Finite Group Databasebeta

← #33057441

Subgroups — #33057441

Order 18688.

638 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618883C211yesmaximal, characteristic
2Q8×#11924011C211yesmaximal, characteristic
3C4×Q16×D73C211yesmaximal, characteristic
4#22608826C211yesmaximal, characteristic, complemented
5#22608825C211yesmaximal, characteristic, complemented
6#22608783C211yesmaximal, characteristic, complemented
7#22608782C211yesmaximal, characteristic, complemented
8#22608780C211yesmaximal, characteristic, complemented
9C8×#11924018C211yesmaximal, characteristic, complemented
10#22608735C211yesmaximal, characteristic, complemented
11#22608688C211yesmaximal, characteristic
12#22608358C211yesmaximal, characteristic
13#22608358C211yesmaximal, characteristic
14#22608210C211yesmaximal, characteristic
15C73×#1584C211yesmaximal, characteristic, complemented
16C2×Q16×D73C411yescharacteristic
17C4×Q8×D73C2211yescharacteristic
18C4×Q8×D73C2211yescharacteristic
19#17988718C2211yescharacteristic
20#17988717—21no—
21#17988717C2211yescharacteristic
22#17988716—21no—
23#17988715—21no—
24#17988714—21no—
25#17988714C2211yescharacteristic
26Q8×#10596831C2211yescharacteristic
27C8×Dic146C2211yescharacteristic
28#17988711C2211yescharacteristic
29C4×Dic292C2211yescharacteristic
30Q16×Dic73C411yescharacteristic
31#17988681C411yescharacteristic
32C4×#11924018C2211yescharacteristic
33C4×#11924018C2211yescharacteristic
34#17988679C411yescharacteristic
35#17988679C411yescharacteristic
36#17988638C2211yescharacteristic
37#107×D73C411yescharacteristic
38#17988609—21no—
39#17988609—21no—
40#17988609C2211yescharacteristic
41#17988609C2211yescharacteristic
42#103×D73C411yescharacteristic
43#103×D73C411yescharacteristic
44#17988539—21no—
45C4×#11924011—21no—
46C4×#11924011C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic
49#17988339C2211yescharacteristic
50#17988339C2211yescharacteristic
51#17988319C2211yescharacteristic
52#17988319C2211yescharacteristic
53Q8⋊#10596831C2211yescharacteristic
54Q8⋊#10596831C2211yescharacteristic
55#17988317C2211yescharacteristic
56#17988317C2211yescharacteristic
57#17988309C2211yescharacteristic
58C8×#10596831C2211yescharacteristic
59C8×#10596831C2211yescharacteristic
60#446×C73C2211yescharacteristic
61C8×Q8×C73C2211yescharacteristic
62C4×Q16×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65Q8⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×Q8×D73C2×C411yescharacteristic
68C2×Q8×D73C2×C411yescharacteristic
69C2×Dic292C2×C411yescharacteristic
70C2×#11924018C2×C411yescharacteristic
71C2×#11924018C2×C411yescharacteristic
72Q16×D73—24no—
73C2×#11924011—21no—
74C2×#11924011—21no—
75C2×#11924011D411yescharacteristic
76C2×#11924011D411yescharacteristic
77C2×C8×D73C2×C411yescharacteristic
78C2×C8×D73C2×C411yescharacteristic
79Q8×Dic73C2×C411yescharacteristic
80Q8×Dic73C2×C411yescharacteristic
81C4×Dic146—21no—
82C4×Dic146—21no—
83C4×Dic146C2311yescharacteristic
84C4×Dic146C2311yescharacteristic
85#14556556—21no—
86#14556556—21no—
87#14556556C2×C411yescharacteristic
88#14556556C2×C411yescharacteristic
89C4⋊C4×D73—21no—
90C4⋊C4×D73—21no—
91C4⋊C4×D73C2×C411yescharacteristic
92C4⋊C4×D73C2×C411yescharacteristic
93C42×D73—21no—
94C42×D73—21no—
95C42×D73C2311yescharacteristic
96#14556525C2×C411yescharacteristic
97#14556524C2×C411yescharacteristic
98#14556524C2×C411yescharacteristic
99C4⋊#10596831—21no—
100C4⋊#10596831—21no—
