Finite Group Databasebeta

← #32155416

Subgroups — #32155416

Order 18176.

472 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
1#22601208C211yesmaximal, characteristic, complemented
2#22590627C211yesmaximal, characteristic, complemented
3#22590623C211yesmaximal, characteristic, complemented
4#22590616C211yesmaximal, characteristic, complemented
5#22590497C211yesmaximal, characteristic, complemented
6#22590497C211yesmaximal, characteristic, complemented
7#22590494C211yesmaximal, characteristic, complemented
8#22590494C211yesmaximal, characteristic, complemented
9#22590477C211yesmaximal, characteristic, complemented
10#22590474C211yesmaximal, characteristic, complemented
11#22590249C211yesmaximal, characteristic, complemented
12#22590214C211yesmaximal, characteristic
13#22590214C211yesmaximal, characteristic
14C4×#14556069C211yesmaximal, characteristic
15C71×#3380C211yesmaximal, characteristic, complemented
16#17986709—21no—
17C2×#14556069C2211yescharacteristic
18#17986404C2211yescharacteristic
19#17986401C2211yescharacteristic
20#17986401C2211yescharacteristic
21#17986300—21no—
22#17986300—21no—
23#17986290—21no—
24C4×D568C2211yescharacteristic
25C4×D568C2211yescharacteristic
26C4×D568C2211yescharacteristic
27#17985826C2211yescharacteristic
28#17985826C2211yescharacteristic
29#17985826C2211yescharacteristic
30#17985826C2211yescharacteristic
31#17985824C2211yescharacteristic
32#17985824C2211yescharacteristic
33#17985824C2211yescharacteristic
34#17985824C2211yescharacteristic
35#17985824C2211yescharacteristic
36C4⋊(C71⋊C16)C2211yescharacteristic
37C4⋊(C71⋊C16)C2211yescharacteristic
38C4⋊(C71⋊C16)C2211yescharacteristic
39#17985800C2211yescharacteristic
40#17985800C2211yescharacteristic
41#17985800C2211yescharacteristic
42#17985799C2211yescharacteristic
43#17985799C2211yescharacteristic
44#17985799C2211yescharacteristic
45#17985775C2211yescharacteristic
46#17985775C2211yescharacteristic
47C4×C71⋊C16C2211yescharacteristic
48#554×C71—21no—
49#500×C71C2211yescharacteristic
50#499×C71C2211yescharacteristic
51#499×C71C2211yescharacteristic
52#478×C71—21no—
53#478×C71—21no—
54#476×C71—21no—
55C8×Q8×C71C2211yescharacteristic
56C4×D8×C71C2211yescharacteristic
57C4×D8×C71C2211yescharacteristic
58C4×D8×C71C2211yescharacteristic
59C2×D568C2311yescharacteristic
60#14556128—21no—
61#14556120—21no—
62#14556120—21no—
63#14556106—41no—
64#14556106—41no—
65#14556104—41no—
66C4×D284—21no—
67C4×D284—21no—
68C4×D284—21no—
69C4⋊C4×D71—41no—
70#14556069Q814yescomplemented
71C2×C71⋊C16C2311yescharacteristic
72C2×C71⋊C16C2311yescharacteristic
73C2×C71⋊C16C2311yescharacteristic
74C2×C71⋊C16C2311yescharacteristic
75#14556062—21no—
76#14556062C2311yescharacteristic
77#14556062C2311yescharacteristic
78#14556062C2311yescharacteristic
