Finite Group Databasebeta

← #33119915

Subgroups — #33119915

Order 18688.

1350 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988869C211yesmaximal, characteristic, complemented
2C2×C4×#11924016C211yesmaximal, characteristic, complemented
3#22614853C211yesmaximal, characteristic, complemented
4#22611117C212yesmaximal, complemented
5#22610934C212yesmaximal, complemented
6#22610893C211yesmaximal, characteristic, complemented
7#22610726C211yesmaximal, characteristic, complemented
8#22610238C212yesmaximal, complemented
9#22610124C212yesmaximal, complemented
10#22609931C211yesmaximal, characteristic, complemented
11C2×#547×C73C211yesmaximal, characteristic
12C22×#11924016C2211yescharacteristic
13C2×#14556570C2212yes—
14C2×C4×Dic146C2211yescharacteristic
15#17988869—24no—
16C4×#11924016—24no—
17C2×#14556522C2212yes—
18C2×#14556522C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×#14556514C2212yes—
21C2×#14556514C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556512C2212yes—
24C2×C22⋊#10596831C2212yes—
25C2×C22⋊#10596831C2212yes—
26C2×C4×#10596831C2211yescharacteristic
27#17988510—22no—
28#17988490—21no—
29#17988483—21no—
30#17988457C2211yescharacteristic
31#17988447C2212yes—
32#17988349C2212yes—
33#17988349C2212yes—
34#17988349C2212yes—
35C22×#41×C73—21no—
36#547×C73—24no—
37#547×C73—24no—
38C2×C4⋊Q8×C73C2211yescharacteristic
39C2×#123×C73—22no—
40C2×C22⋊Q8×C73—21no—
41C2×#121×C73C2212yes—
42C2×C4×D4×C73C2211yescharacteristic
43C2×C4×D4×C73C2211yescharacteristic
44C22×C4⋊C4×C73C2212yes—
45C22×Dic146C2311yescharacteristic
46C2×#11924016—22no—
47C2×#11924016—22no—
48C2×#11924016—22no—
49C22×#10596831C2312yes—
50C22×#10596831C2311yescharacteristic
51C22×#10596831C2311yescharacteristic
52#14556570—24no—
53#14556570—42no—
54C4×Dic146—22no—
55C4×Dic146—22no—
56C2×C4⋊Dic73—21no—
57C2×C4⋊Dic73—21no—
58C2×C4⋊Dic73C2311yescharacteristic
59C2×C4⋊Dic73C2312yes—
60C2×#11924007—22no—
61C2×#11924007—21no—
62C2×C22⋊Dic73—22no—
63C2×C22⋊Dic73—22no—
64C2×C4×Dic73—21no—
65#14556522—24no—
66#14556522—22no—
67C4⋊#10596831—22no—
68#14556514—24no—
69#14556514—22no—
70#14556512—22no—
71#14556512—42no—
72C22⋊#10596831—24no—
73C22⋊#10596831—42no—
74C4×#10596831—22no—
75#14556498—22no—
76C2×#41×C73—28no—
77C2×#41×C73—44no—
78C2×#41×C73—44no—
79C22×Q8×C73—21no—
80C22×D4×C73—22no—
81C22×D4×C73D411yescharacteristic
82C22×D4×C73C2311yescharacteristic
83C23×C4×C73—22no—
84C23×C4×C73—21no—
85C23×C4×C73C2312yes—
86C4⋊Q8×C73—22no—
87C4⋊Q8×C73D414yes—
88#123×C73—216no—
89C22⋊Q8×C73—28no—
90#121×C73—24no—
91#121×C73—24no—
92C4×D4×C73—22no—
93C4×D4×C73—22no—
94C4×D4×C73—22no—
95C4×D4×C73D414yes—
96C2×C4⋊C4×C73—22no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—24no—
99C2×C4⋊C4×C73—24no—
100C2×C4⋊C4×C73D411yescharacteristic
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C4⋊C4×C73C2312yes—
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—22no—
105C2×C22⋊C4×C73D412yes—
106C2×C42×C73C2311yescharacteristic
107C2×Dic146—21no—
108C2×Dic146—21no—
109C2×Dic146—21no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112C2×Dic146#4112yescomplemented
