Finite Group Databasebeta

← #33240251

Subgroups — #33240251

Order 18688.

718 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989017C212yesmaximal, complemented
2#22620359C211yesmaximal, characteristic, complemented
3#22611993C212yesmaximal, complemented
4#22611986C212yesmaximal, complemented
5#22611981C211yesmaximal, characteristic, complemented
6#22611976C211yesmaximal, characteristic, complemented
7#22611971C211yesmaximal, characteristic, complemented
8#22611966C211yesmaximal, characteristic, complemented
9C2×C4⋊#11924003C211yesmaximal, characteristic, complemented
10#22611636C212yesmaximal, complemented
11C4×#135×C73C211yesmaximal, characteristic, complemented
12C2×#14556613—21no—
13C2×#14556613C2212yes—
14#17989137—21no—
15#17989017—28no—
16C2×#14556537C2212yes—
17C2×#14556535C2212yes—
18C2×#14556534C2212yes—
19C2×#14556532C2212yes—
20C22×#11924003C2212yes—
21#17988971—21no—
22#17988971C2212yes—
23#17988967C2211yescharacteristic
24#17988964C2211yescharacteristic
25#17988960—21no—
26C4⋊#11924003C2211yescharacteristic
27C4⋊#11924003C2211yescharacteristic
28C4⋊#11924003C2212yes—
29#17988390C2212yes—
30#17988390C2212yes—
31#17988387C2212yes—
32#17988387C2212yes—
33C2×#135×C73C2211yescharacteristic
34C4×#41×C73—21no—
35C4×Q16×C73C2211yescharacteristic
36C4×Q16×C73C2211yescharacteristic
37C4×SD16×C73—21no—
38C4×SD16×C73—21no—
39C4×D8×C73C2211yescharacteristic
40C4×D8×C73C2211yescharacteristic
41#427×C73C2211yescharacteristic
42#416×C73—21no—
43C2×C4×C8×C73C2211yescharacteristic
44C2×#11924028—22no—
45#14556613—24no—
46#14556613—24no—
47#14556613—44no—
48C2×Dic292—21no—
49C2×Dic292C2312yes—
50C2×#11924017—22no—
51C2×#11924017—22no—
52C2×D584—21no—
53C2×D584C2312yes—
54#14556591—21no—
55#14556588—21no—
56#14556579—22no—
57#14556565—41no—
58#14556565—41no—
59C4×D292—41no—
60C4×D292—41no—
61#14556537—24no—
62#14556535—24no—
63#14556534—24no—
64#14556532—24no—
65C2×#11924003—24no—
66C2×#11924003C2312yes—
67C2×#11924003C2312yes—
68C2×#41×C73—21no—
69#135×C73—22no—
70#135×C73D414yescomplemented
71C2×Q16×C73C2311yescharacteristic
72C2×SD16×C73—21no—
73C2×D8×C73C2311yescharacteristic
74C22×C8×C73D411yescharacteristic
75C22×C8×C73C2311yescharacteristic
76C4×Q8×C73—21no—
77C4×Q8×C73—21no—
78C4×D4×C73—21no—
79C4×D4×C73—21no—
80C4×D4×C73—41no—
81C4×D4×C73—41no—
82#117×C73—21no—
83#117×C73—41no—
84C2×C42×C73—41no—
85C2×C42×C73D411yescharacteristic
86#107×C73C2311yescharacteristic
87#107×C73C2311yescharacteristic
88#106×C73—21no—
89#103×C73—21no—
90#103×C73—21no—
91#102×C73—21no—
92#102×C73—21no—
93C4×C8×C73D411yescharacteristic
94C4×C8×C73D411yescharacteristic
95C4×C8×C73C2311yescharacteristic
96C4×C8×C73C2311yescharacteristic
97#11924028—44no—
98#11924028—44no—
99C2×Dic146—22no—
100C2×D292—22no—
101C2×D292—41no—
102C2×D292—41no—
103C2×C22⋊D73—42no—
104C2×C4×D73—42no—
105Dic292—22no—
106Dic292—22no—
107Dic292—42no—
108#11924017—44no—
109#11924017—44no—
110D584—22no—
111D584—22no—
112D584—42no—
113C4⋊Dic73—41no—
114C4⋊Dic73—41no—
115#11924005—42no—
116#11924005—42no—
117#11924003—22no—
118#11924003—22no—
119#41×C73—24no—
120#41×C73—42no—
121C2×Q8×C73—21no—
122C2×D4×C73—21no—
123C2×D4×C73—41no—
124C22×C4×C73—41no—
125C22×C4×C73—41no—
126C22×C4×C73—41no—
127C22×C4×C73D811yescharacteristic
128C22×C4×C73D811yescharacteristic
129C22×C4×C73C2×D411yescharacteristic
130Q16×C73—21no—
131Q16×C73C2×D412yes—
132SD16×C73—24no—
133D8×C73—21no—
134D8×C73C2×D412yes—
135C2×C8×C73—22no—
136C2×C8×C73C2×D411yescharacteristic
137C2×C8×C73C2×D411yescharacteristic
138C2×C8×C73C2×D411yescharacteristic
