Finite Group Databasebeta

← #33119926

Subgroups — #33119926

Order 18688.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988879C211yesmaximal, characteristic, complemented
2C2×C4×#11924016C211yesmaximal, characteristic, complemented
3#22614853C211yesmaximal, characteristic, complemented
4#22611110C212yesmaximal, complemented
5#22610934C212yesmaximal, complemented
6#22610893C211yesmaximal, characteristic, complemented
7#22610726C211yesmaximal, characteristic, complemented
8#22610243C212yesmaximal, complemented
9#22610130C212yesmaximal, complemented
10#22609931C211yesmaximal, characteristic, complemented
11C2×#555×C73C211yesmaximal, characteristic
12C22×#11924016C2211yescharacteristic
13C2×#14556570C2212yes—
14C2×C4×Dic146C2211yescharacteristic
15#17988879—24no—
16C4×#11924016—24no—
17C2×#14556524C2212yes—
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—
35#555×C73—24no—
36#555×C73—24no—
37C2×C4⋊Q8×C73C2211yescharacteristic
38C2×#124×C73—22no—
39C2×#121×C73—21no—
40C2×#121×C73C2212yes—
41C2×C4×D4×C73C2211yescharacteristic
42C2×C4×D4×C73C2211yescharacteristic
43C2×#117×C73—21no—
44C2×#117×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#14556524—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—
76C22×Q8×C73—21no—
77C22×D4×C73—22no—
78C22×D4×C73D411yescharacteristic
79C22×D4×C73C2311yescharacteristic
80C23×C4×C73—21no—
81C23×C4×C73C2312yes—
82C4⋊Q8×C73—22no—
83C4⋊Q8×C73D414yes—
84#124×C73—216no—
85#121×C73—28no—
86#121×C73—24no—
87#121×C73—24no—
88C4×D4×C73—22no—
89C4×D4×C73—22no—
90C4×D4×C73—22no—
91C4×D4×C73D414yes—
92#117×C73—28no—
93#117×C73—24no—
94#117×C73—24no—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73D411yescharacteristic
97C2×C4⋊C4×C73C2311yescharacteristic
98C2×C4⋊C4×C73C2312yes—
99C2×C22⋊C4×C73—22no—
100C2×C22⋊C4×C73—22no—
101C2×C22⋊C4×C73—21no—
102C2×C22⋊C4×C73D412yes—
103C2×C42×C73—22no—
104C2×C42×C73C2311yescharacteristic
105C2×Dic146—21no—
106C2×Dic146—21no—
107C2×Dic146—21no—
108C2×Dic146—21no—
109C2×Dic146—21no—
110C2×Dic146#4112yes—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—22no—
114#11924016—28no—
115C2×#10596831—21no—
116C2×#10596831—21no—
117C2×#10596831—21no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—21no—
121C2×#10596831—22no—
122C2×#10596831—21no—
123C2×#10596831#4112yes—
124C4⋊Dic73—21no—
125C4⋊Dic73—21no—
126C4⋊Dic73—22no—
127C4⋊Dic73—22no—
128C4⋊Dic73—42no—
129C4⋊Dic73—81no—
130#11924007—44no—
131#11924007—42no—
132C22⋊Dic73—44no—
133C22⋊Dic73—82no—
134C4×Dic73—42no—
135C24×C73—21no—
136C24×C73C2×D412yes—
137C2×Q8×C73—24no—
138C2×Q8×C73—81no—
139C2×Q8×C73—81no—
140C2×D4×C73—28no—
141C2×D4×C73—22no—
142C2×D4×C73—22no—
143C2×D4×C73—22no—
144C2×D4×C73—21no—
145C2×D4×C73—21no—
146C2×D4×C73—21no—
147C2×D4×C73—21no—
148C2×D4×C73—21no—
149C2×D4×C73—82no—
150C2×D4×C73—82no—
151C2×D4×C73C2×D412yes—
152C22×C4×C73—21no—
153C22×C4×C73—22no—
154C22×C4×C73—24no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—42no—
162C22×C4×C73—42no—
163C22×C4×C73C2×D412yes—
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D411yescharacteristic
166C22×C4×C73C2411yescharacteristic
167C4⋊C4×C73—24no—
168C4⋊C4×C73—22no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—22no—
171C4⋊C4×C73C2×D412yes—
172C4⋊C4×C73C2×D414yes—
173C22⋊C4×C73—28no—
174C22⋊C4×C73—28no—
175C22⋊C4×C73—24no—
176C22⋊C4×C73—24no—
177C42×C73—28no—
178C42×C73C2×D412yes—
179C42×C73C2×D412yes—
180C23×C73—22no—
181C23×C73—22no—
182C23×C73—22no—
183C23×C73—22no—
184C23×C73—22no—
185C23×C73—22no—
186C23×C73—22no—
187C23×C73—22no—
188C23×C73—41no—
189C23×C73—41no—
190C23×C73—41no—
191C23×C73—41no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73C22×D411yescharacteristic
