Finite Group Databasebeta

← #33068880

Subgroups — #33068880

Order 18688.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620305C211yesmaximal, characteristic
2#22619970C211yesmaximal, characteristic, complemented
3Q8×#11924012C211yesmaximal, characteristic, complemented
4#22609497C212yesmaximal, complemented
5#22609469C212yesmaximal, complemented
6#22609373C212yesmaximal, complemented
7#22609345C212yesmaximal, complemented
8#22608681C211yesmaximal, characteristic, complemented
9#22608619C211yesmaximal, characteristic, complemented
10#22608599C211yesmaximal, characteristic
11C2×#460×C73C211yesmaximal, characteristic
12#17989144C2211yescharacteristic
13#17988951—22no—
14#17988933C2212yes—
15#17988933C2212yes—
16#17988922—21no—
17#17988882C2211yescharacteristic, complemented
18#17988882C2211yescharacteristic, complemented
19#17988797C2211yescharacteristic, complemented
20#17988797C2211yescharacteristic, complemented
21#17988743—21no—
22#17988718—22no—
23#17988718C2212yescomplemented
24Q8×#10596831C2212yescomplemented
25C4×#11924018—21no—
26C4×#11924018—21no—
27#17988640C2211yescharacteristic
28#17988611—21no—
29#17988611C2211yescharacteristic
30#17988592—21no—
31#17988592—21no—
32C4×#11924012—21no—
33C4×#11924012C2211yescharacteristic
34#17988339C2212yescomplemented
35#17988337C2212yescomplemented
36#17988335C2212yescomplemented
37#17988334C2212yescomplemented
38#17988331C2212yes—
39#17988331C2212yes—
40C22×SD16×C73—21no—
41C2×#127×C73C2211yescharacteristic
42C2×C4×Q8×C73C2211yescharacteristic
43#460×C73—21no—
44#460×C73—21no—
45#460×C73C2212yes—
46#460×C73C2212yes—
47C2×C4⋊C8×C73C2211yescharacteristic
48C2×#102×C73—21no—
49C2×#11924018—21no—
50C2×#11924018—21no—
51C2×#11924012—21no—
52C2×#11924012D411yescharacteristic
53C2×#11924012D411yescharacteristic
54#14556591C2311yescharacteristic
55#14556591C2311yescharacteristic
56#14556583—21no—
57#14556570—21no—
58#14556570—21no—
59C4×Dic146—21no—
60C4×Dic146—21no—
61#14556550—21no—
62#14556524—21no—
63#14556524—21no—
64C4⋊#10596831—22no—
65C4⋊#10596831—22no—
66C4⋊#10596831—21no—
67C4⋊#10596831C2312yes—
68C4⋊#10596831C2311yescharacteristic
69C4⋊#10596831C2311yescharacteristic
70#14556515—21no—
71#14556515—21no—
72#14556514—21no—
73#14556514—21no—
74#14556512—22no—
75#14556512—22no—
76#14556512—22no—
77#14556512—22no—
78C22⋊#10596831—21no—
79#14556502—22no—
80#14556502D412yes—
81C4×#10596831—22no—
82C4×#10596831—21no—
83C4×#10596831C2312yes—
84C22×Q8×C73D411yescharacteristic
85C22×D4×C73—21no—
86C22×D4×C73—21no—
87C22×D4×C73—21no—
88C2×SD16×C73—22no—
89C2×SD16×C73—22no—
90C2×SD16×C73—44no—
91C22×C8×C73—21no—
92#127×C73—21no—
93#127×C73—21no—
94#127×C73—21no—
95#127×C73C2312yes—
96C4×Q8×C73—21no—
97C4×Q8×C73—21no—
98C4×Q8×C73D411yescharacteristic
99C4×Q8×C73D411yescharacteristic
100C4×Q8×C73C2311yescharacteristic
101C4×Q8×C73C2311yescharacteristic
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73D411yescharacteristic
104C2×C42×C73—21no—
105C2×C42×C73C2311yescharacteristic
106C4⋊C8×C73—21no—
107C4⋊C8×C73C2312yes—
108#102×C73—22no—
109#102×C73—22no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112#11924018—42no—
113#11924018—42no—
114#11924012—22no—
115#11924012—22no—
116#11924012—42no—
117C2×#10596831—22no—
118C2×#10596831—21no—
119C2×#10596831—21no—
120C2×#10596831C2×D412yes—
121C2×#10596831C2×D412yes—
122C4⋊Dic73—21no—
123C4⋊Dic73—21no—
124C4⋊Dic73—21no—
125C4⋊Dic73—21no—
126#11924007—41no—
127#11924007—41no—
128C22⋊Dic73—41no—
129C4×Dic73—41no—
130C24×C73—41no—
131C2×Q8×C73—21no—
132C2×Q8×C73—22no—
133C2×Q8×C73—41no—
134C2×Q8×C73C2×D411yescharacteristic
135C2×Q8×C73C2×D411yescharacteristic
136C2×D4×C73—22no—
137C2×D4×C73—22no—
138C2×D4×C73—22no—
139C2×D4×C73—42no—
