Finite Group Databasebeta

← #34016088

Subgroups — #34016088

Order 18688.

984 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626784C211yesmaximal, characteristic, complemented
2#22626780C211yesmaximal, characteristic, complemented
3#22625212C212yesmaximal, complemented
4#22625209C212yesmaximal, complemented
5#22625204C212yesmaximal, complemented
6#22625197C212yesmaximal, complemented
7#22625167C212yesmaximal, complemented
8C2×#17989432C212yesmaximal, complemented
9C2×#17989181C211yesmaximal, characteristic
10#17989616—24no—
11#17989614—24no—
12#17989608C2212yes—
13#17989604C2212yes—
14#17989558C2212yes—
15#17989555C2212yes—
16C2×C22⋊#10596833C2212yes—
17C2×C22⋊#10596833C2212yes—
18#17989432C2214yes—
19#17989432C2214yes—
20#17989432C2214yes—
21#17989430C2214yes—
22#17989181—24no—
23#17989181—24no—
24C2×#14556586C411yescharacteristic
25C2×#14556584C2211yescharacteristic
26C2×#14556582C2211yescharacteristic
27C2×#14556581C411yescharacteristic
28C2×#14556575C411yescharacteristic
29C2×#14556574C411yescharacteristic
30C2×#14556573C2211yescharacteristic
31C2×#14556571C2211yescharacteristic
32C2×#14556568C411yescharacteristic
33C2×#14556568C411yescharacteristic
34C2×#14556563C411yescharacteristic
35C2×D4×Dic73C2211yescharacteristic
36C2×#14556551C2211yescharacteristic
37C22×C4⋊Dic73C2211yescharacteristic
38C2×#123×C73C411yescharacteristic
39C2×#11924038—22no—
40C2×#11924036—22no—
41C22×#10596833—22no—
42C22×#10596833—22no—
43#14556658—22no—
44C4⋊#10596833—22no—
45#14556648—22no—
46#14556648—22no—
47C22⋊#10596833—22no—
48C22⋊#10596833—22no—
49C22⋊#10596833C2314yes—
50C22⋊#10596833C2314yes—
51#14556646—22no—
52C4×#10596833—22no—
53C22×Dic146C2×C411yescharacteristic
54C22×C22⋊D73C2×C411yescharacteristic
55C22×C22⋊D73C2×C411yescharacteristic
56C23×Dic73C2×C411yescharacteristic
57C23×Dic73C2×C411yescharacteristic
58C23×Dic73C2311yescharacteristic
59C22×C4×D73C2311yescharacteristic
60#14556586—24no—
61#14556584—22no—
62#14556584—22no—
63#14556582—22no—
64#14556582—22no—
65#14556581—24no—
66#14556575—24no—
67#14556574—24no—
68#14556573—22no—
69#14556573—22no—
70#14556571—22no—
71#14556571—22no—
72#14556568—24no—
73#14556568—24no—
74#14556563—24no—
75D4×Dic73—22no—
76D4×Dic73—22no—
77#14556551D414yes—
78#14556551D414yes—
79C2×C4⋊Dic73—22no—
80C2×C4⋊Dic73—22no—
81C2×C4⋊Dic73C2×C411yescharacteristic
82C2×C4⋊Dic73C2×C411yescharacteristic
83C2×C4⋊Dic73C2×C411yescharacteristic
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×C4⋊Dic73C2311yescharacteristic
87C2×#11924007C2×C411yescharacteristic
88C2×#11924007C2×C411yescharacteristic
89C2×#11924007C2×C411yescharacteristic
90C2×#11924007C2×C411yescharacteristic
91C2×#11924007C2×C411yescharacteristic
92C2×#11924007C2×C411yescharacteristic
93C2×C22⋊Dic73C2×C411yescharacteristic
94C2×C22⋊Dic73C2×C411yescharacteristic
95C2×C22⋊Dic73C2311yescharacteristic
96C2×#11924005C2×C411yescharacteristic
97C2×#11924005C2×C411yescharacteristic
98C2×#11924005C2×C411yescharacteristic
99C2×#11924005C2×C411yescharacteristic
100C2×C4×Dic73C2×C411yescharacteristic
101C2×C4×Dic73C2311yescharacteristic
