
215C2×C4×C73—21no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—24no—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—21no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—42no—
234C2×C4×C73—84no—
235C2×C4×C73—84no—
236C2×C4×C73#12012yes—
237C2×C4×C73C2×Q1612yes—
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73C22×D411yescharacteristic
240C2×C4×C73C2×#4112yes—
241C2×C4×C73H(5,𝔽2)12yes—
242C2×Dic73—24no—
243C2×Dic73—48no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249Dic146—24no—
250Dic146—24no—
251Dic146—48no—
252Dic146—42no—
253Dic146—42no—
254#10596831—42no—
255#10596831—42no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73—24no—
259C22×C73—24no—
260C22×C73—22no—
261C22×C73—24no—
262C22×C73—41no—
263C22×C73—41no—
264C22×C73—42no—
265C22×C73#45114yes—
266C22×C73C2×#12011yescharacteristic
267C22×C73#54612yes—
268C22×C73C22×Q1611yescharacteristic
269C22×C73C2×#13711yescharacteristic
270C22×C73#57712yes—
271C4×C73—22no—
272C4×C73—22no—
273C4×C73—41no—
274C4×C73—42no—
275C4×C73—41no—
276C4×C73—42no—
277C4×C73—42no—
278C4×C73—42no—
279C4×C73#52511yescharacteristic
280C4×C73#53511yescharacteristic
281C4×C73#54612yes—
282Dic73—44no—
283Dic73—82no—
284#23970—731nomaximal, Sylow
285C2×C73—24no—
286C2×C73—42no—
287C2×C73#240411yescharacteristic
288C2×C73C2×#45111yescharacteristic
289C2×C73#300911yescharacteristic
290C2×C73#329912yes—
291C2×C73#331612yes—
292C2×#554D7311yescharacteristic
293C2×#451—731no—
294C22×#103—731no—
295#2409—731no—
296#2024—732no—
297#2001—732no—
298#1892—731no—
299#1878—732no—
300#1872—732no—
301#1869—731no—
302#1824—731no—
303C73#2397011yescharacteristic, complemented, Sylow
304C23×Q8—731no—
305C22×Q16—732no—
306C23×C8—731no—
307#554—24no—
308#554—24no—
309C2×C4⋊Q8C2×D7311yescharacteristic
310C2×#125—21no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8—21no—
313C2×C22⋊Q8—732no—
314C2×C22⋊Q8C2×D7311yescharacteristic
315C2×C22⋊Q8C2×D7311yescharacteristic
316C2×C4×Q8C2×D7311yescharacteristic
317C2×C4×D4—21no—
318C2×C4×D4C2×D7311yescharacteristic
319C22×C4⋊C4C2×D7311yescharacteristic
320C22×C4⋊C4C2×D7311yescharacteristic
321#451—1464no—
322C2×#103—732no—
323C2×#103—732no—
324C2×#103—732no—
325C2×#103—732no—
326C2×#103—1464no—
327C2×C22⋊C8—732no—
328C2×C22⋊C8—731no—
329#395—1461no—
330#393—1461no—
331#391—732no—
332#386—1461no—
333#385—1461no—
334#380—731no—
335#340—732no—
336#340—732no—
337#340—732no—
338#336—732no—
339C22×Q8—21no—
340C22×Q8—2922no—
341C22×Q8—732no—
342C22×Q8—1464no—
343C22×Q8—1462no—
344C22×Q8—1462no—
345C22×Q8—1461no—
346C22×Q8C22×D7311yescharacteristic
347C22×D4C22⋊D7311yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—1461no—
351C23×C4C22×D7311yescharacteristic
352C23×C4C22×D7311yescharacteristic
353C2×Q16—2922no—
354C2×Q16—1464no—
355C2×Q16—1464no—
356C22×C8—2922no—
357C22×C8—732no—
358C22×C8—732no—
359C22×C8—1462no—
360C4⋊Q8—22no—
361C4⋊Q8—22no—
362#125—28no—
363C22⋊Q8—28no—
364C22⋊Q8—28no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—2922no—
369C22⋊Q8—1464no—
370C22⋊Q8C22⋊D7314yes—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×D4—28no—
374C4×D4—22no—
375C4×D4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—732no—
386C2×C4⋊C4—1462no—
387C2×C4⋊C4C22⋊D7311yescharacteristic
388C2×C4⋊C4C22⋊D7314yes—
389C2×C4⋊C4C22×D7311yescharacteristic
390C2×C4⋊C4C22×D7311yescharacteristic
391C2×C4⋊C4C22×D7311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1462no—
395C2×C22⋊C4—1461no—
396C2×C22⋊C4C22⋊D7311yescharacteristic
397C2×C22⋊C4C22⋊D7311yescharacteristic
