Finite Group Databasebeta

← #33068523

Subgroups — #33068523

Order 18688.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620441C211yesmaximal, characteristic, complemented
2#22619939C211yesmaximal, characteristic
3C2×#17988665C211yesmaximal, characteristic
4#22609452C212yesmaximal, complemented
5#22609442C212yesmaximal, complemented
6#22609432C212yesmaximal, complemented
7#22609428C212yesmaximal, complemented
8#22608677C211yesmaximal, characteristic, complemented
9#22608588C211yesmaximal, characteristic, complemented
10#22608586C211yesmaximal, characteristic, complemented
11C73×#3101C211yesmaximal, characteristic
12C22×#11924012—21no—
13#17989144C2211yescharacteristic
14#17988990C2211yescharacteristic, complemented
15#17988990C2211yescharacteristic, complemented
16#17988986—21no—
17#17988986—21no—
18#17988983C2211yescharacteristic, complemented
19#17988983C2211yescharacteristic, complemented
20#17988917—21no—
21#17988877C2212yes—
22#17988792C2212yes—
23#17988738—21no—
24C4×#11924016—22no—
25#17988665—21no—
26#17988665—21no—
27#17988665C2212yes—
28#17988665C2212yes—
29#17988640C2211yescharacteristic
30#17988623—21no—
31#17988623—21no—
32C2×C4⋊#10596831C2211yescharacteristic
33C2×C22⋊#10596831—21no—
34C2×C4×#10596831C2211yescharacteristic
35#17988339C2212yescomplemented
36#17988337C2212yescomplemented
37#17988336C2212yescomplemented
38#17988335C2212yescomplemented
39#17988334C2212yescomplemented
40#17988333C2212yescomplemented
41#533×C73C2211yescharacteristic
42C2×C4×D4×C73C2211yescharacteristic
43#483×C73—21no—
44#476×C73C2212yes—
45#461×C73C2212yes—
46#450×C73—21no—
47C4×SD16×C73—22no—
48#431×C73C2211yescharacteristic
49C2×#11924016—22no—
50C2×#11924012—22no—
51C2×#11924012—22no—
52C2×#11924012—44no—
53C22×#10596831—21no—
54C22×#10596831D411yescharacteristic
55C22×#10596831D411yescharacteristic
56#14556591C2311yescharacteristic
57#14556591C2311yescharacteristic
58#14556583—21no—
59#14556570—21no—
60#14556570—21no—
61C4×Dic146—21no—
62C4×Dic146—21no—
63#14556550—21no—
64#14556524—21no—
65#14556524—21no—
66#14556524—21no—
67#14556524—21no—
68#14556522—21no—
69#14556522—21no—
70#14556522—21no—
71#14556522—21no—
72#14556522—22no—
73C4⋊#10596831—21no—
74C4⋊#10596831C2312yes—
75C4⋊#10596831C2312yes—
76#14556514—22no—
77#14556512—22no—
78C22⋊#10596831—22no—
79C22⋊#10596831—22no—
80C22⋊#10596831—21no—
81C4×#10596831—21no—
82C4×#10596831D411yescharacteristic
83C4×#10596831D411yescharacteristic
84C4×#10596831C2311yescharacteristic
85C4×#10596831C2311yescharacteristic
86C22×D4×C73D411yescharacteristic
87C23×C4×C73—21no—
88C2×SD16×C73—22no—
89C4⋊Q8×C73C2311yescharacteristic
90C4⋊Q8×C73C2311yescharacteristic
91#125×C73—21no—
92C22⋊Q8×C73—21no—
93C22⋊Q8×C73—21no—
94C4×Q8×C73—21no—
95C4×Q8×C73—21no—
96C4×D4×C73—21no—
97C4×D4×C73—21no—
98C4×D4×C73D412yes—
99C4×D4×C73C2312yes—
100#117×C73—21no—
101C2×C4⋊C4×C73D411yescharacteristic
102C2×C22⋊C4×C73—21no—
103C2×C42×C73C2311yescharacteristic
104#106×C73—22no—
105C4⋊C8×C73C2312yes—
106#103×C73—22no—
107#102×C73—22no—
108C22⋊C8×C73—21no—
109C4×C8×C73—21no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112#11924016—44no—
113#11924012—44no—
114#11924012—44no—
115C2×#10596831—22no—
116C2×#10596831—22no—
117C2×#10596831—22no—
118C2×#10596831—42no—
119C2×#10596831SD1614yescomplemented
120C2×#10596831C2×D411yescharacteristic
121C2×#10596831C2×D411yescharacteristic
122C2×#10596831C2×D411yescharacteristic
123C2×#10596831C2×D411yescharacteristic
124C4⋊Dic73—21no—
125C4⋊Dic73—21no—
126C4⋊Dic73—21no—
127C4⋊Dic73—21no—
128#11924007—41no—
129#11924007—41no—
130C22⋊Dic73—41no—
131C4×Dic73—41no—