79#14556060—21no—
80#14556060—21no—
81#14556050—21no—
82#14556050—21no—
83#14556050—21no—
84C2×D8×C71C2311yescharacteristic
85C4⋊Q8×C71—21no—
86#125×C71—21no—
87#125×C71—21no—
88C22⋊Q8×C71—41no—
89C22⋊Q8×C71—41no—
90C22⋊Q8×C71—41no—
91C4×Q8×C71D411yescharacteristic
92C4×D4×C71—21no—
93C4×D4×C71—21no—
94C4×D4×C71—21no—
95C2×C4⋊C4×C71—41no—
96#107×C71—21no—
97#107×C71C2311yescharacteristic
98#107×C71C2311yescharacteristic
99#107×C71C2311yescharacteristic
100#106×C71—21no—
101#106×C71—21no—
102C4⋊C8×C71D411yescharacteristic
103C4⋊C8×C71D411yescharacteristic
104C4⋊C8×C71D411yescharacteristic
105#102×C71—21no—
106#102×C71—21no—
107#102×C71—21no—
108C4×C8×C71C2311yescharacteristic
109C4×C8×C71C2311yescharacteristic
110C4×C8×C71C2311yescharacteristic
111C2×Dic142—41no—
112C2×D284—21no—
113C2×C4×D71—41no—
114C2×C4×D71—41no—
115C2×C4×D71—41no—
116D568—21no—
117D568C2×Q812yes—
118C4⋊Dic71—21no—
119C4⋊Dic71—21no—
120C4⋊Dic71—21no—
121C4⋊Dic71—41no—
122Dic71⋊C4—41no—
123Dic71⋊C4—41no—
124Dic71⋊C4—41no—
125Dic71⋊C4—41no—
126(C2×D71)⋊C4—41no—
127(C2×D71)⋊C4—41no—
128(C2×D71)⋊C4—41no—
129C71⋊C16—21no—
130C71⋊C16—21no—
131C71⋊C16—21no—
132C71⋊C16C2×Q812yes—
133C2×Q8×C71—41no—
134C2×Q8×C71D811yescharacteristic
135C2×D4×C71—21no—
136C22×C4×C71—41no—
137C22×C4×C71—41no—
138C22×C4×C71—41no—
139D8×C71—21no—
140D8×C71C2×Q812yes—
141C2×C8×C71C2×D411yescharacteristic
142C2×C8×C71C2×D411yescharacteristic
143C2×C8×C71C2×D411yescharacteristic
144C2×C8×C71C2411yescharacteristic
145C4⋊C4×C71—21no—
146C4⋊C4×C71—21no—
147C4⋊C4×C71—21no—
148C4⋊C4×C71—41no—
149C4⋊C4×C71—41no—
150C4⋊C4×C71—41no—
151C4⋊C4×C71—41no—
152C4⋊C4×C71—41no—
153C4⋊C4×C71D811yescharacteristic
154C4⋊C4×C71D811yescharacteristic
155C4⋊C4×C71D811yescharacteristic
156C22⋊C4×C71—41no—
157C22⋊C4×C71—41no—
158C22⋊C4×C71—41no—
159C42×C71C2×D411yescharacteristic
160C42×C71C2×D411yescharacteristic
161C42×C71C2×D411yescharacteristic
162C23×C71—41no—
163C22×D71—41no—
164Q8×C71—41no—
165Q8×C71—161no—
166D4×C71—22no—
167D4×C71—41no—
168C2×C4×C71—41no—
169C2×C4×C71—41no—
170C2×C4×C71—41no—
171C2×C4×C71—82no—
172C2×C4×C71—161no—
173C2×C4×C71—161no—
174C2×C4×C71C2×D811yescharacteristic
175C2×C4×C71C2×D811yescharacteristic
176C2×C4×C71C2×D811yescharacteristic
177C2×C4×C71C2×D811yescharacteristic
178C2×C4×C71C2×D811yescharacteristic
179C2×C4×C71C2×D811yescharacteristic
180C2×C4×C71C22×D411yescharacteristic
181C2×Dic71—41no—
182C2×Dic71—41no—
183C2×Dic71—41no—
184D284—22no—
185D284—41no—
186C4×D71—82no—
187C4×D71—161no—
188C4×D71—161no—
189Dic142—161no—
190C8×C71—21no—
191C8×C71—21no—
192C8×C71—21no—
193C8×C71C22×Q811yescharacteristic
194C8×C712−1+411yescharacteristic
195C22×C71—42no—
196C22×C71—81no—
197C22×C71—81no—
198C22×C71C22×D811yescharacteristic
199C2×D71—42no—