398C2×C42—21no—
399C2×C42C22×D7311yescharacteristic
400#103—2922no—
401#103—1464no—
402#103—1464no—
403#103—1464no—
404C22⋊C8—2922no—
405C22⋊C8—1462no—
406#95—1461no—
407#95—1462no—
408#95—1462no—
409C24C2×C22⋊D7311yescharacteristic
410C24C2×C22⋊D7311yescharacteristic
411C2×Q8—24no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—82no—
416C2×Q8—2924no—
417C2×Q8—2928no—
418C2×Q8—2924no—
419C2×Q8—2922no—
420C2×Q8—734no—
421C2×Q8—1468no—
422C2×Q8—1462no—
423C2×Q8—1462no—
424C2×Q8—1462no—
425C2×Q8—1462no—
426C2×Q8—1462no—
427C2×Q8C2×C22⋊D7312yes—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4#1192401814yescomplemented
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—2922no—
446C22×C4—2924no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4C2×C22⋊D7311yescharacteristic
450C22×C4C2×C22⋊D7311yescharacteristic
451C22×C4C2×C22⋊D7311yescharacteristic
452C22×C4C2×C22⋊D7312yes—
453C22×C4C2×C22⋊D7312yes—
454C22×C4C2×C22⋊D7311yescharacteristic
455C22×C4C23×D7311yescharacteristic
456Q16—2928no—
457C2×C8—2924no—
458C2×C8—1462no—
459C2×C8—1462no—
460C2×C8—1462no—
461C2×C8—1462no—
462C2×C8—1462no—
463C2×C8—1462no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—2924no—
475C4⋊C4—1462no—
476C4⋊C4—1462no—
477C4⋊C4C2×C22⋊D7312yes—
478C4⋊C4C2×C22⋊D7312yes—
479C4⋊C4C2×C22⋊D7312yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—2922no—
485C22⋊C4—2924no—
486C42—24no—
487C42—21no—
488C42—21no—
489C23—22no—
490C23—22no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—21no—
495C23#1455656012yes—
496C23C2×#1192401812yes—
497C23C2×#1192401812yes—
498C23#1455662212yes—
499C23C22×C22⋊D7311yescharacteristic
500Q8—24no—
501Q8—41no—
502Q8—41no—
503Q8—84no—
504Q8—2928no—
505Q8—2922no—
506Q8—2922no—
507Q8—1464no—
508Q8—1464no—
509D4—28no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—2928no—
534C2×C4—2922no—
535C2×C4—2922no—
536C2×C4—2922no—
537C2×C4—2922no—
538C2×C4—2922no—
539C2×C4—1464no—
540C2×C4#1455656012yes—
541C2×C4C2×#1192401812yes—
542C2×C4C22×C22⋊D7311yescharacteristic
543C2×C4C22×C22⋊D7311yescharacteristic
544C2×C4#41×D7312yes—
545C2×C4#1455663712yes—
546C8—2922no—
547C8—2922no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—24no—
553C22—22no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22#1798874314yes—
558C22C2×#1455656011yescharacteristic
559C22#1798922512yes—
560C22C22×#1192401811yescharacteristic
561C22C2×#1455662211yescharacteristic
562C22#1798939812yes—
563C4—22no—
564C4—22no—
565C4—41no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—2924no—
572C4—5842no—
573C4#1798909411yescharacteristic
574C4#1798915111yescharacteristic
575C4#1798922512yes—
576C2—24no—
577C2—42no—
578C2#2261324611yescharacteristic
579C2C2×#1798874311yescharacteristic
580C2#2261920811yescharacteristic
581C2#2262180112yes—
582C2#2262203612yes—
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#554×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73D4
hypercenterC2×#554D73
hypocenterC73#23970
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#17988742C2semidirect
2. C22×#14556514C2semidirect
4. #22611069C2semidirect
5. #22610939C2semidirect
6. #22610355C2semidirect
7. #22610272C2semidirect
8. #22610238C2semidirect
10. #22610021C2semidirect
155. C2×D4×C73Q16semidirect
303. C73#23970semidirect
430. C2×D4#11924018semidirect

Maximal subgroups

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

subgroupindex
C2×#179887422
C22×#145565142
#226132962
#226110692
#226109392
#226103552
#226102722
#226102382
#226102242
#226100212
C2×#554×C732
#2397073