Finite Group Databasebeta

← #32015729

Subgroups — #32015729

Order 18176.

1122 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986109C212yesmaximal
2C2×C4×#11765850C211yesmaximal, characteristic
3C4⋊C4×D284C211yesmaximal, characteristic, complemented
4#22589075C212yesmaximal, complemented
5#22588601C212yesmaximal, complemented
6#22588600C212yesmaximal, complemented
7#22588496C211yesmaximal, characteristic, complemented
8#22588340C211yesmaximal, characteristic, complemented
9#22588337C211yesmaximal, characteristic, complemented
10#22588209C211yesmaximal, characteristic, complemented
11D4×C4⋊C4×C71C211yesmaximal, characteristic, complemented
12C22×#11765850C2211yescharacteristic
13C2×C4×D284C2212yes—
14C2×C4×D284C2211yescharacteristic
15C2×C4⋊C4×D71—21no—
16#17986109—28no—
17C4×#11765850C418yescomplemented
18C2×#14556061C2211yescharacteristic
19C2×#14556059C2212yes—
20C2×#14556059C2212yes—
21C2×C4⋊(C71⋊C8)C2211yescharacteristic
22C2×#14556049C2212yes—
23C2×#14556049C2211yescharacteristic
24C2×#14556048C2212yes—
25C2×#14556048C2211yescharacteristic
26C2×#14556039C2212yes—
27C2×C4×C71⋊C8C2211yescharacteristic
28#17985897C2212yes—
29#17985873—22no—
30#17985869C2211yescharacteristic
31#17985856—21no—
32#17985761C2212yes—
33#17985761C2212yes—
34#17985754C2212yes—
35C2×C4×D4×C71C2212yes—
36C2×C4×D4×C71C2211yescharacteristic
37C22×C4⋊C4×C71—21no—
38#389×C71C2212yes—
39#385×C71—22no—
40#384×C71C2211yescharacteristic
41#380×C71—21no—
42C4×C4⋊C4×C71C2211yescharacteristic
43C22×D284C2311yescharacteristic
44C22×C4×D71—21no—
45C22×C4×D71—22no—
46C2×#11765850C2×C414yes—
47C2×#11765850D414yes—
48C2×#11765850Q814yes—
49C22×C71⋊C8C2312yes—
50C22×C71⋊C8C2311yescharacteristic
51C22×C71⋊C8C2311yescharacteristic
52C4×D284—24no—
53C4×D284—22no—
54C4×D284—42no—
55C4×D284C2×C414yescomplemented
56C2×C4⋊Dic71—21no—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71C2312yes—
59C2×C4⋊Dic71C2311yescharacteristic
60C2×Dic71⋊C4—22no—
61C4⋊C4×D71—44no—
62C2×(C2×D71)⋊C4—21no—
63C2×(C2×D71)⋊C4—22no—
64#14556061C2×C414yescomplemented
65#14556059—24no—
66#14556059—42no—
67C4⋊(C71⋊C8)—22no—
68#14556049—24no—
69#14556049C2×C414yescomplemented
70#14556048—24no—
71#14556048C2×C414yescomplemented
72#14556039—24no—
73C4×C71⋊C8C2×C414yescomplemented
74C71⋊#95—22no—
75C22×D4×C71C2311yescharacteristic
76C23×C4×C71—21no—
77C23×C4×C71—22no—
78C4×D4×C71—24no—
79C4×D4×C71—22no—
80C4×D4×C71—42no—
81C4×D4×C71C2×C414yescomplemented
82C2×C4⋊C4×C71—21no—
83C2×C4⋊C4×C71—22no—
84C2×C4⋊C4×C71—21no—
85C2×C4⋊C4×C71—44no—
86C2×C4⋊C4×C71D411yescharacteristic
87C2×C4⋊C4×C71D411yescharacteristic
88C2×C4⋊C4×C71C2312yes—
89C2×C4⋊C4×C71C2311yescharacteristic
90C2×C22⋊C4×C71—21no—
91C2×C22⋊C4×C71—22no—
92C2×C42×C71C2312yes—
93C2×C42×C71C2311yescharacteristic
94#95×C71—22no—
95#95×C71D412yes—
96C23×D71—21no—
97C2×D284—21no—
98C2×D284—21no—
99C2×D284—21no—
100C2×D284C22×C412yes—
101C2×D284C2×D412yes—
102C2×D284C2×Q812yes—
103C22×Dic71—21no—
104C22×Dic71—22no—
105C2×C4×D71—24no—
106C2×C4×D71—42no—
107C2×C4×D71—42no—
108C2×C4×D71—44no—
109C2×C4×D71—44no—
110C2×C4×D71—82no—
111#11765850C4⋊C4116yescomplemented
112C2×C71⋊C8—21no—
113C2×C71⋊C8—21no—