195Q8×C73—84no—
196D4×C73—22no—
197D4×C73—22no—
198D4×C73—41no—
199D4×C73—41no—
200D4×C73—44no—
201D4×C73—88no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—24no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—41no—
210C2×C4×C73—41no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—48no—
218C2×C4×C73#12012yes—
219C2×C4×C73#12012yes—
220C2×C4×C73#13512yes—
221C2×C4×C73#13712yes—
222C2×C4×C73C22×D411yescharacteristic
223C2×C4×C73C22×D411yescharacteristic
224C2×C4×C73C2×#4111yescharacteristic
225C2×C4×C73C2×#4111yescharacteristic
226C2×C4×C73H(5,𝔽2)11yescharacteristic
227C2×C4×C732−1+411yescharacteristic
228C2×Dic73—22no—
229C2×Dic73—41no—
230C2×Dic73—41no—
231C2×Dic73—41no—
232C2×Dic73—42no—
233C2×Dic73—42no—
234C2×Dic73—41no—
235C2×Dic73—42no—
236C2×Dic73—41no—
237Dic146—22no—
238Dic146—22no—
239Dic146—41no—
240Dic146—41no—
241#10596831—22no—
242#10596831—41no—
243#10596831—42no—
244C22×C73—42no—
245C22×C73—42no—
246C22×C73—42no—
247C22×C73—42no—
248C22×C73—42no—
249C22×C73—42no—
250C22×C73—42no—
251C22×C73—42no—
252C22×C73—42no—
253C22×C73—42no—
254C22×C73—81no—
255C22×C73—81no—
256C22×C73—81no—
257C22×C73—81no—
258C22×C73C2×#12011yescharacteristic
259C22×C73#54611yescharacteristic
260C22×C73#54711yescharacteristic
261C22×C73C2×#13511yescharacteristic
262C22×C73C2×#13711yescharacteristic
263C22×C73#57611yescharacteristic
264C22×C73#57711yescharacteristic
265C4×C73—22no—
266C4×C73—42no—
267C4×C73—42no—
268C4×C73—44no—
269C4×C73#45214yes—
270C4×C73#52511yescharacteristic
271C4×C73#53711yescharacteristic
272C4×C73#54811yescharacteristic
273C4×C73#55011yescharacteristic
274Dic73—42no—
275Dic73—81no—
276Dic73—82no—
277#24121—731nomaximal, Sylow
278C2×C73—82no—
279C2×C73—82no—
280C2×C73#240711yescharacteristic
281C2×C73C2×#45211yescharacteristic
282C2×C73#301211yescharacteristic
283C2×C73#330511yescharacteristic
284C2×C73#332911yescharacteristic
285C2×C73#334011yescharacteristic
286C2×C73#335711yescharacteristic
287C2×#555D7311yescharacteristic
288C2×#477—731no—
289C2×C4×SD16—731no—
290#2538—731no—
291#2033—732no—
292#2000—732no—
293#1990—731no—
294#1954—731no—
295#1873—732no—
296#1854—732no—
297#1802—731no—
298C73#2412111yescharacteristic, complemented, Sylow
299C22×SD16—731no—
300#555—24no—
301#555—24no—
302C2×C4⋊Q8C2×D7311yescharacteristic
303C2×#124—22no—
304C2×C22⋊Q8—732no—
305C2×#121—21no—
306C2×#121C2×D7312yes—
307C2×C4×Q8—731no—
308C2×C4×D4C2×D7311yescharacteristic
309C2×C4×D4C2×D7311yescharacteristic
310C2×#117—21no—
311C2×#117C2×D7312yes—
312#477—1464no—
313C4×SD16—1464no—
314C2×#107—732no—
315C2×#106—731no—
316C2×C4⋊C8—731no—
317C2×#103—732no—
318C2×#103—731no—
319C2×#102—732no—
320C2×#102—731no—
321C2×C22⋊C8—732no—
322C2×C22⋊C8—732no—
323C2×C4×C8—731no—
324#396—1462no—
325#390—1461no—
326#389—1461no—
327#384—731no—
328#380—732no—
329#340—732no—
330#340—732no—
331#340—732no—
332C22×Q8—21no—
333C22×Q8—731no—
334C22×D4—22no—
335C22×D4C22⋊D7311yescharacteristic
336C22×D4C22×D7311yescharacteristic
337C23×C4—21no—
338C23×C4C22×D7312yes—
339C2×SD16—1462no—
340C2×SD16—1462no—
341C2×SD16—1462no—
342C22×C8—731no—
343C22×C8—731no—
344C22×C8—732no—
345C4⋊Q8—22no—
346C4⋊Q8C22⋊D7314yes—
347#124—216no—
348C22⋊Q8—2922no—
349C22⋊Q8—1464no—
350#121—28no—
351#121—24no—
352#121—24no—
353C4×Q8—1462no—
354C4×Q8—1462no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4C22⋊D7314yes—
359#117—28no—
360#117—24no—
361#117—24no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—731no—
364C2×C4⋊C4—732no—
365C2×C4⋊C4—1461no—
366C2×C4⋊C4—1462no—
367C2×C4⋊C4—1461no—
368C2×C4⋊C4—1461no—
369C2×C4⋊C4C22⋊D7311yescharacteristic
370C2×C4⋊C4C22×D7311yescharacteristic
371C2×C4⋊C4C22×D7312yes—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—1462no—