132C24×C73—21no—
133C2×Q8×C73—21no—
134C2×Q8×C73—21no—
135C2×D4×C73—21no—
136C2×D4×C73—42no—
137C2×D4×C73C2×D412yes—
138C22×C4×C73—21no—
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—42no—
142C22×C4×C73—42no—
143C22×C4×C73C2×D411yescharacteristic
144C22×C4×C73C2×D411yescharacteristic
145C22×C4×C73#4111yescharacteristic
146SD16×C73—44no—
147C2×C8×C73—22no—
148C4⋊C4×C73—21no—
149C4⋊C4×C73—21no—
150C4⋊C4×C73—21no—
151C4⋊C4×C73—21no—
152C4⋊C4×C73—21no—
153C4⋊C4×C73—41no—
154C4⋊C4×C73—41no—
155C4⋊C4×C73C2×D412yes—
156C22⋊C4×C73—22no—
157C22⋊C4×C73—22no—
158C22⋊C4×C73—41no—
159C42×C73—41no—
160C42×C73C2×D411yescharacteristic
161C42×C73C2×D411yescharacteristic
162C42×C73#4111yescharacteristic
163C42×C73C2411yescharacteristic
164C23×C73—22no—
165C23×C73—42no—
166C23×C73—42no—
167C23×C73—42no—
168C23×C73#12111yescharacteristic
169Q8×C73—42no—
170Q8×C73—81no—
171D4×C73—22no—
172D4×C73—22no—
173D4×C73—42no—
174C2×C4×C73—22no—
175C2×C4×C73—22no—
176C2×C4×C73—41no—
177C2×C4×C73—42no—
178C2×C4×C73—42no—
179C2×C4×C73—42no—
180C2×C4×C73—44no—
181C2×C4×C73—41no—
182C2×C4×C73—41no—
183C2×C4×C73#12111yescharacteristic
184C2×C4×C73#12111yescharacteristic
185C2×C4×C73#12111yescharacteristic
186C2×C4×C73C2×SD1612yes—
187C2×C4×C73#13712yes—
188C2×C4×C73C22×D411yescharacteristic
189C2×C4×C73C22×D411yescharacteristic
190C2×C4×C73C2×#4111yescharacteristic
191C2×Dic73—41no—
192C2×Dic73—41no—
193Dic146—42no—
194Dic146—81no—
195C8×C73—42no—
196#10596831—21no—
197#10596831—21no—
198#10596831—42no—
199#10596831—42no—
200#10596831C2×SD1612yes—
201#10596831C2×SD1612yes—
202C22×C73—21no—
203C22×C73—21no—
204C22×C73—42no—
205C22×C73—42no—
206C22×C73—42no—
207C22×C73—84no—
208C22×C73#46112yes—
209C22×C73C2×#12111yescharacteristic
210C4×C73—21no—
211C4×C73—21no—
212C4×C73—21no—
213C4×C73—42no—
214C4×C73—81no—
215C4×C73—81no—
216C4×C73#46112yes—
217C4×C73C22×SD1611yescharacteristic
218C4×C73C2×#13711yescharacteristic
219C4×C73#57512yes—
220Dic73—81no—
221Dic73—81no—
222#19137—731nomaximal, Sylow
223C2×C73—22no—
224C2×C73—82no—
225C2×C73C2×#46111yescharacteristic
226C2×C73#305411yescharacteristic
227C2×C73#306711yescharacteristic
228#3160—731no—
229#3101—731no—
230#3101D7311yescharacteristic
231C2×#436—731no—
232#1695—732no—
233#1693—732no—
234#1691—732no—
235#1691—732no—
236#1564—731no—
237#1542—731no—
238#1542—731no—
239C73#1913711yescharacteristic, complemented, Sylow
240C22×M16—1461no—
241#533—731no—
242#533C2×D7311yescharacteristic
243C2×C4×D4C2×D7311yescharacteristic
244#500—731no—
245#500—731no—
246#499—1461no—
247#499—1461no—
248#498—731no—
249#498—731no—
250#483—21no—
251#483—1461no—
252#476—732no—
253#476C2×D7312yes—
254#461—732no—
255#461C2×D7312yes—
256#450—21no—
257#450—1461no—
258C4×SD16—22no—
259C4×SD16—1462no—
260#436—732no—
261#436—732no—
262#436—1461no—
263#436—1461no—
264#431—731no—
265#431C2×D7311yescharacteristic
266#427—1461no—
267#427—1461no—
268C2×C4⋊C8—731no—
269C2×C22⋊C8—1461no—
270C2×C4×C8—731no—
271#335—732no—
272#334—732no—
273#333—732no—
274#333—732no—
275#332—732no—
276#332—732no—
277C22×D4C22⋊D7311yescharacteristic
278C23×C4—21no—
279C2×SD16—22no—
280C2×SD16—1462no—
281C2×M16—2924no—
282C2×M16—1462no—
283C2×M16—1462no—
284C22×C8—731no—
285C22×C8—731no—
286C22×C8—1461no—
287C4⋊Q8—731no—
288C4⋊Q8—731no—
289C4⋊Q8C22×D7311yescharacteristic
290C4⋊Q8C22×D7311yescharacteristic
291#125—21no—
292#125—1461no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—1461no—
296C22⋊Q8—1461no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—1461no—
300C4×Q8—1461no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4C22⋊D7312yes—