114C2×C71⋊C8—22no—
115C2×C71⋊C8—22no—
116C2×C71⋊C8—22no—
117C2×C71⋊C8C22×C412yes—
118C2×C71⋊C8C22×C412yes—
119C2×C71⋊C8C2×D412yes—
120C2×C71⋊C8C2×Q812yes—
121C4⋊Dic71—21no—
122C4⋊Dic71—22no—
123C4⋊Dic71—22no—
124C4⋊Dic71—42no—
125C4⋊Dic71—81no—
126C4⋊Dic71C22×C412yes—
127Dic71⋊C4—44no—
128(C2×D71)⋊C4—24no—
129(C2×D71)⋊C4—44no—
130C24×C71—21no—
131C2×D4×C71—21no—
132C2×D4×C71—21no—
133C2×D4×C71—21no—
134C2×D4×C71C22×C412yes—
135C2×D4×C71C2×D412yes—
136C2×D4×C71C2×Q812yes—
137C22×C4×C71—21no—
138C22×C4×C71—24no—
139C22×C4×C71—22no—
140C22×C4×C71—44no—
141C22×C4×C71—44no—
142C22×C4×C71—42no—
143C22×C4×C71—42no—
144C22×C4×C71—82no—
145C22×C4×C71C2×D412yes—
146C22×C4×C71C2×D412yes—
147C22×C4×C71C2×D411yescharacteristic
148C22×C4×C71C2×D411yescharacteristic
149C22×C4×C71C2411yescharacteristic
150C4⋊C4×C71—22no—
151C4⋊C4×C71—22no—
152C4⋊C4×C71—21no—
153C4⋊C4×C71—22no—
154C4⋊C4×C71—22no—
155C4⋊C4×C71—44no—
156C4⋊C4×C71—42no—
157C4⋊C4×C71—81no—
158C4⋊C4×C71C22×C412yes—
159C22⋊C4×C71—24no—
160C22⋊C4×C71—44no—
161C42×C71—22no—
162C42×C71—22no—
163C42×C71C22×C412yes—
164C42×C71#4112yes—
165C23×C71—22no—
166C23×C71—22no—
167C23×C71—22no—
168C23×C71—41no—
169C23×C71—41no—
170C23×C71—41no—
171C23×C71—41no—
172C23×C71C22×D411yescharacteristic
173C22×D71—22no—
174C22×D71—22no—
175C22×D71—22no—
176C22×D71—41no—
177C22×D71—41no—
178C22×D71—41no—
179C22×D71—41no—
180D4×C71—22no—
181D4×C71—22no—
182D4×C71—22no—
183D4×C71C2×C4⋊C414yescomplemented
184C2×C4×C71—21no—
185C2×C4×C71—21no—
186C2×C4×C71—22no—
187C2×C4×C71—22no—
188C2×C4×C71—22no—
189C2×C4×C71—22no—
190C2×C4×C71—22no—
191C2×C4×C71—22no—
192C2×C4×C71—22no—
193C2×C4×C71—41no—
194C2×C4×C71—41no—
195C2×C4×C71—48no—
196C2×C4×C71—42no—
197C2×C4×C71—42no—
198C2×C4×C71—42no—
199C2×C4×C71—88no—
200C2×C4×C71C4×D412yes—
201C2×C4×C71C4×D412yes—
202C2×C4×C71AGL(1,ℤ/8)12yes—
203C2×C4×C71AGL(1,ℤ/8)12yes—
204C2×C4×C71C23×C411yescharacteristic
205C2×C4×C71C22×D411yescharacteristic
206C2×C4×C71C22×Q811yescharacteristic
207C2×C4×C71C2×#4111yescharacteristic
208C2×C4×C71H(5,𝔽2)11yescharacteristic
209C2×C4×C712−1+411yescharacteristic
210C2×Dic71—22no—
211C2×Dic71—41no—
212C2×Dic71—41no—
213C2×Dic71—42no—
214C2×Dic71—42no—
215C2×Dic71—42no—
216D284—22no—
217D284—22no—
218D284—22no—
219D284C2×C4⋊C414yescomplemented
220C4×D71—48no—
221C4×D71—88no—
222C71⋊C8—22no—
223C71⋊C8—42no—
224C71⋊C8C2×C4⋊C414yescomplemented
225C22×C71—24no—
226C22×C71—42no—
227C22×C71—42no—
228C22×C71—42no—
229C22×C71—42no—
230C22×C71—42no—
231C22×C71—42no—
232C22×C71C2×C4×D411yescharacteristic
233C22×C71D4211yescharacteristic
234C22×C71D4×Q811yescharacteristic
235C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
236C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C71#57711yescharacteristic
238C22×C71#57711yescharacteristic
239C2×D71—24no—
240C2×D71—42no—
241C2×D71—42no—
242C2×D71—42no—
243C2×D71—42no—
244C2×D71—42no—
245C2×D71—42no—
246C4×C71—22no—
247C4×C71—42no—
248C4×C71—42no—
249C4×C71—42no—
250C4×C71—82no—
251C4×C71#44214yescomplemented
252C4×C71C22×C4⋊C411yescharacteristic
253C4×C71#52011yescharacteristic
254C4×C71#54711yescharacteristic
255C4×C71#55411yescharacteristic
256Dic71—42no—