140C2×D4×C73—42no—
141C2×D4×C73—41no—
142C2×D4×C73—41no—
143C2×D4×C73—41no—
144C2×D4×C73—42no—
145C2×D4×C73—42no—
146C2×D4×C73—82no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73C2×D411yescharacteristic
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73#4111yescharacteristic
152SD16×C73—44no—
153SD16×C73—44no—
154C2×C8×C73—22no—
155C2×C8×C73—42no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73—21no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73C2×D411yescharacteristic
162C4⋊C4×C73C2×D411yescharacteristic
163C42×C73—21no—
164C42×C73—21no—
165C42×C73—21no—
166C42×C73—21no—
167C42×C73C2×D411yescharacteristic
168C42×C73C2×D411yescharacteristic
169C42×C73#4111yescharacteristic
170C42×C73C2411yescharacteristic
171C23×C73—42no—
172C23×C73—82no—
173C23×C73—82no—
174C23×C73—82no—
175C23×C73#12111yescharacteristic
176Q8×C73—21no—
177Q8×C73—21no—
178Q8×C73—21no—
179Q8×C73—42no—
180Q8×C73AGL(1,ℤ/8)12yes—
181D4×C73—42no—
182D4×C73—42no—
183D4×C73—42no—
184D4×C73—42no—
185D4×C73—82no—
186D4×C73—82no—
187D4×C73—82no—
188D4×C73—82no—
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—42no—
197C2×C4×C73—81no—
198C2×C4×C73#12111yescharacteristic
199C2×C4×C73#12111yescharacteristic
200C2×C4×C73#12111yescharacteristic
201C2×C4×C73#13512yes—
202C2×C4×C73#13712yes—
203C2×C4×C73C22×D411yescharacteristic
204C2×C4×C73C22×D411yescharacteristic
205C2×C4×C73C2×#4111yescharacteristic
206C2×Dic73—41no—
207C2×Dic73—41no—
208Dic146—41no—
209Dic146—41no—
210Dic146—81no—
211C8×C73—42no—
212#10596831—22no—
213#10596831—22no—
214#10596831—42no—
215#10596831—41no—
216#10596831—41no—
217C22×C73—21no—
218C22×C73—21no—
219C22×C73—82no—
220C22×C73—82no—
221C22×C73—82no—
222C22×C73—164no—
223C22×C73#46412yes—
224C22×C73C2×#12111yescharacteristic
225C4×C73—21no—
226C4×C73—22no—
227C4×C73—21no—
228C4×C73—21no—
229C4×C73—41no—
230C4×C73—41no—
231C4×C73—41no—
232C4×C73#46412yes—
233C4×C73C2×#13511yescharacteristic
234C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
235C4×C73C2×#13711yescharacteristic
236C4×C73#57511yescharacteristic
237Dic73—81no—
238Dic73—81no—
239#19174—731nomaximal, Sylow
240C2×C73—22no—
241C2×C73—162no—
242C2×C73C2×#46411yescharacteristic
243C2×C73#305511yescharacteristic
244C2×C73#306811yescharacteristic
245#3138—731no—
246#3104—731no—
247C2×#460D7311yescharacteristic
248Q8×M16—731no—
249#1702—732no—
250#1698—732no—
251#1682—732no—
252#1678—732no—
253#1565—731no—
254#1550—731no—
255#1545—731no—
256C73#1917411yescharacteristic, complemented, Sylow
257C22×SD16—21no—
258#533—731no—
259C2×#127C2×D7311yescharacteristic
260C2×C4×Q8C2×D7311yescharacteristic
261#490—1462no—
262#486—732no—
263#486—732no—
264#484—1461no—
265#477—731no—
266#477—731no—
267#462—731no—
268#462—731no—
269#460—21no—
270#460—21no—
271#460C2×D7312yes—
272#460C2×D7312yes—
273#451—1461no—
274#446—732no—
275#446—1462no—
276C8×Q8—732no—
277C4×Q16—1461no—
278C4×Q16—1461no—
279#431—731no—
280#423—731no—
281#423—1461no—
282C2×C4⋊C8C2×D7311yescharacteristic
283#418—1461no—
284#418—1461no—
285C2×#102—21no—
286C4×M16—731no—
287C4×M16—1461no—
288#335—732no—
289#334—732no—
290#333—732no—
291#332—732no—
292#331—732no—
293#331—732no—
294C22×Q8C22⋊D7311yescharacteristic
295C22×D4—21no—
296C22×D4—21no—
297C22×D4—21no—
298C2×Q16—1461no—
299C2×Q16—1461no—
300C2×SD16—22no—
301C2×SD16—22no—
302C2×SD16—44no—
303C2×M16—731no—
304C2×M16—731no—
305C2×M16—1461no—
306C22×C8—21no—
307C4⋊Q8—731no—
308C4⋊Q8—731no—
309#127—21no—
310#127—21no—
311#127—21no—
312#127C22×D7312yes—
313#125—1461no—
314C22⋊Q8—1461no—
315C22⋊Q8—1461no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—1461no—
319C4×Q8—1461no—
320C4×Q8C22⋊D7311yescharacteristic
321C4×Q8C22⋊D7311yescharacteristic