200C2×D71—81no—
201C2×D71—81no—
202C4×C71—21no—
203C4×C71—21no—
204C4×C71—21no—
205C4×C71—21no—
206C4×C71—21no—
207C4×C71—21no—
208C4×C71—81no—
209C4×C71—81no—
210C4×C71—81no—
211C4×C71D4×Q811yescharacteristic
212C4×C71#57811yescharacteristic
213Dic71—81no—
214Dic71—81no—
215Dic71—81no—
216#33956—711nomaximal, Sylow
217C2×C71—82no—
218C2×C71Q8×D811yescharacteristic
219C2×C71#341411yescharacteristic
220C2×C71#341511yescharacteristic
221D71—82no—
222#3380—711no—
223#3380D7111yescharacteristic, complemented
224#2255—711no—
225#2255—711no—
226#2253—711no—
227#2228—711no—
228#2228—711no—
229#2228—711no—
230#2228—711no—
231#2224—711no—
232#2224—711no—
233#2183—711no—
234#2177—711no—
235#2177—711no—
236C4×D16—711no—
237C71#3395611yescharacteristic, complemented, Sylow
238#554—21no—
239#554—1421no—
240C2×D16—711no—
241#500—711no—
242#500C2×D7111yescharacteristic
243#499—711no—
244#499—711no—
245#499C2×D7111yescharacteristic
246#499C2×D7111yescharacteristic
247#478—21no—
248#478—21no—
249#478—1421no—
250#478—1421no—
251#476—21no—
252#476—1421no—
253C8×Q8C2×D7111yescharacteristic
254C4×D8—711no—
255C4×D8—711no—
256C4×D8—711no—
257C4×D8C2×D7111yescharacteristic
258C4×D8C2×D7111yescharacteristic
259C4×D8C2×D7111yescharacteristic
260#367—711no—
261#367—711no—
262#367—711no—
263#367—711no—
264#366—711no—
265#366—711no—
266#366—711no—
267#366—711no—
268#366—711no—
269C4⋊C16—711no—
270C4⋊C16—711no—
271C4⋊C16—711no—
272#357—711no—
273#357—711no—
274#357—711no—
275#357—711no—
276#357—711no—
277#357—711no—
278#346—711no—
279#346—711no—
280C4×C16—711no—
281C2×D8—711no—
282C2×D8C22×D7111yescharacteristic
283C4⋊Q8—21no—
284C4⋊Q8—1421no—
285#125—21no—
286#125—21no—
287#125—1421no—
288#125—1421no—
289C22⋊Q8—41no—
290C22⋊Q8—41no—
291C22⋊Q8—41no—
292C22⋊Q8—2841no—
293C22⋊Q8—2841no—
294C22⋊Q8—2841no—
295C4×Q8C22⋊D7111yescharacteristic
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—1421no—
300C4×D4—1421no—
301C4×D4—1421no—
302C2×C4⋊C4—41no—
303C2×C4⋊C4—2841no—
304D16—714no—
305C2×C16—711no—
306C2×C16—711no—
307C2×C16—711no—
308C2×C16—711no—
309#107—21no—
310#107—711no—
311#107—711no—
312#107—711no—
313#107—1421no—
314#107C22×D7111yescharacteristic
315#107C22×D7111yescharacteristic
316#107C22×D7111yescharacteristic
317#106—21no—
318#106—21no—
319#106—1421no—
320#106—1421no—
321C4⋊C8C22⋊D7111yescharacteristic
322C4⋊C8C22⋊D7111yescharacteristic
323C4⋊C8C22⋊D7111yescharacteristic
324#102—21no—
325#102—21no—
326#102—21no—
327#102—1421no—
328#102—1421no—
329#102—1421no—
330C4×C8C22×D7111yescharacteristic
331C4×C8C22×D7111yescharacteristic
332C4×C8C22×D7111yescharacteristic
333C2×Q8—41no—
334C2×Q8—2841no—
335C2×Q8#1176585011yescharacteristic
336C2×D4—21no—
337C2×D4—1421no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—2841no—