257Dic71—82no—
258#22670—711nomaximal, Sylow
259C2×C71—44no—
260C2×C71D4×C4⋊C411yescharacteristic
261C2×C71C2×#44211yescharacteristic
262C2×C71#295211yescharacteristic
263C2×C71#327211yescharacteristic
264C2×C71#327211yescharacteristic
265C2×C71#338411yescharacteristic
266C2×C71#338411yescharacteristic
267D71—44no—
268C2×#442—712no—
269C2×C4×D8—711no—
270D4×C4⋊C4—711no—
271D4×C4⋊C4D7111yescharacteristic, complemented
272#1943—712no—
273#1864—712no—
274#1864—712no—
275#1844—711no—
276#1802—711no—
277#1802—711no—
278#1770—711no—
279C71#2267011yescharacteristic, complemented, Sylow
280C22×D8—711no—
281C2×C4×D4—712no—
282C2×C4×D4—711no—
283C2×C4×D4C2×D7112yes—
284C2×C4×D4C2×D7111yescharacteristic
285C22×C4⋊C4—21no—
286C22×C4⋊C4—1421no—
287#442—1428no—
288C4×D8—718no—
289C2×#107—711no—
290C2×#106—712no—
291C2×#106—712no—
292C2×C4⋊C8—711no—
293C2×#102—711no—
294C2×#102—711no—
295C2×#102—712no—
296C2×#102—712no—
297C2×#97—712no—
298C2×C4×C8—711no—
299#389—712no—
300#389C2×D7112yes—
301#385—22no—
302#385—1422no—
303#384—711no—
304#384C2×D7111yescharacteristic
305#380—21no—
306#380—1421no—
307C4×C4⋊C4C2×D7111yescharacteristic
308#340—712no—
309#340—712no—
310#336—712no—
311C22×D4—711no—
312C22×D4C22×D7111yescharacteristic
313C23×C4—21no—
314C23×C4—22no—
315C23×C4—1422no—
316C23×C4—1421no—
317C2×D8—714no—
318C2×D8—714no—
319C2×D8—714no—
320C22×C8—711no—
321C22×C8—711no—
322C22×C8—712no—
323C4×D4—24no—
324C4×D4—22no—
325C4×D4—42no—
326C4×D4—2842no—
327C4×D4—714no—
328C4×D4—1424no—
329C4×D4—1422no—
330C4×D4C4×D7114yescomplemented
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—2844no—
336C2×C4⋊C4—711no—
337C2×C4⋊C4—712no—
338C2×C4⋊C4—1421no—
339C2×C4⋊C4—1422no—
340C2×C4⋊C4—1421no—
341C2×C4⋊C4C22⋊D7111yescharacteristic
342C2×C4⋊C4C22⋊D7111yescharacteristic
343C2×C4⋊C4C22×D7112yes—
344C2×C4⋊C4C22×D7111yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—1422no—
348C2×C22⋊C4—1421no—
349C2×C42C22×D7112yes—
350C2×C42C22×D7111yescharacteristic
351#107—714no—
352#106—2842no—
353#106—1424no—
354C4⋊C8—1422no—
355#102—714no—
356#102—714no—
357#102—1424no—
358#102—1424no—
359#97—1424no—
360C4×C8—714no—
361#95—22no—
362#95—1422no—
363#95C22⋊D7112yes—
364C24—21no—
365C24—1421no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—712no—
370C2×D4—712no—
371C2×D4—712no—
372C2×D4—1421no—
373C2×D4—1421no—
374C2×D4—1421no—
375C2×D4C2×C4×D7112yes—
376C2×D4D4×D7112yes—
377C2×D4Q8×D7112yes—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—82no—
386C22×C4—2844no—
387C22×C4—2844no—
388C22×C4—2842no—
389C22×C4—2842no—
390C22×C4—5682no—
391C22×C4—1422no—
392C22×C4—1421no—
393C22×C4—1424no—
394C22×C4C2×C22⋊D7112yes—
395C22×C4C2×C22⋊D7112yes—
396C22×C4C2×C22⋊D7111yescharacteristic
397C22×C4C2×C22⋊D7111yescharacteristic
398C22×C4C23×D7111yescharacteristic
399D8—7116no—
400C2×C8—712no—
401C2×C8—712no—
402C2×C8—712no—
403C2×C8—712no—
404C2×C8—1422no—
405C2×C8—1422no—
406C2×C8—1422no—
407C2×C8—1421no—
408C2×C8—1421no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—2844no—
418C4⋊C4—2842no—
419C4⋊C4—5681no—
420C4⋊C4—712no—
421C4⋊C4—1422no—
422C4⋊C4—1422no—
423C4⋊C4—1421no—
424C4⋊C4C2×C4×D7112yes—
425C22⋊C4—24no—