102C22×D4×C73C2×C411yescharacteristic
103C23×C4×C73C2×C411yescharacteristic
104#123×C73—24no—
105C2×C4⋊C4×C73C2×C411yescharacteristic
106C2×C4⋊C4×C73C2×C411yescharacteristic
107C2×C22⋊C4×C73C2×C411yescharacteristic
108C2×C22⋊C4×C73C2×C411yescharacteristic
109C2×C22⋊C4×C73C2×C411yescharacteristic
110#11924038—82no—
111#11924036—82no—
112C2×#10596833—24no—
113C2×#10596833—24no—
114C2×#10596833—82no—
115C2×#10596833—82no—
116C23×D73C22×C411yescharacteristic
117C2×Dic146—22no—
118C2×Dic146—81no—
119C2×C22⋊D73—24no—
120C2×C22⋊D73—22no—
121C2×C22⋊D73—81no—
122C2×C22⋊D73C22⋊C414yes—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—81no—
126C22×Dic73—81no—
127C22×Dic73—81no—
128C22×Dic73C22×C411yescharacteristic
129C22×Dic73C22×C411yescharacteristic
130C22×Dic73C22×C411yescharacteristic
131C22×Dic73C22×C411yescharacteristic
132C22×Dic73C22×C411yescharacteristic
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C2×D412yes—
135C22×Dic73C2×D412yes—
136C22×Dic73C2411yescharacteristic
137C2×C4×D73—81no—
138C2×C4×D73C2×D412yes—
139C2×C4×D73C2×D412yes—
140C4⋊Dic73—22no—
141C4⋊Dic73—22no—
142C4⋊Dic73—22no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149#11924007—22no—
150#11924007—22no—
151#11924007—22no—
152#11924007—22no—
153#11924007—22no—
154#11924007C22⋊C414yes—
155C22⋊Dic73—22no—
156C22⋊Dic73—22no—
157C22⋊Dic73—21no—
158C22⋊Dic73—21no—
159#11924005—22no—
160#11924005—22no—
161#11924005—22no—
162#11924005C22⋊C414yes—
163C4×Dic73—22no—
164C4×Dic73C2×D412yes—
165C4×Dic73C2×D412yes—
166C24×C73C22×C411yescharacteristic
167C24×C73C22×C411yescharacteristic
168C2×D4×C73—22no—
169C2×D4×C73—24no—
170C22×C4×C73—22no—
171C22×C4×C73—24no—
172C22×C4×C73C22×C411yescharacteristic
173C22×C4×C73C22×C411yescharacteristic
174C22×C4×C73C22×C411yescharacteristic
175C22×C4×C73C22×C411yescharacteristic
176C22×C4×C73C22×C411yescharacteristic
177C4⋊C4×C73—22no—
178C4⋊C4×C73—22no—
179C22⋊C4×C73—22no—
180C22⋊C4×C73—22no—
181C22⋊C4×C73C22⋊C414yes—
182C23×C73—21no—
183C23×C73—22no—
184C23×C73—41no—
185C23×C73—41no—
186C23×C73—41no—
187C23×C73—41no—
188C23×C73C2×C22⋊C412yes—
189C23×C73#13114yes—
190C23×C73C23×C411yescharacteristic
191C22×D73—22no—
192C22×D73—41no—
193C22×D73—41no—
194C22×D73C2×C22⋊C412yes—
195D4×C73—44no—
196C2×C4×C73—21no—
197C2×C4×C73—21no—
198C2×C4×C73—21no—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—41no—
203C2×C4×C73—41no—
204C2×C4×C73—44no—
205C2×C4×C73C2×C22⋊C412yes—
206C2×C4×C73C2×C22⋊C412yes—
207C22⋊D73—28no—
208C22⋊D73—82no—
209C2×Dic73—22no—
210C2×Dic73—21no—
211C2×Dic73—21no—
212C2×Dic73—21no—
213C2×Dic73—21no—
214C2×Dic73—41no—
215C2×Dic73—41no—
216C2×Dic73—42no—
217C2×Dic73—42no—
218C2×Dic73—41no—
219C2×Dic73—41no—
220C2×Dic73—41no—
221C2×Dic73—41no—
222C2×Dic73—82no—
223C2×Dic73—82no—
224C2×Dic73C2×C22⋊C412yes—
225C2×Dic73C2×C22⋊C412yes—
226C2×Dic73C22×D411yescharacteristic
227C2×Dic73C22×D411yescharacteristic
228C4×D73—42no—
229C4×D73—42no—
230Dic146—82no—
231#10596833—44no—
232#10596833—44no—
233C22×C73—21no—
234C22×C73—22no—
235C22×C73—21no—
236C22×C73—22no—
237C22×C73—44no—
238C22×C73—41no—
239C22×C73—41no—
