Finite Group Databasebeta

← #32150797

Subgroups — #32150797

Order 18176.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986434C211yesmaximal, characteristic
2#22597216C211yesmaximal, characteristic, complemented
3#22597191C211yesmaximal, characteristic, complemented
4#22590213C212yesmaximal, complemented
5#22590207C212yesmaximal, complemented
6#22590189C212yesmaximal, complemented
7#22590184C212yesmaximal, complemented
8#22590129C211yesmaximal, characteristic, complemented
9C2×#17985801C211yesmaximal, characteristic, complemented
10#22589936C211yesmaximal, characteristic, complemented
11C71×#2138C211yesmaximal, characteristic, complemented
12C22×D568C2211yescharacteristic
13#17986482—21no—
14#17986473—21no—
15#17986434C418yescomplemented
16C2×#14556071C2212yes—
17C2×#14556069C2212yes—
18C2×#14556068C2211yescharacteristic
19#17986110—22no—
20#17986100—22no—
21Q16×Dic71C2212yescomplemented
22D8×Dic71C2212yescomplemented
23#17986081C2212yes—
24#17986081C2212yes—
25C2×#14556062C2211yescharacteristic
26#17986003—21no—
27#17986000—21no—
28#17985954C2211yescharacteristic
29#17985826C2212yescomplemented
30#17985824C2212yescomplemented
31#17985801C2212yes—
32#17985801C2212yes—
33#17985774C2212yescomplemented
34C16×Dic71C2212yescomplemented
35C2×#135×C71C2211yescharacteristic
36C22×C16×C71C2211yescharacteristic
37#427×C71C2211yescharacteristic
38#358×C71C2211yescharacteristic, complemented
39#358×C71C2211yescharacteristic, complemented
40#357×C71C2211yescharacteristic, complemented
41#357×C71C2211yescharacteristic, complemented
42C22×D284—21no—
43C2×D568—21no—
44C2×D568—22no—
45C2×D568C2×C414yescomplemented
46C2×D568C2312yes—
47C2×#11765849D411yescharacteristic
48Q8×Dic71—22no—
49C4×D284—41no—
50C4×D284—41no—
51D4×Dic71—22no—
52D4×Dic71—42no—
53#14556090—22no—
54#14556090—22no—
55#14556087—41no—
56#14556087D411yescharacteristic
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—41no—
59C22⋊C4×D71—42no—
60#14556071C2×C418yescomplemented
61#14556069C2×C418yescomplemented
62#14556068C2×C414yescomplemented
63C2×C71⋊C16C2312yes—
64#14556062—21no—
65#14556062C2312yes—
66#14556052—22no—
67#14556049—22no—
68#14556048—22no—
69#14556048—22no—
70#14556038D412yes—
71C8×Dic71C2312yes—
72C2×#41×C71—21no—
73#135×C71—22no—
74#135×C71C2×C414yescomplemented
75C2×Q16×C71C2311yescharacteristic
76C2×SD16×C71—21no—
77C2×D8×C71C2311yescharacteristic
78C22×C8×C71C2311yescharacteristic
79#117×C71—21no—
80C2×C16×C71—22no—
81C2×C16×C71C2311yescharacteristic
82C2×C16×C71C2311yescharacteristic
83#107×C71C2311yescharacteristic
84#107×C71C2311yescharacteristic
85#106×C71—21no—
86C23×D71—41no—
87C2×D284—24no—
88C2×D284—22no—
89C2×D284—41no—
90C2×D284—42no—
91C22×Dic71—41no—
92C22×Dic71—42no—
93C2×C4×D71—42no—
94C2×C4×D71—42no—
95D568—24no—
96D568—22no—
97D568C22×C414yes—
98#11765849—22no—
99C2×C71⋊C8C2×D412yes—
100C4⋊Dic71—22no—
101C4⋊Dic71—41no—
102C4⋊Dic71—42no—
103C4⋊Dic71D811yescharacteristic
104C4⋊Dic71D811yescharacteristic
105C22⋊Dic71—42no—
106C22⋊Dic71D812yes—
107(C2×D71)⋊C4—42no—
108(C2×D71)⋊C4—42no—
109C4×Dic71—42no—
110C4×Dic71C2×D412yes—
111C71⋊C16C22×C414yes—
112#41×C71—24no—
113#41×C71—42no—
114C2×Q8×C71—21no—
115C2×D4×C71—21no—
116C2×D4×C71—41no—
117C22×C4×C71—41no—
118C22×C4×C71C2×D411yescharacteristic
119Q16×C71—21no—
120Q16×C71C22×C412yes—
