Finite Group Databasebeta

← #34021157

Subgroups — #34021157

Order 18688.

1318 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626408C212yesmaximal, complemented
2#22625829C212yesmaximal, complemented
3#22625793C212yesmaximal, complemented
4#22625792C212yesmaximal, complemented
5#22625760C212yesmaximal
6#22625678C212yesmaximal, complemented
7C4×#14556650C212yesmaximal, complemented
8C2×#17989067C211yesmaximal, characteristic
9C2×#14556650C2214yes—
10C2×#14556649C2214yes—
11#17989536—28no—
12#17989530C2214yes—
13#17989529C2214yes—
14C4×C22⋊#64408C2214yes—
15#17989465C2214yes—
16#17989465C2214yes—
17C22×#11924028C2211yescharacteristic
18#17989067—28no—
19C2×C4×Dic146C411yescharacteristic, complemented
20C2×C4×Dic146C411yescharacteristic, complemented
21C2×C4×D292C2211yescharacteristic
22C2×C4×D292C2211yescharacteristic
23C2×#14556551C411yescharacteristic, complemented
24C2×#14556551C411yescharacteristic, complemented
25C2×#14556551C2211yescharacteristic
26C2×#14556551C2211yescharacteristic
27C2×#14556549C411yescharacteristic, complemented
28C2×#14556549C2211yescharacteristic
29C22×C4⋊Dic73C411yescharacteristic, complemented
30C2×C4⋊C4×D73C411yescharacteristic, complemented
31C2×C4⋊C4×D73C2211yescharacteristic
32C2×#117×C73C411yescharacteristic, complemented
33C2×C4⋊#64408—24no—
34C2×C22⋊#64408C2318yes—
35C2×C4×#64408—24no—
36#14556650—28no—
37#14556649—28no—
38(C2×#64408)⋊C4—24no—
39(C2×#64408)⋊C4—24no—
40(C2×#64408)⋊C4—28no—
41C2×#11924028—24no—
42C2×#11924028—24no—
43C2×#11924028—24no—
44C22×Dic146C2×C411yescharacteristic
45C22×D292C2311yescharacteristic
46C22×C22⋊D73C2×C411yescharacteristic
47C22×C22⋊D73C2311yescharacteristic
48C23×Dic73C2×C411yescharacteristic
49C23×Dic73C2×C411yescharacteristic
50C22×C4×D73C2×C411yescharacteristic
51C22×C4×D73C2×C411yescharacteristic
52C22×C4×D73C2×C411yescharacteristic
53C22×C4×D73C2311yescharacteristic
54C22×C4×D73C2311yescharacteristic
55C22×C4×D73C2311yescharacteristic
56C4×Dic146—24no—
57C4×Dic146—24no—
58C4×D292—24no—
59C4×D292—24no—
60#14556551—24no—
61#14556551—24no—
62#14556551—24no—
63#14556551—24no—
64#14556549—24no—
65#14556549—24no—
66C2×C4⋊Dic73—24no—
67C2×C4⋊Dic73C2×C411yescharacteristic
68C2×C4⋊Dic73C2×C411yescharacteristic
69C2×C4⋊Dic73C2×C411yescharacteristic
70C2×C4⋊Dic73C2×C411yescharacteristic
71C2×#11924007C2×C411yescharacteristic
72C2×#11924007C2×C411yescharacteristic
73C2×#11924007C2×C411yescharacteristic
74C2×#11924007C2×C411yescharacteristic
75C4⋊C4×D73—24no—
76C4⋊C4×D73D418yes—
77C2×#11924005C2×C411yescharacteristic
78C2×#11924005C2×C411yescharacteristic
79C2×#11924005C2311yescharacteristic
80C2×#11924005C2311yescharacteristic
81C2×C4×Dic73C2×C411yescharacteristic
82C2×C4×Dic73C2×C411yescharacteristic
83C2×C4×Dic73C2×C411yescharacteristic
84C2×C4×Dic73C2×C411yescharacteristic
85C23×C4×C73C2×C411yescharacteristic
86#117×C73—24no—
87C2×C4⋊C4×C73C2×C411yescharacteristic
88C2×C4⋊C4×C73C2×C411yescharacteristic
89C2×C22⋊C4×C73C2×C411yescharacteristic
90C2×C22⋊C4×C73C2×C411yescharacteristic
91C2×C42×C73C2×C411yescharacteristic
92C2×C42×C73C2×C411yescharacteristic
93C22×#64408—28no—
94C4⋊#64408—84no—