240C22×C73—41no—
241C22×C73—82no—
242C22×C73#40814yes—
243C22×C73C22×C22⋊C411yescharacteristic
244C22×C73C2×#13112yes—
245C2×D73—22no—
246C2×D73—22no—
247C2×D73—22no—
248C2×D73—82no—
249C4×C73—22no—
250C4×C73—22no—
251C4×C73—41no—
252C4×C73—41no—
253C4×C73—41no—
254Dic73—22no—
255Dic73—21no—
256Dic73—21no—
257Dic73—22no—
258Dic73—41no—
259Dic73—41no—
260Dic73—41no—
261Dic73—41no—
262#17387—731nomaximal, Sylow
263C2×C73—22no—
264C2×C73—24no—
265C2×C73—81no—
266C2×C73C2×#40811yescharacteristic
267C2×C73#288311yescharacteristic
268C2×C73#288911yescharacteristic
269D73—42no—
270C2×#539—731no—
271#2973—731no—
272#2972—731no—
273#1473—732no—
274#1472—732no—
275#1470—732no—
276#1468—732no—
277#1459—732no—
278C2×#324—732no—
279C73#1738711yescharacteristic, complemented, Sylow
280#539—1464no—
281#539—1464no—
282C2×#126—731no—
283C2×#126—731no—
284C2×#125—731no—
285C2×#125—731no—
286C2×#123—731no—
287C2×#123—731no—
288C2×#123—731no—
289C2×#123#6440811yescharacteristic
290C2×C22⋊Q8—731no—
291C2×C22⋊Q8—731no—
292C2×C22⋊Q8—731no—
293C2×#121—731no—
294C2×C4×D4—731no—
295C2×C4×D4—731no—
296C22×C4⋊C4—731no—
297#435—1464no—
298C8×D4—1464no—
299#432—732no—
300#431—732no—
301#407—732no—
302C2×C22⋊C8—732no—
303C2×C22⋊C8—732no—
304C2×C22⋊C8—732no—
305#324—734no—
306#324—734no—
307#324—734no—
308#323—734no—
309C22×Q8—731no—
310C22×D4—731no—
311C22×D4—731no—
312C22×D4C2×#6440811yescharacteristic
313C23×C4—731no—
314C23×C4—731no—
315C23×C4—731no—
316C23×C4—731no—
317C23×C4C2×#6440811yescharacteristic
318C2×M16—1462no—
319C22×C8—1462no—
320C22×C8—1462no—
321C22×C8—1462no—
322#126—1462no—
323#126—1462no—
324#126—1464no—
325#125—1462no—
326#125—1462no—
327#125—1464no—
328#123—24no—
329#123—1464no—
330#123—1462no—
331#123—1462no—
332#123—1464no—
333C22⋊Q8—1464no—
334C22⋊Q8—1462no—
335C22⋊Q8—1462no—
336C22⋊Q8—1464no—
337#121—1464no—
338C4×D4—734no—
339C4×D4—734no—
340C4×D4—1462no—
341C4×D4—1462no—
342C2×C4⋊C4—731no—
343C2×C4⋊C4—731no—
344C2×C4⋊C4—731no—
345C2×C4⋊C4—731no—
346C2×C4⋊C4—731no—
347C2×C4⋊C4—731no—
348C2×C4⋊C4—731no—
349C2×C4⋊C4—731no—
350C2×C4⋊C4—731no—
351C2×C4⋊C4—731no—
352C2×C4⋊C4—731no—
353C2×C4⋊C4—731no—
354C2×C4⋊C4—1462no—
355C2×C4⋊C4—1462no—
356C2×C4⋊C4C2×#6440811yescharacteristic
357C2×C4⋊C4C2×#6440811yescharacteristic
358C2×C22⋊C4—731no—
359C2×C22⋊C4—731no—
360C2×C22⋊C4—731no—
361C2×C22⋊C4—731no—
362C2×C22⋊C4—731no—
363C2×C22⋊C4—731no—
364C2×C22⋊C4—731no—
365C2×C22⋊C4C2×#6440811yescharacteristic
366C2×C22⋊C4C2×#6440811yescharacteristic
367C2×C22⋊C4C2×#6440811yescharacteristic
368C2×C42—731no—
369C2×C42—731no—
370C4⋊C8—1462no—
371C4⋊C8—1462no—
372C22⋊C8—734no—
373C22⋊C8—734no—
374C22⋊C8—1462no—
375C22⋊C8—1462no—
376C22⋊C8—1462no—
377C22⋊C8—1462no—
378#97—1462no—
379C4×C8—1462no—
380C24—731no—
381C24C22×#6440811yescharacteristic
382C24C22×#6440811yescharacteristic
383C2×Q8—5841no—
384C2×Q8—1462no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—5841no—
388C2×D4—734no—
389C2×D4—1462no—
390C2×D4—1464no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—5841no—
394C22×C4—5841no—
395C22×C4—5841no—
396C22×C4—5841no—