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—22no—
116#11924016—28no—
117C2×#10596831—21no—
118C2×#10596831—21no—
119C2×#10596831—21no—
120C2×#10596831—22no—
121C2×#10596831—22no—
122C2×#10596831—21no—
123C2×#10596831—22no—
124C2×#10596831—21no—
125C2×#10596831#4112yescomplemented
126C4⋊Dic73—21no—
127C4⋊Dic73—21no—
128C4⋊Dic73—22no—
129C4⋊Dic73—22no—
130C4⋊Dic73—42no—
131C4⋊Dic73—81no—
132#11924007—44no—
133#11924007—42no—
134C22⋊Dic73—44no—
135C22⋊Dic73—82no—
136C4×Dic73—42no—
137C24×C73—21no—
138C24×C73C2×D412yes—
139#41×C73—432no—
140C2×Q8×C73—24no—
141C2×Q8×C73—42no—
142C2×Q8×C73—42no—
143C2×D4×C73—28no—
144C2×D4×C73—22no—
145C2×D4×C73—22no—
146C2×D4×C73—21no—
147C2×D4×C73—21no—
148C2×D4×C73—21no—
149C2×D4×C73—21no—
150C2×D4×C73—21no—
151C2×D4×C73—44no—
152C2×D4×C73—44no—
153C2×D4×C73SD1614yescomplemented
154C2×D4×C73C2×D412yes—
155C22×C4×C73—28no—
156C22×C4×C73—21no—
157C22×C4×C73—22no—
158C22×C4×C73—24no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—44no—
166C22×C4×C73—44no—
167C22×C4×C73—42no—
168C22×C4×C73—42no—
169C22×C4×C73C2×D412yes—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2411yescharacteristic
173C4⋊C4×C73—28no—
174C4⋊C4×C73—24no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—22no—
178C4⋊C4×C73C2×D412yes—
179C4⋊C4×C73C2×D414yes—
180C22⋊C4×C73—28no—
181C22⋊C4×C73—24no—
182C22⋊C4×C73—24no—
183C42×C73C2×D412yes—
184C42×C73C2×D412yes—
185C23×C73—22no—
186C23×C73—22no—
187C23×C73—22no—
188C23×C73—22no—
189C23×C73—22no—
190C23×C73—22no—
191C23×C73—22no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73C2×SD1614yes—
199C23×C73C22×D411yescharacteristic
200Q8×C73—48no—
201D4×C73—22no—
202D4×C73—22no—
203D4×C73—28no—
204D4×C73—416no—
205D4×C73—41no—
206D4×C73—41no—
207C2×C4×C73—21no—
208C2×C4×C73—21no—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—24no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—416no—
215C2×C4×C73—41no—
216C2×C4×C73—42no—
217C2×C4×C73—41no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—48no—
224C2×C4×C73#12012yes—
225C2×C4×C73#12012yes—
226C2×C4×C73C2×SD1612yes—
227C2×C4×C73AGL(1,ℤ/8)12yes—
228C2×C4×C73C22×D411yescharacteristic
229C2×C4×C73C22×D411yescharacteristic
230C2×C4×C73C2×#4111yescharacteristic
231C2×C4×C73C2×#4111yescharacteristic
232C2×C4×C73H(5,𝔽2)11yescharacteristic
233C2×C4×C732−1+411yescharacteristic
234C2×Dic73—22no—
235C2×Dic73—41no—
236C2×Dic73—41no—
237C2×Dic73—41no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—41no—
241C2×Dic73—42no—
242C2×Dic73—41no—
243Dic146—22no—
244Dic146—22no—
245Dic146—41no—
246Dic146—41no—
247#10596831—22no—
248#10596831—41no—
249#10596831—42no—
250C22×C73—24no—
251C22×C73—24no—
252C22×C73—24no—
253C22×C73—42no—
254C22×C73—42no—
255C22×C73—42no—
256C22×C73—42no—
257C22×C73—42no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73—81no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73C2×#12011yescharacteristic
265C22×C73#54611yescharacteristic
266C22×C73#54711yescharacteristic
267C22×C73C22×SD1611yescharacteristic
268C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C73#57711yescharacteristic
270C22×C73#57811yescharacteristic
271C4×C73—22no—