95C22⋊#64408—28no—
96C22⋊#64408—28no—
97C4×#64408—84no—
98C23×D73C22×C411yescharacteristic
99C23×D73C2411yescharacteristic
100#11924028—232no—
101C2×Dic146—22no—
102C2×Dic146—22no—
103C2×Dic146—22no—
104C2×D292—22no—
105C2×D292—22no—
106C2×D292#4114yes—
107C2×C22⋊D73—22no—
108C2×C22⋊D73—22no—
109C2×C22⋊D73—22no—
110C2×C22⋊D73—22no—
111C2×C22⋊D73—22no—
112C2×C22⋊D73#4114yes—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—81no—
116C22×Dic73—81no—
117C22×Dic73C22×C411yescharacteristic
118C22×Dic73C22×C411yescharacteristic
119C22×Dic73C22×C411yescharacteristic
120C22×Dic73C22×C411yescharacteristic
121C22×Dic73C22×C411yescharacteristic
122C22×Dic73C22×C411yescharacteristic
123C2×C4×D73—22no—
124C2×C4×D73—22no—
125C2×C4×D73—22no—
126C2×C4×D73—22no—
127C2×C4×D73—22no—
128C2×C4×D73—22no—
129C2×C4×D73—22no—
130C2×C4×D73—81no—
131C2×C4×D73—81no—
132C2×C4×D73—81no—
133C2×C4×D73—81no—
134C2×C4×D73C2×D414yes—
135C2×C4×D73C2×D414yes—
136C2×C4×D73C2×D414yes—
137C4⋊Dic73—22no—
138C4⋊Dic73—22no—
139C4⋊Dic73—22no—
140C4⋊Dic73C22⋊C414yes—
141#11924007—22no—
142#11924007—22no—
143#11924007C22⋊C414yes—
144#11924007C22⋊C414yes—
145#11924005—22no—
146#11924005—22no—
147#11924005—22no—
148#11924005—22no—
149C4×Dic73—22no—
150C4×Dic73—22no—
151C4×Dic73—22no—
152C4×Dic73—22no—
153C24×C73C22×C411yescharacteristic
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73C22×C411yescharacteristic
158C22×C4×C73C22×C411yescharacteristic
159C22×C4×C73C22×C411yescharacteristic
160C22×C4×C73C22×C411yescharacteristic
161C22×C4×C73C22×C411yescharacteristic
162C22×C4×C73C22×C411yescharacteristic
163C4⋊C4×C73—22no—
164C4⋊C4×C73C22⋊C414yes—
165C22⋊C4×C73—22no—
166C22⋊C4×C73—22no—
167C42×C73—22no—
168C42×C73—22no—
169C23×C73—21no—
170C23×C73—21no—
171C23×C73—41no—
172C23×C73—41no—
173C23×C73#11712yes—
174C23×C73C23×C411yescharacteristic
175C22×D73—21no—
176C22×D73—21no—
177C22×D73—41no—
178C22×D73—41no—
179C22×D73—41no—
180C22×D73—41no—
181C22×D73#11712yes—
182C22×D73C22×D412yes—
183C22×D73C2×#4112yes—
184C22×D73C2×#4112yes—
185C2×#64408—48no—
186C2×#64408—48no—
187C2×#64408—48no—
188C2×C4×C73—28no—
189C2×C4×C73—21no—
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73—21no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73C2×C22⋊C412yes—
200C2×C4×C73C2×C22⋊C412yes—
201C2×C4×C73C2×C22⋊C412yes—
202C2×C4×C73#11712yes—
203C22⋊D73—28no—
204C22⋊D73—28no—
205C2×Dic73—21no—
206C2×Dic73—21no—
207C2×Dic73—21no—
208C2×Dic73—21no—
209C2×Dic73—21no—
210C2×Dic73—21no—
211C2×Dic73—41no—
212C2×Dic73—41no—
213C2×Dic73—41no—
214C2×Dic73—41no—
215C2×Dic73—82no—
216C2×Dic73—82no—
217C2×Dic73C2×C22⋊C412yes—
218C2×Dic73C2×C22⋊C412yes—
219C2×Dic73C2×C22⋊C412yes—
220C2×Dic73#11712yes—
221D292—28no—
222C4×D73—28no—
223C4×D73—28no—
224C4×D73—44no—
225C4×D73—44no—
226C4×D73—82no—
227C4×D73—82no—
228Dic146—28no—
229C22×C73—21no—
230C22×C73—21no—
231C22×C73—22no—
232C22×C73—42no—
233C22×C73—42no—
234C22×C73—41no—
235C22×C73—41no—
236C22×C73C22×C22⋊C411yescharacteristic
237C22×C73C2×#11711yescharacteristic
238C22×C73C2×#11711yescharacteristic
239C2×D73—22no—
240C2×D73—22no—
241C2×D73—22no—
242C2×D73—42no—