397C22×C4—731no—
398C22×C4—731no—
399C22×C4—732no—
400C22×C4—732no—
401C22×C4—732no—
402C22×C4—732no—
403C22×C4—731no—
404C22×C4—731no—
405C22×C4—731no—
406C22×C4—731no—
407C22×C4—731no—
408C22×C4—1462no—
409C22×C4—1462no—
410C22×C4C22×#6440811yescharacteristic
411C22×C4C22×#6440811yescharacteristic
412C22×C4C22×#6440811yescharacteristic
413C22×C4C22×#6440811yescharacteristic
414C22×C4C22×#6440811yescharacteristic
415M16—5842no—
416C2×C8—5842no—
417C2×C8—5842no—
418C2×C8—5842no—
419C2×C8—1464no—
420C2×C8—1464no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—734no—
424C4⋊C4—1462no—
425C4⋊C4—1462no—
426C4⋊C4—1462no—
427C4⋊C4—1461no—
428C4⋊C4—1461no—
429C4⋊C4—1462no—
430C4⋊C4—1461no—
431C4⋊C4—1461no—
432C4⋊C4—1462no—
433C4⋊C4—1461no—
434C4⋊C4—1461no—
435C4⋊C4—1462no—
436C4⋊C4—1462no—
437C4⋊C4—1462no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—734no—
441C22⋊C4—1462no—
442C22⋊C4—1462no—
443C22⋊C4—1462no—
444C22⋊C4—1461no—
445C22⋊C4—1461no—
446C22⋊C4—1462no—
447C22⋊C4—1462no—
448C22⋊C4C22⋊#6440814yes—
449C42—732no—
450C42—732no—
451C42—1462no—
452C23—21no—
453C23—22no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—2921no—
459C23—2921no—
460C23—732no—
461C23—1462no—
462C23C2×C22⋊#6440812yes—
463C23#1455667814yes—
464C23C23×#6440811yescharacteristic
465Q8—5842no—
466D4—44no—
467D4—5842no—
468D4—1468no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—2922no—
479C2×C4—2922no—
480C2×C4—2921no—
481C2×C4—2921no—
482C2×C4—2921no—
483C2×C4—2921no—
484C2×C4—2922no—
485C2×C4—2922no—
486C2×C4—2921no—
487C2×C4—2921no—
488C2×C4—5842no—
489C2×C4—5842no—
490C2×C4—732no—
491C2×C4—731no—
492C2×C4—731no—
493C2×C4—732no—
494C2×C4—1461no—
495C2×C4—1461no—
496C2×C4—1461no—
497C2×C4—1461no—
498C2×C4—1462no—
499C2×C4C2×C22⋊#6440812yes—
500C2×C4C2×C22⋊#6440812yes—
501C8—2924no—
502C8—2924no—
503C22—21no—
504C22—22no—
505C22—21no—
506C22—22no—
507C22—44no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—82no—
512C22—5842no—
513C22—1462no—
514C22—1462no—
515C22—1462no—
516C22#1798956114yes—
517C22C22×C22⋊#6440811yescharacteristic
518C22C2×#1455667812yes—
519C4—22no—
520C4—22no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—2921no—
525C4—2921no—
526C4—2921no—
527C4—2921no—
528C4—1462no—
529C4—1461no—
530C4—1461no—
531C4—1462no—
532C2—22no—
533C2—24no—
534C2—81no—
535C2—2922no—
536C2C2×#1798956111yescharacteristic
537C2#2262653111yescharacteristic
538C2#2262654311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×#123×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC2×#14556563C4
cosocleC22×Dic73C24
hypercenterC2×#123#64408
hypocenterC73#17387
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #22626784C2semidirect
2. #22626780C2semidirect
3. #22625212C2semidirect
4. #22625209C2semidirect
5. #22625204C2semidirect
6. #22625197C2semidirect
7. #22625167C2semidirect
8. C2×#17989432C2semidirect
279. C73#17387semidirect

Maximal subgroups

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

subgroupindex
#226267842
#226267802
#226252122
#226252092
#226252042
#226251972
#226251672
C2×#179894322
C2×#179891812
#1738773