101C4⋊#10596831C2311yescharacteristic
102C4⋊#10596831C2311yescharacteristic
103#14556520—21no—
104#14556520—21no—
105#14556520—21no—
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556520—21no—
110#14556516C2×C411yescharacteristic
111#14556516C2×C411yescharacteristic
112#14556515C2×C411yescharacteristic
113#14556515C2×C411yescharacteristic
114#14556514C2×C411yescharacteristic
115#14556514C2×C411yescharacteristic
116#14556503—21no—
117#14556502—21no—
118#14556501—21no—
119#14556501—21no—
120#14556501D411yescharacteristic
121#14556501D411yescharacteristic
122C4×#10596831—21no—
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73C2×C411yescharacteristic
129C8×Dic73C2×C411yescharacteristic
130C2×Q16×C73C2×C411yescharacteristic
131C4×Q8×C73C2311yescharacteristic
132C4×Q8×C73C2311yescharacteristic
133#107×C73C2×C411yescharacteristic
134C4⋊C8×C73—21no—
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2311yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138#103×C73C2×C411yescharacteristic
139#103×C73C2×C411yescharacteristic
140#97×C73—21no—
141C4×C8×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C2×Dic146—21no—
146C2×Dic146—21no—
147C2×Dic146C22×C411yescharacteristic
148C2×Dic146C22×C411yescharacteristic
149Q8×D73—22no—
150Q8×D73—22no—
151Q8×D73—41no—
152Q8×D73—41no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—21no—
156C2×C4×D73—21no—
157C2×C4×D73C22×C411yescharacteristic
158C2×C4×D73C2×D411yescharacteristic
159C2×C4×D73#4111yescharacteristic
160Dic292—22no—
161#11924018—22no—
162#11924018M1614yescomplemented
163#11924011—22no—
164#11924011—42no—
165#11924011—42no—
166#11924011Q1614yescomplemented
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831C22×C411yescharacteristic
171C2×#10596831C22×C411yescharacteristic
172C2×#10596831C2×D411yescharacteristic
173C2×#10596831C2×D411yescharacteristic
174C8×D73—22no—
175C8×D73—41no—
176C4⋊Dic73—21no—
177C4⋊Dic73—21no—
178C4⋊Dic73C22×C411yescharacteristic
179C4⋊Dic73C22×C411yescharacteristic
180#11924007—21no—
181#11924007—21no—
182#11924007—21no—
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73C22×C411yescharacteristic
193C4×Dic73C2×D411yescharacteristic
194C4×Dic73#4111yescharacteristic
195C2×Q8×C73C22×C411yescharacteristic
196C2×Q8×C73C22×C411yescharacteristic
197Q16×C73—22no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73C22×C411yescharacteristic
202C2×C8×C73C22×C411yescharacteristic
203C2×C8×C73C2×D411yescharacteristic
204C2×C8×C73C2×D411yescharacteristic
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73C22×C411yescharacteristic
208C4⋊C4×C73C22×C411yescharacteristic
209C42×C73—21no—
210C42×C73—21no—
211C42×C73C2411yescharacteristic
212C22×D73C4×D411yescharacteristic
213Q8×C73—21no—
214Q8×C73—21no—
215Q8×C73—21no—
216Q8×C73C2×M1612yes—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73C23×C411yescharacteristic
222C2×C4×C73C22×D411yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73C4×D411yescharacteristic
229C2×Dic73C4×D411yescharacteristic
230C2×Dic73C4×D411yescharacteristic
231C4×D73—21no—
232C4×D73—21no—
233C4×D73—41no—
234C4×D73—42no—
235C4×D73—42no—
236C4×D73—81no—
237C4×D73—81no—
238C4×D73C2×Q1612yes—
239C4×D73#13512yes—
240Dic146—21no—
241Dic146—21no—
242Dic146—21no—
243Dic146—42no—
244Dic146—42no—
245Dic146C2×M1612yes—
246C8×C73—21no—
247C8×C73—21no—
248C8×C73—21no—
249C8×C73—41no—
250C8×C73—41no—
251C8×C73—41no—