139C2×C8×C73C2×D412yes—
140C2×C8×C73#4112yes—
141C2×C8×C73C2411yescharacteristic
142C4⋊C4×C73—21no—
143C4⋊C4×C73—21no—
144C4⋊C4×C73—41no—
145C4⋊C4×C73—41no—
146C22⋊C4×C73—41no—
147C22⋊C4×C73—41no—
148C22⋊C4×C73—41no—
149C42×C73—41no—
150C42×C73—41no—
151C42×C73—41no—
152C42×C73D811yescharacteristic
153C42×C73D811yescharacteristic
154C42×C73C2×D411yescharacteristic
155C42×C73C2×D411yescharacteristic
156C23×C73—41no—
157C23×C73C2×D811yescharacteristic
158C22×D73—42no—
159Q8×C73—22no—
160Q8×C73—41no—
161D4×C73—22no—
162D4×C73—44no—
163D4×C73—41no—
164C2×C4×C73—22no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—44no—
168C2×C4×C73—41no—
169C2×C4×C73—41no—
170C2×C4×C73—82no—
171C2×C4×C73—82no—
172C2×C4×C73#12112yes—
173C2×C4×C73C2×D811yescharacteristic
174C2×C4×C73C2×D811yescharacteristic
175C2×C4×C73C2×D811yescharacteristic
176C2×C4×C73C2×D811yescharacteristic
177C2×C4×C73C2×D811yescharacteristic
178C2×C4×C73AGL(1,ℤ/8)12yes—
179C2×C4×C73C22×D411yescharacteristic
180C22⋊D73—84no—
181C2×Dic73—42no—
182D292—42no—
183D292—42no—
184D292—82no—
185D292—82no—
186C4×D73—84no—
187Dic146—42no—
188Dic146—42no—
189C8×C73—21no—
190C8×C73—21no—
191C8×C73#12112yes—
192C8×C73C22×D411yescharacteristic
193C8×C73C2×#4111yescharacteristic
194C22×C73—21no—
195C22×C73—21no—
196C22×C73—42no—
197C22×C73—81no—
198C22×C73—81no—
199C22×C73#45912yes—
200C22×C73C22×D811yescharacteristic
201C2×D73—82no—
202C2×D73—82no—
203C2×D73—82no—
204C4×C73—21no—
205C4×C73—21no—
206C4×C73—21no—
207C4×C73—42no—
208C4×C73—81no—
209C4×C73—81no—
210C4×C73#45912yes—
211C4×C73#50812yescomplemented
212C4×C73C2×#12111yescharacteristic
213C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
214Dic73—82no—
215#33579—731nomaximal, Sylow
216C2×C73—22no—
217C2×C73—82no—
218C2×C73C2×#45911yescharacteristic
219C2×C73C2×#50811yescharacteristic
220C2×C73#341411yescharacteristic
221D73—162no—
222C2×#508—732no—
223#3147—731no—
224C4×#135D7311yescharacteristic, complemented
225#2211—732no—
226#2210—732no—
227#2209—731no—
228#2208—731no—
229#2207—731no—
230#2206—731no—
231C2×C4⋊C16—731no—
232#2138—732no—
233C73#3357911yescharacteristic, complemented, Sylow
234C2×#135—732no—
235C2×#135—1461no—
236C2×#135C2×D7311yescharacteristic
237#532—1461no—
238C4×#41—21no—
239#508—1468no—
240C2×Q32—732no—
241C2×SD32—732no—
242C2×SD32—732no—
243C2×D16—732no—
244C22×C16—732no—
245#495—732no—
246#495—1461no—
247#494—731no—
248#493—731no—
249#492—1461no—
250C4×Q16C2×D7311yescharacteristic
251C4×Q16C2×D7311yescharacteristic
252C4×SD16—21no—
253C4×SD16—21no—
254C4×D8C2×D7311yescharacteristic
255C4×D8C2×D7311yescharacteristic
256#427C2×D7311yescharacteristic
257#416—21no—
258C2×C4×C8C2×D7311yescharacteristic
259C4⋊C16—731no—
260C4⋊C16—731no—
261C4⋊C16—732no—
262#358—732no—
263#358—732no—
264#357—732no—
265#357—732no—
266C2×#41—21no—
267C2×#41—1462no—
268#135—22no—
269#135—2924no—
270#135—1464no—
271#135—1464no—
272#135D29214yescomplemented
273C2×Q16—732no—
274C2×Q16—1461no—
275C2×Q16C22×D7311yescharacteristic
276C2×SD16—21no—
277C2×SD16—1462no—
278C2×SD16—1462no—
279C2×D8—732no—
280C2×D8—1461no—
281C2×D8C22×D7311yescharacteristic
282C22×C8C22⋊D7311yescharacteristic
283C22×C8C22×D7311yescharacteristic
284C4⋊Q8—1461no—
285#127—1461no—
286#124—1462no—
287#121—2921no—
288#121—2921no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—41no—
294C4×D4—41no—
295C4×D4—2921no—
296C4×D4—2921no—
297#117—21no—
298#117—41no—
299C2×C42—41no—
300C2×C42C22⋊D7311yescharacteristic
301Q32—1464no—
302SD32—1464no—
303SD32—1464no—
304D16—1464no—
305C2×C16—732no—
306C2×C16—732no—