272C4×C73—42no—
273C4×C73—42no—
274C4×C73—44no—
275C4×C73#45014yes—
276C4×C73#52511yescharacteristic
277C4×C73#53711yescharacteristic
278C4×C73#54811yescharacteristic
279C4×C73#55011yescharacteristic
280Dic73—42no—
281Dic73—82no—
282Dic73—81no—
283#24120—731nomaximal, Sylow
284C2×C73—44no—
285C2×C73—82no—
286C2×C73#240711yescharacteristic
287C2×C73C2×#45011yescharacteristic
288C2×C73#301311yescharacteristic
289C2×C73#329811yescharacteristic
290C2×C73#333211yescharacteristic
291C2×C73#333511yescharacteristic
292C2×C73#335311yescharacteristic
293C2×#547D7311yescharacteristic
294C2×#475—731no—
295C2×C4×SD16—731no—
296#2538—731no—
297#2034—732no—
298#2000—732no—
299#1990—731no—
300#1954—731no—
301#1872—732no—
302#1853—732no—
303#1802—731no—
304C73#2412011yescharacteristic, complemented, Sylow
305C22×#41—21no—
306C22×SD16—731no—
307#547—24no—
308#547—24no—
309C2×C4⋊Q8C2×D7311yescharacteristic
310C2×#123—22no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8—732no—
313C2×#121C2×D7312yes—
314C2×C4×Q8—731no—
315C2×C4×D4C2×D7311yescharacteristic
316C2×C4×D4C2×D7311yescharacteristic
317C22×C4⋊C4C2×D7312yes—
318#475—1464no—
319C4×SD16—1464no—
320C2×#106—732no—
321C2×#106—731no—
322C2×C4⋊C8—731no—
323C2×#103—732no—
324C2×#103—731no—
325C2×#102—732no—
326C2×#102—731no—
327C2×C22⋊C8—732no—
328C2×C22⋊C8—732no—
329C2×C4×C8—731no—
330#396—1462no—
331#390—1461no—
332#389—1461no—
333#384—731no—
334#380—732no—
335#340—732no—
336#340—732no—
337#340—732no—
338C2×#41—28no—
339C2×#41—44no—
340C2×#41—44no—
341C22×Q8—21no—
342C22×Q8—731no—
343C22×D4—22no—
344C22×D4C22⋊D7311yescharacteristic
345C22×D4C22×D7311yescharacteristic
346C23×C4—22no—
347C23×C4—21no—
348C23×C4C22×D7312yes—
349C2×SD16—1462no—
350C2×SD16—1462no—
351C2×SD16—1462no—
352C22×C8—731no—
353C22×C8—731no—
354C22×C8—732no—
355C4⋊Q8—22no—
356C4⋊Q8C22⋊D7314yes—
357#123—216no—
358C22⋊Q8—28no—
359C22⋊Q8—2922no—
360C22⋊Q8—1464no—
361#121—24no—
362#121—24no—
363C4×Q8—1462no—
364C4×Q8—1462no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4C22⋊D7314yes—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—24no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—732no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1462no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4—1461no—
379C2×C4⋊C4C22⋊D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C4⋊C4C22×D7312yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1462no—
385C2×C22⋊C4—1462no—
386C2×C22⋊C4C22⋊D7312yes—
387C2×C42—1461no—
388C2×C42C22×D7311yescharacteristic
389#106—1462no—
390#106—1464no—
391C4⋊C8—1462no—
392#103—1462no—
393#103—1464no—
394#102—2922no—
395#102—1462no—
396C22⋊C8—2922no—
397C22⋊C8—1464no—
398C4×C8—1462no—
399#95—1462no—
400C24—21no—
401C24C2×C22⋊D7312yes—
402#41—432no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—732no—
407C2×Q8—1461no—
408C2×Q8—1461no—
409C2×Q8—1461no—
410C2×Q8—1461no—
411C2×Q8—1461no—
412C2×D4—28no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4#1192401614yescomplemented
423C2×D4C2×C22⋊D7312yes—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—1461no—
439C22×C4—1462no—
440C22×C4—1461no—