252C8×C73C2×Q1612yescomplemented
253#10596831—21no—
254#10596831—21no—
255#10596831—41no—
256#10596831—41no—
257#10596831—41no—
258#10596831C2×M1612yes—
259#10596831C2×Q1612yescomplemented
260C22×C73C2×C4×D411yescharacteristic
261C2×D73—21no—
262C2×D73—21no—
263C2×D73C4×Q1612yes—
264C4×C73—22no—
265C4×C73—21no—
266C4×C73—41no—
267C4×C73—41no—
268C4×C73—41no—
269C4×C73C22×M1611yescharacteristic
270C4×C73#56811yescharacteristic
271C4×C73C22×Q1611yescharacteristic
272C4×C73C2×#13511yescharacteristic
273Dic73—22no—
274Dic73—21no—
275Dic73—21no—
276Dic73—41no—
277Dic73—41no—
278Dic73—41no—
279Dic73C4×Q1612yes—
280M16×Q16—731nomaximal, Sylow
281C2×C73D4×M1611yescharacteristic
282C2×C73C2×C4×Q1611yescharacteristic
283C2×C73#295711yescharacteristic
284D73—22no—
285Q8×M16—731no—
286Q8×M16—731no—
287C2×C4×Q16—731no—
288#1591—731no—
289#1591—731no—
290#1584—731no—
291#1584—731no—
292#1584—731no—
293#1584D7311yescharacteristic, complemented
294C8×Q16—731no—
295C8×Q16—731no—
296#1567—731no—
297#1487—731no—
298#1487—731no—
299C8×M16—731no—
300C73M16×Q1611yescharacteristic, complemented, Sylow
301C22×Q16—731no—
302C2×C4×Q8—731no—
303C2×C4×Q8—731no—
304#446—731no—
305#446—731no—
306#446—731no—
307#446—1461no—
308#446—1461no—
309#446—1461no—
310#446—1461no—
311#446C2×D7311yescharacteristic
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—731no—
315C8×Q8C2×D7311yescharacteristic
316C4×Q16—731no—
317C4×Q16—731no—
318C4×Q16—731no—
319C4×Q16—731no—
320C4×Q16—731no—
321C4×Q16—731no—
322C4×Q16—731no—
323C4×Q16C2×D7311yescharacteristic
324#431—731no—
325C2×#107—731no—
326#423—731no—
327#423—731no—
328#423—1461no—
329#423—1461no—
330C2×#103—731no—
331C2×#103—731no—
332C4×M16—731no—
333C4×M16—1461no—
334C4×M16—1461no—
335C2×C4×C8—731no—
336#335—731no—
337#335—731no—
338#335—731no—
339#335C2×D7311yescharacteristic
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343Q8⋊C8—731no—
344Q8⋊C8—731no—
345Q8⋊C8—731no—
346Q8⋊C8C2×D7311yescharacteristic
347Q8⋊C8C2×D7311yescharacteristic
348#322—731no—
349#322C2×D7311yescharacteristic
350C82—731no—
351C82—731no—
352C22×Q8—731no—
353C22×Q8—731no—
354C2×Q16—731no—
355C2×Q16—731no—
356C2×Q16—731no—
357C2×Q16—1464no—
358C2×Q16C4×D7311yescharacteristic
359C2×M16—731no—
360C2×M16—731no—
361C2×M16—1461no—
362C2×M16—1461no—
363C22×C8—731no—
364C22×C8—731no—
365C4×Q8—731no—
366C4×Q8—731no—
367C4×Q8—731no—
368C4×Q8—731no—
369C4×Q8—731no—
370C4×Q8—731no—
371C4×Q8—1461no—
372C4×Q8—1461no—
373C4×Q8—1461no—
374C4×Q8—1461no—
375C4×Q8C22×D7311yescharacteristic
376C4×Q8C22×D7311yescharacteristic
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—1461no—
380C2×C4⋊C4—1461no—
381C2×C42—731no—
382C2×C42—1461no—
383C2×C42—1461no—
384#107—731no—
385#107—731no—
386#107—731no—
387#107C4×D7311yescharacteristic
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—731no—
391C4⋊C8—731no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396C4⋊C8—1461no—
397C4⋊C8—1461no—
398C4⋊C8—1461no—
399C4⋊C8—1461no—
400C4⋊C8—1461no—
401C4⋊C8C22×D7311yescharacteristic
402C4⋊C8C22×D7311yescharacteristic
403#103—731no—
404#103—731no—
405#103—731no—
406#103—731no—
407#103—731no—
408#103—731no—
409#103C4×D7311yescharacteristic
410#103C4×D7311yescharacteristic
411C22⋊C8—1461no—
412#97—21no—
413#97—731no—
414#97—731no—
415#97—1461no—
416#97—1461no—
417#97—1461no—
418C4×C8—21no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—731no—
422C4×C8—731no—
423C4×C8—731no—