243C2×D73—42no—
244C2×D73—42no—
245C2×D73—41no—
246C2×D73—41no—
247C2×D73—42no—
248C2×D73#52514yes—
249#64408—88no—
250C4×C73—22no—
251C4×C73—22no—
252C4×C73—22no—
253C4×C73—22no—
254C4×C73—41no—
255C4×C73—41no—
256Dic73—22no—
257Dic73—22no—
258Dic73—22no—
259Dic73—22no—
260Dic73—41no—
261Dic73—41no—
262#20725—731nomaximal, Sylow
263C2×C73—21no—
264C2×C73—21no—
265C2×C73—42no—
266C2×C73#228011yescharacteristic
267C2×C73#288311yescharacteristic
268C2×C73#288411yescharacteristic
269D73—24no—
270D73—42no—
271C2×#520—731no—
272#2313—732no—
273#1820—732no—
274#1799—732no—
275#1799—732no—
276#1782—732no—
277C4×#99—732no—
278C4×#99—732no—
279C73#2072511yescharacteristic, complemented, Sylow
280C22×#41—731no—
281#520—1468no—
282C2×C4×Q8—731no—
283C2×C4×Q8—731no—
284C2×C4×D4—731no—
285C2×C4×D4—731no—
286C2×C4×D4—731no—
287C2×C4×D4—731no—
288C2×C4×D4—731no—
289C2×C4×D4—731no—
290C2×#117—731no—
291C2×#117—731no—
292C2×#117#6440811yescharacteristic, complemented
293C22×C4⋊C4—731no—
294C22×C4⋊C4—731no—
295C22×C4⋊C4—731no—
296C2×#99—734no—
297C2×#99—734no—
298#387—1468no—
299#381—734no—
300#380—734no—
301C4×C22⋊C4—734no—
302#342—734no—
303#342—734no—
304C2×#41—1464no—
305C2×#41—1464no—
306C2×#41—1464no—
307C22×Q8—731no—
308C22×D4—731no—
309C22×D4—731no—
310C22×D4—731no—
311C23×C4—731no—
312C23×C4—731no—
313C23×C4—731no—
314C23×C4—731no—
315C23×C4—731no—
316C23×C4—731no—
317C23×C4—731no—
318C23×C4—731no—
319C23×C4C2×#6440811yescharacteristic
320C4×Q8—1464no—
321C4×Q8—1464no—
322C4×D4—1464no—
323C4×D4—1464no—
324C4×D4—1464no—
325C4×D4—1464no—
326C4×D4—1464no—
327C4×D4—1464no—
328#117—24no—
329#117—1464no—
330#117—1464no—
331C2×C4⋊C4—731no—
332C2×C4⋊C4—731no—
333C2×C4⋊C4—731no—
334C2×C4⋊C4—738no—
335C2×C4⋊C4—731no—
336C2×C4⋊C4—731no—
337C2×C4⋊C4—731no—
338C2×C4⋊C4—731no—
339C2×C4⋊C4—731no—
340C2×C4⋊C4—1464no—
341C2×C4⋊C4—1464no—
342C2×C4⋊C4—1464no—
343C2×C4⋊C4C2×#6440811yescharacteristic
344C2×C4⋊C4C2×#6440811yescharacteristic
345C2×C22⋊C4—731no—
346C2×C22⋊C4—731no—
347C2×C22⋊C4—731no—
348C2×C22⋊C4—731no—
349C2×C22⋊C4—738no—
350C2×C22⋊C4C2×#6440811yescharacteristic
351C2×C22⋊C4C2×#6440811yescharacteristic
352C2×C42—731no—
353C2×C42—731no—
354C2×C42—731no—
355C2×C42—731no—
356C2×C42—1464no—
357C2×C42C2×#6440811yescharacteristic
358C2×C42C2×#6440811yescharacteristic
359#99—1468no—
360#99—1468no—
361#95—1464no—
362#95—1464no—
363#95—1468no—
364C24—731no—
365C24—731no—
366C24C22×#6440811yescharacteristic
367#41—14632no—
368C2×Q8—1462no—
369C2×Q8—1462no—
370C2×Q8—1462no—
371C2×D4—734no—
372C2×D4—734no—
373C2×D4—1462no—
374C2×D4—1462no—
375C2×D4—1462no—
376C2×D4—1462no—
377C2×D4—1462no—
378C2×D4—1462no—
379C2×D4—1462no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—5841no—
384C22×C4—5841no—
385C22×C4—5841no—
386C22×C4—5841no—
387C22×C4—5841no—
388C22×C4—5841no—
389C22×C4—731no—
390C22×C4—731no—
391C22×C4—734no—
392C22×C4—734no—
393C22×C4—731no—
394C22×C4—731no—
395C22×C4—731no—
396C22×C4—731no—
397C22×C4—734no—
398C22×C4—1462no—
399C22×C4—1462no—
400C22×C4—1462no—
401C22×C4—1462no—
402C22×C4—1468no—
403C22×C4—1462no—
404C22×C4—1462no—
405C22×C4—1462no—
406C22×C4—1462no—