121SD16×C71—24no—
122D8×C71—21no—
123D8×C71C22×C412yes—
124C2×C8×C71C22×C412yes—
125C2×C8×C71C2×D411yescharacteristic
126C2×C8×C71#4112yes—
127C2×C8×C71C2411yescharacteristic
128C4⋊C4×C71—21no—
129C4⋊C4×C71—21no—
130C22⋊C4×C71—41no—
131C42×C71—41no—
132C16×C71—21no—
133C16×C71—21no—
134C23×C71—41no—
135C23×C71C2×D811yescharacteristic
136C22×D71—42no—
137C22×D71—44no—
138C22×D71—161no—
139C22×D71—161no—
140Q8×C71—22no—
141Q8×C71—41no—
142D4×C71—22no—
143D4×C71—44no—
144D4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—44no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71C4×D412yes—
150C2×C4×C71C2×D811yescharacteristic
151C2×C4×C71#13512yes—
152C2×C4×C71C22×D411yescharacteristic
153C2×Dic71—42no—
154C2×Dic71—162no—
155C2×Dic71—161no—
156C2×Dic71C2×D812yes—
157C2×Dic71C2×D812yes—
158D284—24no—
159D284—42no—
160D284—44no—
161C4×D71—162no—
162C4×D71—162no—
163C8×C71C4×D412yes—
164C8×C71C23×C411yescharacteristic
165C8×C71C2×#4111yescharacteristic
166C71⋊C8—22no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—42no—
170C22×C71—81no—
171C22×C71—81no—
172C22×C71C4×D812yes—
173C22×C71C22×D811yescharacteristic
174C2×D71—44no—
175C2×D71—82no—
176C2×D71—82no—
177C2×D71—164no—
178C4×C71—42no—
179C4×C71—81no—
180C4×C71—81no—
181C4×C71C4×D812yes—
182C4×C71C2×C4×D411yescharacteristic
183C4×C71C2×#13511yescharacteristic
184Dic71—22no—
185Dic71—22no—
186Dic71—82no—
187#33454—711nomaximal, Sylow
188C2×C71—22no—
189C2×C71—82no—
190C2×C71C2×C4×D811yescharacteristic
191C2×C71#341511yescharacteristic
192C2×C71#341611yescharacteristic
193D71—84no—
194C2×#509—711no—
195#2951—711no—
196#2948—711no—
197#2177—712no—
198#2175—712no—
199C4×SD32—712no—
200C4×D16—712no—
201#2159—711no—
202#2138D7111yescharacteristic, complemented
203C2×#357—711no—
204#2108—711no—
205C71#3345411yescharacteristic, complemented, Sylow
206C2×#135C2×D7111yescharacteristic
207C22×D8—711no—
208#520—1421no—
209#518—1421no—
210#509—718no—
211C2×SD32—712no—
212C2×D16—712no—
213C2×M32—711no—
214C22×C16C2×D7111yescharacteristic
215#442—1422no—
216#440—1422no—
217C4×Q16—712no—
218C4×D8—712no—
219C4×D8—712no—
220C4×D8—712no—
221#427C2×D7111yescharacteristic
222C2×#107—711no—
223#418—1421no—
224#417—1421no—
225#405—711no—
226#367—712no—
227#366—712no—
228#358C2×D7111yescharacteristic, complemented
229#358C2×D7111yescharacteristic, complemented
230#357—712no—
231#357—712no—
232#357C2×D7111yescharacteristic, complemented
233#357C2×D7111yescharacteristic, complemented
234#346—712no—
235C4×C16—712no—
236C2×#41—21no—
237C22×D4—1421no—
238#135—22no—
239#135C4×D7114yescomplemented
240C2×Q16C22×D7111yescharacteristic
241C2×SD16—21no—
242C2×D8—714no—
243C2×D8—712no—
244C2×D8—1421no—
245C2×D8—1422no—
246C2×D8C22×D7111yescharacteristic
247C2×M16—711no—
248C22×C8C22×D7111yescharacteristic
249C4×Q8—1422no—
250C4×D4—2841no—
251C4×D4—2841no—
252C4×D4—2842no—
253C4×D4—1422no—
254C4×D4—1422no—
255C4×D4—1422no—
256#117—21no—
257#117—2841no—
258#117—711no—
259C2×C4⋊C4—2841no—
260C2×C4⋊C4—1421no—
261C2×C22⋊C4—2842no—
262SD32—718no—
263D16—718no—
264M32—714no—
265C2×C16—22no—
266C2×C16—712no—
267C2×C16C22×D7111yescharacteristic
268C2×C16C22×D7111yescharacteristic
269#107—712no—
270#107—1421no—
271#107C22×D7111yescharacteristic
272#107C22×D7111yescharacteristic
273#106—21no—
274#103—1422no—
275#102—1422no—