307C2×C16—1464no—
308#107C22×D7311yescharacteristic
309#107C22×D7311yescharacteristic
310#106—21no—
311#103—21no—
312#103—21no—
313#102—21no—
314#102—21no—
315C4×C8C22⋊D7311yescharacteristic
316C4×C8C22⋊D7311yescharacteristic
317C4×C8C22×D7311yescharacteristic
318C4×C8C22×D7311yescharacteristic
319#41—24no—
320#41—42no—
321#41—2924no—
322#41—2924no—
323C2×Q8—21no—
324C2×Q8—1462no—
325C2×D4—21no—
326C2×D4—41no—
327C2×D4—2922no—
328C2×D4—2921no—
329C2×D4—2921no—
330C2×D4—1462no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—2922no—
335C22×C4#1192401311yescharacteristic
336C22×C4#1192401311yescharacteristic
337C22×C4C2×C22⋊D7311yescharacteristic
338Q16—21no—
339Q16—2922no—
340Q16—1462no—
341Q16—1462no—
342Q16C2×D29212yes—
343SD16—24no—
344SD16—2924no—
345SD16—2924no—
346D8—21no—
347D8—2922no—
348D8—1462no—
349D8—1462no—
350D8C2×D29212yes—
351C2×C8—22no—
352C2×C8C2×C22⋊D7311yescharacteristic
353C2×C8C2×C22⋊D7311yescharacteristic
354C2×C8C2×C22⋊D7311yescharacteristic
355C2×C8C2×D29212yes—
356C2×C8#1192402812yes—
357C2×C8C23×D7311yescharacteristic
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—2921no—
363C4⋊C4—2921no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—2922no—
368C22⋊C4—2922no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42#1192401311yescharacteristic
373C42#1192401311yescharacteristic
374C42C2×C22⋊D7311yescharacteristic
375C42C2×C22⋊D7311yescharacteristic
376C16—1462no—
377C16—1462no—
378C23—41no—
379C23—2922no—
380C23C2×#1192401311yescharacteristic
381Q8—22no—
382Q8—41no—
383Q8—2922no—
384Q8—2922no—
385D4—22no—
386D4—44no—
387D4—41no—
388D4—2922no—
389D4—2922no—
390D4—5842no—
391D4—5844no—
392D4—5842no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—2922no—
402C2×C4—5844no—
403C2×C4#1455656512yes—
404C2×C4C2×#1192401311yescharacteristic
405C2×C4C2×#1192401311yescharacteristic
406C2×C4C2×#1192401311yescharacteristic
407C2×C4C2×#1192401311yescharacteristic
408C2×C4C2×#1192401311yescharacteristic
409C2×C4#1455661612yes—
410C2×C4C22×C22⋊D7311yescharacteristic
411C8—21no—
412C8—21no—
413C8#1455656512yes—
414C8C22×D29211yescharacteristic
415C8C2×#1192402811yescharacteristic
416C22—21no—
417C22—21no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—5842no—
422C22—5842no—
423C22—5842no—
424C22#1798878212yes—
425C22C22×#1192401311yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—81no—
431C4—81no—
432C4—5842no—
433C4#1798878212yes—
434C4#1798901712yescomplemented
435C4C2×#1455656511yescharacteristic
436C4C2×#1455661611yescharacteristic
437C2—22no—
438C2—82no—
439C2—11682no—
440C2C2×#1798878211yescharacteristic
441C2C2×#1798901711yescharacteristic
442C2#2262306511yescharacteristic
subgroupG/H
centerC2×C4C2×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC4×#135×C73C2
socleC22×C73C22×D8
hypercenterC4×#135D73
hypocenterC73#33579
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. C2×#17989017C2semidirect
2. #22620359C2semidirect
3. #22611993C2semidirect
4. #22611986C2semidirect
5. #22611981C2semidirect
6. #22611976C2semidirect
7. #22611971C2semidirect
8. #22611966C2semidirect
9. C2×C4⋊#11924003C2semidirect
10. #22611636C2semidirect
11. C4×#135×C73C2semidirect
70. #135×C73D4semidirect
211. C4×C73#508semidirect
224. C4×#135D73semidirect
233. C73#33579semidirect
272. #135D292semidirect
434. C4#17989017semidirect

Maximal subgroups

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

subgroupindex
C2×#179890172
#226203592
#226119932
#226119862
#226119812
#226119762
#226119712
#226119662
C2×C4⋊#119240032
#226116362
C4×#135×C732
#3357973