304C4×D4C22×D7312yes—
305#117—21no—
306#117—1461no—
307C2×C4⋊C4C22⋊D7311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C42C22×D7311yescharacteristic
310#107—1461no—
311#107—1461no—
312#107—1461no—
313#107—1461no—
314#106—22no—
315#106—1461no—
316#106—1461no—
317#106—1461no—
318#106—1461no—
319#106—1462no—
320C4⋊C8—732no—
321C4⋊C8—732no—
322C4⋊C8—1461no—
323C4⋊C8C22×D7312yes—
324#103—22no—
325#103—1462no—
326#102—22no—
327#102—1462no—
328C22⋊C8—21no—
329C22⋊C8—1462no—
330C22⋊C8—1462no—
331C22⋊C8—1461no—
332C4×C8—21no—
333C4×C8—731no—
334C4×C8—731no—
335C4×C8—731no—
336C4×C8—731no—
337C4×C8—1461no—
338C24—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—1461no—
342C2×Q8—1461no—
343C2×D4—21no—
344C2×D4—42no—
345C2×D4C2×C22⋊D7312yes—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4D4×D7311yescharacteristic
352C22×C4C2×C22⋊D7311yescharacteristic
353C22×C4#1192402711yescharacteristic
354SD16—44no—
355SD16—2924no—
356M16—2924no—
357M16—2924no—
358C2×C8—22no—
359C2×C8—2922no—
360C2×C8—734no—
361C2×C8—731no—
362C2×C8—731no—
363C2×C8—731no—
364C2×C8—731no—
365C2×C8—1462no—
366C2×C8—1462no—
367C2×C8—1462no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—2921no—
376C4⋊C4—2921no—
377C4⋊C4—1461no—
378C4⋊C4—1461no—
379C4⋊C4—1461no—
380C4⋊C4—1461no—
381C4⋊C4C2×C22⋊D7312yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—41no—
385C22⋊C4—2921no—
386C42—41no—
387C42—2921no—
388C42D4×D7311yescharacteristic
389C42C2×C22⋊D7311yescharacteristic
390C42#1192402711yescharacteristic
391C42C23×D7311yescharacteristic
392C23—22no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23#1455656311yescharacteristic
397Q8—42no—
398Q8—81no—
399Q8—2922no—
400Q8—5841no—
401D4—22no—
402D4—22no—
403D4—42no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—2921no—
414C2×C4—2921no—
415C2×C4#1455656311yescharacteristic
416C2×C4#1455656311yescharacteristic
417C2×C4#1455656311yescharacteristic
418C2×C4SD16×D7312yes—
419C2×C4#1455662012yes—
420C2×C4C2×D4×D7311yescharacteristic
421C2×C4C22×C22⋊D7311yescharacteristic
422C2×C4C2×#1192402711yescharacteristic
423C8—42no—
424C8—2922no—
425C8—2922no—
426C8—732no—
427C8—732no—
428C8—1461no—
429C8—1461no—
430C22—21no—
431C22—21no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—84no—
436C22#1798879012yes—
437C22C2×#1455656311yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—5841no—
445C4—5841no—
446C4#1798879012yes—
447C4C2×SD16×D7311yescharacteristic
448C4C2×#1455662011yescharacteristic
449C4#1798938512yes—
450C2—22no—
451C2—82no—
452C2C2×#1798879011yescharacteristic
453C2#2261953711yescharacteristic
454C2#2261964811yescharacteristic
subgroupG/H
centerC2×C4#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3101C2
socleC22×C73C2×#121
prime radicalC22×D4×C73D4
hypercenter#3101D73
hypocenterC73#19137
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
1. #22620441C2semidirect
4. #22609452C2semidirect
5. #22609442C2semidirect
6. #22609432C2semidirect
7. #22609428C2semidirect
8. #22608677C2semidirect
9. #22608588C2semidirect
10. #22608586C2semidirect
14. #17988990C22semidirect
15. #17988990C22semidirect
18. #17988983C22semidirect
19. #17988983C22semidirect
35. #17988339C22semidirect
36. #17988337C22semidirect
37. #17988336C22semidirect
38. #17988335C22semidirect
39. #17988334C22semidirect
40. #17988333C22semidirect
119. C2×#10596831SD16semidirect
239. C73#19137semidirect

Maximal subgroups

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

subgroupindex
#226204412
#226199392
C2×#179886652
#226094522
#226094422
#226094322
#226094282
#226086772
#226085882
#226085862
C73×#31012
#1913773