407C22×C4—1462no—
408C22×C4C22×#6440811yescharacteristic
409C22×C4C22×#6440811yescharacteristic
410C22×C4C22×#6440811yescharacteristic
411C22×C4C22×#6440811yescharacteristic
412C22×C4C22×#6440811yescharacteristic
413C22×C4C22×#6440811yescharacteristic
414C4⋊C4—22no—
415C4⋊C4—5844no—
416C4⋊C4—734no—
417C4⋊C4—734no—
418C4⋊C4—734no—
419C4⋊C4—1462no—
420C4⋊C4—1462no—
421C4⋊C4—1462no—
422C4⋊C4—1462no—
423C4⋊C4—1462no—
424C4⋊C4C22⋊#6440814yes—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—1462no—
428C22⋊C4—1462no—
429C22⋊C4—1462no—
430C22⋊C4—1462no—
431C22⋊C4—1468no—
432C22⋊C4—1468no—
433C42—22no—
434C42—22no—
435C42—5844no—
436C42—1462no—
437C42—1462no—
438C42—1462no—
439C42—1462no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—41no—
444C23—2921no—
445C23—2921no—
446C23—2921no—
447C23—2921no—
448C23—732no—
449C23—732no—
450C23—732no—
451C23—732no—
452C23—1461no—
453C23—1461no—
454C23#1455667012yes—
455C23C23×#6440811yescharacteristic
456Q8—1468no—
457D4—1468no—
458D4—1468no—
459D4—1468no—
460C2×C4—28no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—2924no—
472C2×C4—2924no—
473C2×C4—2921no—
474C2×C4—2921no—
475C2×C4—2928no—
476C2×C4—2928no—
477C2×C4—2928no—
478C2×C4—2921no—
479C2×C4—2921no—
480C2×C4—5842no—
481C2×C4—5842no—
482C2×C4—5842no—
483C2×C4—5842no—
484C2×C4—732no—
485C2×C4—732no—
486C2×C4—732no—
487C2×C4—732no—
488C2×C4—1468no—
489C2×C4—1468no—
490C2×C4—1461no—
491C2×C4—1461no—
492C2×C4—1461no—
493C2×C4—1461no—
494C2×C4—1461no—
495C2×C4—1461no—
496C2×C4C2×C22⋊#6440812yes—
497C2×C4C2×C22⋊#6440812yes—
498C2×C4C2×C22⋊#6440812yes—
499C2×C4#1455667012yes—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—41no—
507C22—2922no—
508C22—2922no—
509C22—2922no—
510C22—2922no—
511C22—2921no—
512C22—2921no—
513C22—734no—
514C22—1462no—
515C22—1462no—
516C22—1462no—
517C22C22×C22⋊#6440811yescharacteristic
518C22C2×#1455667011yescharacteristic
519C22C2×#1455667011yescharacteristic
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—41no—
525C4—41no—
526C4—2921no—
527C4—2921no—
528C4—5848no—
529C4—1462no—
530C4—1462no—
531C4—1462no—
532C4—1462no—
533C2—21no—
534C2—21no—
535C2—42no—
536C2—2922no—
537C2—1464no—
538C2#2262637511yescharacteristic
539C2#2262653111yescharacteristic
540C2#2262653311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×#117×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC22×#11924028C22
cosocleC23×D73C24
hypercenterC2×#117#64408
hypocenterC73#20725
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #22626408C2semidirect
2. #22625829C2semidirect
3. #22625793C2semidirect
4. #22625792C2semidirect
6. #22625678C2semidirect
7. C4×#14556650C2semidirect
19. C2×C4×Dic146C4semidirect
20. C2×C4×Dic146C4semidirect
23. C2×#14556551C4semidirect
24. C2×#14556551C4semidirect
27. C2×#14556549C4semidirect
29. C22×C4⋊Dic73C4semidirect
30. C2×C4⋊C4×D73C4semidirect
32. C2×#117×C73C4semidirect
279. C73#20725semidirect
292. C2×#117#64408semidirect

Maximal subgroups

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

subgroupindex
#226264082
#226258292
#226257932
#226257922
#226257602
#226256782
C4×#145566502
C2×#179890672
#2072573