426C22⋊C4—44no—
427C22⋊C4—2844no—
428C22⋊C4—1424no—
429C42—22no—
430C42—22no—
431C42C2×C4×D7112yes—
432C42#1176586512yes—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—2841no—
441C23—2841no—
442C23—2841no—
443C23—2841no—
444C23—1422no—
445C23—1422no—
446C23—1422no—
447C23C22×C22⋊D7111yescharacteristic
448D4—22no—
449D4—22no—
450D4—22no—
451D4—714no—
452D4—1422no—
453D4—1422no—
454D4—1422no—
455D4C4⋊C4×D7114yescomplemented
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—48no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—88no—
472C2×C4—2841no—
473C2×C4—2841no—
474C2×C4—2848no—
475C2×C4—2842no—
476C2×C4—2842no—
477C2×C4—2842no—
478C2×C4—5688no—
479C2×C4—1422no—
480C2×C4C4×C22⋊D7112yes—
481C2×C4C4×C22⋊D7112yes—
482C2×C4#1455615312yes—
483C2×C4#1455615412yes—
484C2×C4C22×C4×D7111yescharacteristic
485C2×C4C2×D4×D7111yescharacteristic
486C2×C4C2×Q8×D7111yescharacteristic
487C2×C4C2×#1176586511yescharacteristic
488C2×C4#1455617311yescharacteristic
489C2×C4#1455617611yescharacteristic
490C8—2842no—
491C8—714no—
492C8—1422no—
493C22—24no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—2842no—
501C22—2842no—
502C22—2842no—
503C22—2842no—
504C22—2842no—
505C22—2842no—
506C22—1424no—
507C22C2×C4×C22⋊D7111yescharacteristic
508C22D4×C22⋊D7111yescharacteristic
509C22Q8×C22⋊D7111yescharacteristic
510C22C2×#1455615311yescharacteristic
511C22C2×#1455615411yescharacteristic
512C22#1798680711yescharacteristic
513C22#1798681011yescharacteristic
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—82no—
519C4—2842no—
520C4—5682no—
521C4#1798610914yescomplemented
522C4C2×C4⋊C4×D7111yescharacteristic
523C4#1798648011yescharacteristic
524C4#1798665011yescharacteristic
525C4#1798670411yescharacteristic
526C2—44no—
527C2—2844no—
528C2C4⋊C4×C22⋊D7111yescharacteristic
529C2C2×#1798610911yescharacteristic
530C2#2259722111yescharacteristic
531C2#2259993611yescharacteristic
532C2#2259994011yescharacteristic
533C2#2260123511yescharacteristic
534C2#2260123911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingD4×C4⋊C4×C71C2
socleC23×C71C22×D4
prime radicalC22×#11765850C22
cosocleC22×C4×C71C24
hypercenterD4×C4⋊C4D71
hypocenterC71#22670
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×D284C2semidirect
4. #22589075C2semidirect
5. #22588601C2semidirect
6. #22588600C2semidirect
7. #22588496C2semidirect
8. #22588340C2semidirect
9. #22588337C2semidirect
10. #22588209C2semidirect
11. D4×C4⋊C4×C71C2semidirect
17. C4×#11765850C4semidirect
55. C4×D284C2×C4semidirect
64. #14556061C2×C4semidirect
69. #14556049C2×C4semidirect
71. #14556048C2×C4semidirect
73. C4×C71⋊C8C2×C4semidirect
81. C4×D4×C71C2×C4semidirect
111. #11765850C4⋊C4semidirect
183. D4×C71C2×C4⋊C4semidirect
219. D284C2×C4⋊C4semidirect
224. C71⋊C8C2×C4⋊C4semidirect
251. C4×C71#442semidirect
271. D4×C4⋊C4D71semidirect
279. C71#22670semidirect
330. C4×D4C4×D71semidirect
455. D4C4⋊C4×D71semidirect
521. C4#17986109semidirect

Maximal subgroups

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

subgroupindex
C2×#179861092
C2×C4×#117658502
C4⋊C4×D2842
#225890752
#225886012
#225886002
#225884962
#225883402
#225883372
#225882092
D4×C4⋊C4×C712
#2267071