424C4×C8—1461no—
425C4×C8—1461no—
426C4×C8—1461no—
427C4×C8C22×D7311yescharacteristic
428C4×C8C22×D7311yescharacteristic
429C4×C8C22×D7311yescharacteristic
430C2×Q8—2921no—
431C2×Q8—2921no—
432C2×Q8—731no—
433C2×Q8—731no—
434C2×Q8—1461no—
435C2×Q8—1461no—
436C2×Q8—1462no—
437C2×Q8—1462no—
438C2×Q8C2×C4×D7311yescharacteristic
439C2×Q8C2×C4×D7311yescharacteristic
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—731no—
443C22×C4—1461no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1461no—
447Q16—22no—
448Q16—734no—
449Q16—1462no—
450Q16—1462no—
451M16—2922no—
452M16—2922no—
453M16—734no—
454M16—1462no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—2921no—
459C2×C8—731no—
460C2×C8—731no—
461C2×C8—731no—
462C2×C8—731no—
463C2×C8—1462no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8C2×C4×D7311yescharacteristic
468C2×C8C2×C4×D7311yescharacteristic
469C2×C8D4×D7311yescharacteristic
470C2×C8D4×D7311yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—731no—
474C4⋊C4—731no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4C2×C4×D7311yescharacteristic
484C4⋊C4C2×C4×D7311yescharacteristic
485C42—21no—
486C42—21no—
487C42—731no—
488C42—731no—
489C42—731no—
490C42—1461no—
491C42—1461no—
492C42—1461no—
493C42—1461no—
494C42—1461no—
495C42—1461no—
496C42C23×D7311yescharacteristic
497C23—731no—
498Q8—21no—
499Q8—21no—
500Q8—21no—
501Q8—2922no—
502Q8—2922no—
503Q8—732no—
504Q8—1461no—
505Q8—1461no—
506Q8—1461no—
507Q8M16×D7312yes—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—2922no—
513C2×C4—2921no—
514C2×C4—2922no—
515C2×C4—5841no—
516C2×C4—5841no—
517C2×C4—731no—
518C2×C4—732no—
519C2×C4—732no—
520C2×C4—731no—
521C2×C4—731no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4—1461no—
526C2×C4—1461no—
527C2×C4—1461no—
528C2×C4C22×C4×D7311yescharacteristic
529C2×C4C2×D4×D7311yescharacteristic
530C2×C4#41×D7311yescharacteristic
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—2921no—
538C8—2921no—
539C8—2921no—
540C8—732no—
541C8—732no—
542C8—1461no—
543C8—1461no—
544C8Q16×D7312yescomplemented
545C22—732no—
546C22—1461no—
547C22—1461no—
548C22C4×D4×D7311yescharacteristic
549C4—22no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—2921no—
555C4—2921no—
556C4—2921no—
557C4—732no—
558C4—1462no—
559C4—1461no—
560C4—1461no—
561C4C2×M16×D7311yescharacteristic
562C4#1798935211yescharacteristic
563C4C2×Q16×D7311yescharacteristic
564C4#135×D7311yescharacteristic
565C2—1462no—
566C2#2261866411yescharacteristic
567C2C4×Q16×D7311yescharacteristic
568C2#2261884511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1584C2
socleC22×C73C2×C4×D4
prime radicalC22×D73C4×D4
cosocleC42×C73C24
hypercenter#1584D73
hypocenterC73M16×Q16
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #22608826C2semidirect
5. #22608825C2semidirect
6. #22608783C2semidirect
7. #22608782C2semidirect
8. #22608780C2semidirect
9. C8×#11924018C2semidirect
10. #22608735C2semidirect
15. C73×#1584C2semidirect
162. #11924018M16semidirect
166. #11924011Q16semidirect
252. C8×C73C2×Q16semidirect
259. #10596831C2×Q16semidirect
293. #1584D73semidirect
300. C73M16×Q16semidirect
544. C8Q16×D73semidirect

Maximal subgroups

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

subgroupindex
#226188832
Q8×#119240112
C4×Q16×D732
#226088262
#226088252
#226087832
#226087822
#226087802
C8×#119240182
#226087352
#226086882
#226083582
#226082102
C73×#15842
M16×Q1673