322C4×Q8C22×D7311yescharacteristic
323C4×Q8C22×D7311yescharacteristic
324#117—1461no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C22⋊D7311yescharacteristic
327C2×C42—21no—
328C2×C42C22×D7311yescharacteristic
329#107—1461no—
330#107—1461no—
331C4⋊C8—21no—
332C4⋊C8—732no—
333C4⋊C8—731no—
334C4⋊C8—731no—
335C4⋊C8—1462no—
336C4⋊C8—1462no—
337C4⋊C8—1461no—
338C4⋊C8C22×D7312yes—
339#103—1461no—
340#103—1461no—
341#103—1461no—
342#103—1461no—
343#102—22no—
344#102—22no—
345#102—1462no—
346#102—1462no—
347#102—1462no—
348#102—1462no—
349C22⋊C8—1461no—
350#97—732no—
351#97—1462no—
352C4×C8—732no—
353C4×C8—1462no—
354C4×C8—1461no—
355C24—41no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—1461no—
360C2×Q8—1461no—
361C2×Q8C2×C22⋊D7311yescharacteristic
362C2×Q8C2×C22⋊D7311yescharacteristic
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—82no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4D4×D7311yescharacteristic
377C22×C4C2×C22⋊D7311yescharacteristic
378C22×C4#1192402711yescharacteristic
379Q16—2922no—
380Q16—2922no—
381SD16—44no—
382SD16—44no—
383M16—2922no—
384M16—1462no—
385M16—1462no—
386C2×C8—22no—
387C2×C8—42no—
388C2×C8—732no—
389C2×C8—732no—
390C2×C8—1462no—
391C2×C8—1461no—
392C2×C8—1461no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—2921no—
399C4⋊C4—2921no—
400C4⋊C4—1461no—
401C4⋊C4—1461no—
402C4⋊C4—1461no—
403C4⋊C4—1461no—
404C4⋊C4C2×C22⋊D7311yescharacteristic
405C4⋊C4C2×C22⋊D7311yescharacteristic
406C22⋊C4—2921no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—2921no—
412C42D4×D7311yescharacteristic
413C42C2×C22⋊D7311yescharacteristic
414C42#1192402711yescharacteristic
415C42C23×D7311yescharacteristic
416C23—42no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23#1455656311yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—2921no—
426Q8—2921no—
427Q8—5841no—
428Q8#1455661712yes—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—2921no—
447C2×C4—2921no—
448C2×C4#1455656311yescharacteristic
449C2×C4#1455656311yescharacteristic
450C2×C4#1455656311yescharacteristic
451C2×C4#1455661212yes—
452C2×C4#1455662012yes—
453C2×C4C2×D4×D7311yescharacteristic
454C2×C4C22×C22⋊D7311yescharacteristic
455C2×C4C2×#1192402711yescharacteristic
456C8—42no—
457C8—2921no—
458C8—2922no—
459C8—2921no—
460C8—1462no—
461C8—1462no—
462C22—21no—
463C22—21no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22#1798880712yes—
469C22C2×#1455656311yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—5841no—
478C4—5841no—
479C4#1798880712yes—
480C4C2×#1455661211yescharacteristic
481C4C2×#1455661711yescharacteristic
482C4C2×#1455662011yescharacteristic
483C4#1798938511yescharacteristic
484C2—22no—
485C2—162no—
486C2C2×#1798880711yescharacteristic
487C2#2261954611yescharacteristic
488C2#2261965611yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#460×C73C2
socleC22×C73C2×#121
prime radicalC2×#127×C73C22
hypercenterC2×#460D73
hypocenterC73#19174
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #22619970C2semidirect
3. Q8×#11924012C2semidirect
4. #22609497C2semidirect
5. #22609469C2semidirect
6. #22609373C2semidirect
7. #22609345C2semidirect
8. #22608681C2semidirect
9. #22608619C2semidirect
17. #17988882C22semidirect
18. #17988882C22semidirect
19. #17988797C22semidirect
20. #17988797C22semidirect
23. #17988718C22semidirect
24. Q8×#10596831C22semidirect
34. #17988339C22semidirect
35. #17988337C22semidirect
36. #17988335C22semidirect
37. #17988334C22semidirect
256. C73#19174semidirect

Maximal subgroups

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

subgroupindex
#226203052
#226199702
Q8×#119240122
#226094972
#226094692
#226093732
#226093452
#226086812
#226086192
#226085992
C2×#460×C732
#1917473