276#102—1422no—
277#102—1422no—
278#97—712no—
279C4×C8—712no—
280C24—2841no—
281#41—24no—
282#41—42no—
283C2×Q8—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—2841no—
287C2×D4—2842no—
288C2×D4—1422no—
289C2×D4—1424no—
290C22×C4—41no—
291C22×C4—2842no—
292C22×C4—2842no—
293C22×C4—2841no—
294C22×C4—2842no—
295C22×C4D4×D7111yescharacteristic
296Q16—21no—
297Q16C2×C4×D7112yes—
298SD16—24no—
299D8—21no—
300D8—714no—
301D8—1422no—
302D8—1424no—
303D8C2×C4×D7112yes—
304M16—1422no—
305C2×C8—712no—
306C2×C8C2×C4×D7112yes—
307C2×C8D4×D7111yescharacteristic
308C2×C8#1176586412yes—
309C2×C8C23×D7111yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—2842no—
313C4⋊C4—2841no—
314C4⋊C4—711no—
315C4⋊C4—711no—
316C4⋊C4—1422no—
317C22⋊C4—41no—
318C22⋊C4—2842no—
319C22⋊C4—2842no—
320C22⋊C4—2842no—
321C22⋊C4—712no—
322C42—41no—
323C42—2842no—
324C42—712no—
325C16—21no—
326C16—21no—
327C16—714no—
328C23—41no—
329C23—2842no—
330C23—2844no—
331C23—11361no—
332C23—11361no—
333C23D8×D7111yescharacteristic
334Q8—22no—
335Q8—41no—
336D4—22no—
337D4—44no—
338D4—41no—
339D4—2844no—
340D4—2842no—
341D4—1424no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—2842no—
347C2×C4—712no—
348C2×C4—712no—
349C2×C4—11362no—
350C2×C4—11362no—
351C2×C4—11361no—
352C2×C4—11362no—
353C2×C4(C22⋊D71)⋊C412yes—
354C2×C4D8×D7111yescharacteristic
355C2×C4#1455614912yes—
356C2×C4C2×D4×D7111yescharacteristic
357C8—1422no—
358C8(C22⋊D71)⋊C412yes—
359C8C22×C4×D7111yescharacteristic
360C8C2×#1176586411yescharacteristic
361C22—21no—
362C22—21no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—2844no—
367C22—5682no—
368C22—5682no—
369C22—11364no—
370C22#1798607912yes—
371C22C2×D8×D7111yescharacteristic
372C4—42no—
373C4—81no—
374C4—81no—
375C4—5682no—
376C4—1422no—
377C4—1422no—
378C4#1798607912yes—
379C4C2×(C22⋊D71)⋊C411yescharacteristic
380C4C2×#1455614911yescharacteristic
381C2—22no—
382C2—82no—
383C2—5684no—
384C2C2×#1798607911yescharacteristic
385C2#2260147811yescharacteristic
386C2#2260148311yescharacteristic
subgroupG/H
centerC22C2×D8×D71
commutatorC8×C71C23×C4
FrattiniC2×C8C23×D71
FittingC71×#2138C2
socleC22×C71C22×D8
prime radicalC2×#17986434C2
cosocleC2×C8×C71C24
hypercenter#2138D71
hypocenterC71#33454
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. #22597216C2semidirect
3. #22597191C2semidirect
4. #22590213C2semidirect
5. #22590207C2semidirect
6. #22590189C2semidirect
7. #22590184C2semidirect
8. #22590129C2semidirect
9. C2×#17985801C2semidirect
10. #22589936C2semidirect
11. C71×#2138C2semidirect
15. #17986434C4semidirect
21. Q16×Dic71C22semidirect
22. D8×Dic71C22semidirect
29. #17985826C22semidirect
30. #17985824C22semidirect
33. #17985774C22semidirect
34. C16×Dic71C22semidirect
38. #358×C71C22semidirect
39. #358×C71C22semidirect
40. #357×C71C22semidirect
41. #357×C71C22semidirect
45. C2×D568C2×C4semidirect
60. #14556071C2×C4semidirect
61. #14556069C2×C4semidirect
62. #14556068C2×C4semidirect
74. #135×C71C2×C4semidirect
202. #2138D71semidirect
205. C71#33454semidirect
228. #358C2×D71semidirect
229. #358C2×D71semidirect
232. #357C2×D71semidirect
233. #357C2×D71semidirect
239. #135C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864342
#225972162
#225971912
#225902132
#225902072
#225901892
#225901842
#225901292
C2×#179858012
#225899362
C71×#21382
#3345471