441C22×C4C2×C22⋊D7312yes—
442C22×C4C2×C22⋊D7311yescharacteristic
443C22×C4C2×C22⋊D7311yescharacteristic
444C22×C4C23×D7311yescharacteristic
445SD16—1468no—
446C2×C8—732no—
447C2×C8—1461no—
448C2×C8—1462no—
449C2×C8—1462no—
450C2×C8—1461no—
451C2×C8—1462no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8—1461no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—2924no—
461C4⋊C4—2922no—
462C4⋊C4—2922no—
463C4⋊C4—5841no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1462no—
467C4⋊C4—1462no—
468C4⋊C4C2×C22⋊D7312yes—
469C4⋊C4C2×C22⋊D7314yes—
470C22⋊C4—28no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—2924no—
474C22⋊C4—5842no—
475C42—2922no—
476C42C2×C22⋊D7312yes—
477C42C2×C22⋊D7312yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×#1192401614yes—
492C23C22×C22⋊D7311yescharacteristic
493Q8—48no—
494Q8—2921no—
495Q8—2921no—
496Q8—1462no—
497Q8—1462no—
498D4—22no—
499D4—28no—
500D4—22no—
501D4—416no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—416no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2922no—
525C2×C4—2921no—
526C2×C4—2922no—
527C2×C4—2921no—
528C2×C4—2922no—
529C2×C4—1462no—
530C2×C4#1455656012yes—
531C2×C4#1455656012yes—
532C2×C4C2×#1192401612yes—
533C2×C4#1455661712yes—
534C2×C4C22×C22⋊D7311yescharacteristic
535C2×C4C22×C22⋊D7311yescharacteristic
536C2×C4#41×D7311yescharacteristic
537C2×C4#41×D7311yescharacteristic
538C2×C4#1455663711yescharacteristic
539C2×C4#1455663811yescharacteristic
540C8—2921no—
541C8—2922no—
542C8—1462no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1455656011yescharacteristic
558C22#1798922511yescharacteristic
559C22#1798923611yescharacteristic
560C22C22×#1192401611yescharacteristic
561C22C2×#1455661711yescharacteristic
562C22#1798939511yescharacteristic
563C22#1798940211yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—44no—
568C4—2922no—
569C4—5842no—
570C4—5841no—
571C4#1798873814yes—
572C4#1798909411yescharacteristic
573C4#1798916511yescharacteristic
574C4#1798924411yescharacteristic
575C4#1798925611yescharacteristic
576C2—44no—
577C2—82no—
578C2#2261327011yescharacteristic
579C2C2×#1798873811yescharacteristic
580C2#2261924611yescharacteristic
581C2#2262179011yescharacteristic
582C2#2262225211yescharacteristic
583C2#2262228811yescharacteristic
584C2#2262251511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
Fitting, prime radicalC2×#547×C73C2
socleC23×C73C22×D4
hypercenterC2×#547D73
hypocenterC73#24120
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#17988869C2semidirect
2. C2×C4×#11924016C2semidirect
3. #22614853C2semidirect
4. #22611117C2semidirect
5. #22610934C2semidirect
6. #22610893C2semidirect
7. #22610726C2semidirect
8. #22610238C2semidirect
9. #22610124C2semidirect
10. #22609931C2semidirect
112. C2×Dic146#41semidirect
125. C2×#10596831#41semidirect
153. C2×D4×C73SD16semidirect
304. C73#24120semidirect
422. C2×D4#11924016semidirect

Maximal subgroups

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

subgroupindex
C2×#179888692
C2×C4×#119240162
#226148532
#226111172
#226109342
#226108932
#226107262
#226102382
#226101242
#226099312
C2×#547×C732
#2412073