376C2×C22⋊C4—1462no—
377C2×C22⋊C4C22⋊D7312yes—
378C2×C42—22no—
379C2×C42—1461no—
380C2×C42C22×D7311yescharacteristic
381#107—1464no—
382#106—1462no—
383C4⋊C8—1462no—
384#103—1462no—
385#103—1464no—
386#102—2922no—
387#102—1462no—
388C22⋊C8—2922no—
389C22⋊C8—1464no—
390C4×C8—1462no—
391#95—1462no—
392C24—21no—
393C24C2×C22⋊D7312yes—
394C2×Q8—24no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×Q8—732no—
398C2×Q8—1461no—
399C2×Q8—1461no—
400C2×Q8—1461no—
401C2×Q8—1461no—
402C2×Q8—1461no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4C2×C22⋊D7312yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—1461no—
427C22×C4—1462no—
428C22×C4—1461no—
429C22×C4C2×C22⋊D7312yes—
430C22×C4C2×C22⋊D7311yescharacteristic
431C22×C4C2×C22⋊D7311yescharacteristic
432C22×C4C23×D7311yescharacteristic
433SD16—1468no—
434C2×C8—732no—
435C2×C8—1461no—
436C2×C8—1462no—
437C2×C8—1462no—
438C2×C8—1461no—
439C2×C8—1462no—
440C2×C8—1461no—
441C2×C8—1461no—
442C2×C8—1461no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—2924no—
448C4⋊C4—2922no—
449C4⋊C4—2922no—
450C4⋊C4—5841no—
451C4⋊C4—1461no—
452C4⋊C4—1461no—
453C4⋊C4—1462no—
454C4⋊C4—1462no—
455C4⋊C4C2×C22⋊D7312yes—
456C4⋊C4C2×C22⋊D7314yes—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—2924no—
462C22⋊C4—5842no—
463C42—28no—
464C42—2922no—
465C42C2×C22⋊D7312yes—
466C42C2×C22⋊D7312yes—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23C22×C22⋊D7311yescharacteristic
482Q8—84no—
483Q8—2921no—
484Q8—2921no—
485Q8—1462no—
486Q8—1462no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—44no—
492D4—88no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—2921no—
510C2×C4—2921no—
511C2×C4—2921no—
512C2×C4—2922no—
513C2×C4—2921no—
514C2×C4—2922no—
515C2×C4—2921no—
516C2×C4—2922no—
517C2×C4—1462no—
518C2×C4#1455656012yes—
519C2×C4#1455656012yes—
520C2×C4#1455661112yes—
521C2×C4#1455662112yes—
522C2×C4C22×C22⋊D7311yescharacteristic
523C2×C4C22×C22⋊D7311yescharacteristic
524C2×C4#41×D7311yescharacteristic
525C2×C4#41×D7311yescharacteristic
526C2×C4#1455663711yescharacteristic
527C2×C4#1455663811yescharacteristic
528C8—2921no—
529C8—2922no—
530C8—1462no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22C2×#1455656011yescharacteristic
546C22#1798922511yescharacteristic
547C22#1798923611yescharacteristic
548C22C2×#1455661111yescharacteristic
549C22C2×#1455662111yescharacteristic
550C22#1798939011yescharacteristic
551C22#1798939811yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—44no—
556C4—2922no—
557C4—5841no—
558C4—5842no—
559C4#1798874914yes—
560C4#1798909411yescharacteristic
561C4#1798916511yescharacteristic
562C4#1798924411yescharacteristic
563C4#1798925611yescharacteristic
564C2—82no—
565C2—82no—
566C2#2261327011yescharacteristic
567C2C2×#1798874911yescharacteristic
568C2#2261923511yescharacteristic
569C2#2262187511yescharacteristic
570C2#2262221911yescharacteristic
571C2#2262234411yescharacteristic
572C2#2262255111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
Fitting, prime radicalC2×#555×C73C2
socleC23×C73C22×D4
hypercenterC2×#555D73
hypocenterC73#24121
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#17988879C2semidirect
2. C2×C4×#11924016C2semidirect
3. #22614853C2semidirect
4. #22611110C2semidirect
5. #22610934C2semidirect
6. #22610893C2semidirect
7. #22610726C2semidirect
8. #22610243C2semidirect
9. #22610130C2semidirect
10. #22609931C2semidirect
298. C73#24121semidirect

Maximal subgroups

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

subgroupindex
C2×#179888792
C2×C4×#119240162
#226148532
#226111102
#226109342
#226108932
#226107262
#226102432
#226101302
#226099312
C2×#555×C732
#2412173