342C22×C4—2841no—
343C22×C4—2841no—
344D8—21no—
345D8—712no—
346D8—1421no—
347D8Q8×D7112yes—
348C2×C8C2×C22⋊D7111yescharacteristic
349C2×C8C2×C22⋊D7111yescharacteristic
350C2×C8C2×C22⋊D7111yescharacteristic
351C2×C8C23×D7111yescharacteristic
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—2841no—
361C4⋊C4—2841no—
362C4⋊C4—2841no—
363C4⋊C4—2841no—
364C4⋊C4—2841no—
365C4⋊C4—1421no—
366C4⋊C4—1421no—
367C4⋊C4—1421no—
368C4⋊C4#1176585011yescharacteristic
369C4⋊C4#1176585011yescharacteristic
370C4⋊C4#1176585011yescharacteristic
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—2841no—
375C22⋊C4—2841no—
376C22⋊C4—2841no—
377C42C2×C22⋊D7111yescharacteristic
378C42C2×C22⋊D7111yescharacteristic
379C42C2×C22⋊D7111yescharacteristic
380C16—712no—
381C16—1421no—
382C16—1421no—
383C16—1421no—
384C23—41no—
385C23—2841no—
386Q8—41no—
387Q8—161no—
388Q8—11361no—
389D4—22no—
390D4—41no—
391D4—2841no—
392D4—1422no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—82no—
397C2×C4—161no—
398C2×C4—161no—
399C2×C4—2841no—
400C2×C4—2841no—
401C2×C4—2841no—
402C2×C4—5682no—
403C2×C4—11361no—
404C2×C4—11361no—
405C2×C4C2×#1176585011yescharacteristic
406C2×C4C2×#1176585011yescharacteristic
407C2×C4C2×#1176585011yescharacteristic
408C2×C4C2×#1176585011yescharacteristic
409C2×C4C2×#1176585011yescharacteristic
410C2×C4C2×#1176585011yescharacteristic
411C2×C4C22×C22⋊D7111yescharacteristic
412C8—21no—
413C8—21no—
414C8—21no—
415C8C2×Q8×D7111yescharacteristic
416C8#1455617611yescharacteristic
417C22—42no—
418C22—81no—
419C22—81no—
420C22—2842no—
421C22—5681no—
422C22—5681no—
423C22C22×#1176585011yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4—5681no—
434C4—5681no—
435C4—5681no—
436C4Q8×C22⋊D7111yescharacteristic
437C4#1798681411yescharacteristic
438C2—82no—
439C2—5682no—
440C2Q8×#1176585011yescharacteristic
441C2#2260147411yescharacteristic
442C2#2260147911yescharacteristic
subgroupG/H
centerC22C22×#11765850
commutator, cosocleC2×C8×C71C24
FrattiniC2×C8C23×D71
FittingC71×#3380C2
socleC22×C71C22×D8
prime radicalC2×#14556069C22
hypercenter#3380D71
hypocenterC71#33956
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #22601208C2semidirect
2. #22590627C2semidirect
3. #22590623C2semidirect
4. #22590616C2semidirect
5. #22590497C2semidirect
6. #22590497C2semidirect
7. #22590494C2semidirect
8. #22590494C2semidirect
9. #22590477C2semidirect
10. #22590474C2semidirect
11. #22590249C2semidirect
15. C71×#3380C2semidirect
70. #14556069Q8semidirect
223. #3380D71semidirect
237. C71#33956semidirect

Maximal subgroups

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

subgroupindex
#226012082
#225906272
#225906232
#225906162
#225904972
#225904942
#225904772
#225904742
#225902492
#225902142
C4×#145560692
C71×#33802
#3395671