Finite Group Databasebeta

← #31997467

Subgroups — #31997467

Order 18176.

1326 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic142C211yesmaximal, characteristic
2C2×#17986052C211yesmaximal, characteristic
3#22588424C212yesmaximal
4#22588352C214yesmaximal, complemented
5#22588274C212yesmaximal
6C4×#14556051C214yesmaximal, complemented
7C71×#2283C211yesmaximal, characteristic
8C23×Dic142C2211yescharacteristic
9C2×C4×Dic142—24no—
10C2×C4×Dic142—21no—
11C2×C4×Dic142—21no—
12C2×C4×Dic142C418yescomplemented
13C2×C4×Dic142C2212yes—
14C2×C4×Dic142C2212yes—
15C22×C4⋊Dic71C2211yescharacteristic
16C22×Dic71⋊C4—21no—
17C22×C4×Dic71—21no—
18#17986052—24no—
19C2×C4⋊(C71⋊C8)C2212yes—
20C2×#14556051C2218yes—
21C2×C22⋊(C71⋊C8)C2212yes—
22C2×C4×C71⋊C8C2212yes—
23#17985761C2218yes—
24C22×C42×C71C2211yescharacteristic
25#383×C71—21no—
26#382×C71C2212yes—
27#381×C71—21no—
28#380×C71C2212yes—
29C4×C4⋊C4×C71C2212yes—
30C4×C22⋊C4×C71—21no—
31#376×C71—21no—
32C22×Dic142—28no—
33C22×Dic142—21no—
34C22×Dic142—22no—
35C22×Dic142C2×C414yes—
36C22×Dic142C2312yes—
37C23×Dic71—21no—
38C23×Dic71—21no—
39C22×C71⋊C8C2314yes—
40C4×Dic142—28no—
41C4×Dic142—28no—
42C4×Dic142—24no—
43C4×Dic142—24no—
44C4×Dic142C2×C418yes—
45C4×Dic142C2×C418yes—
46C2×C4⋊Dic71—21no—
47C2×C4⋊Dic71—22no—
48C2×C4⋊Dic71C2×C414yes—
49C2×C4⋊Dic71C2312yes—
50C2×Dic71⋊C4—28no—
51C2×Dic71⋊C4—22no—
52C2×Dic71⋊C4—22no—
53C2×C4×Dic71—28no—
54C2×C4×Dic71—22no—
55C2×C4×Dic71—22no—
56C4⋊(C71⋊C8)—24no—
57#14556051—216no—
58C22⋊(C71⋊C8)—24no—
59C4×C71⋊C8—24no—
60C23×C4×C71D411yescharacteristic
61C23×C4×C71D411yescharacteristic
62C23×C4×C71C2311yescharacteristic
63C2×C4⋊C4×C71C2314yes—
64C2×C22⋊C4×C71—22no—
65C2×C42×C71—22no—
66C2×C42×C71C2×C414yes—
67C2×C42×C71D411yescharacteristic
68C2×C42×C71D411yescharacteristic
69C2×C42×C71D414yes—
70C2×C42×C71C2311yescharacteristic
71C2×C42×C71C2311yescharacteristic
72#95×C71—22no—
73#95×C71—22no—
74#95×C71—22no—
75#95×C71—22no—
76C2×Dic142—28no—
77C2×Dic142—216no—
78C2×Dic142—24no—
79C2×Dic142—22no—
80C2×Dic142—24no—
81C2×Dic142—216no—
82C2×Dic142C22×C414yes—
83C2×Dic142#4118yes—
84C22×Dic71—28no—
85C22×Dic71—24no—
86C22×Dic71—22no—
87C22×Dic71—24no—
88C22×Dic71—22no—
89C22×Dic71—44no—
90C2×C71⋊C8—24no—
91C2×C71⋊C8—28no—
92C4⋊Dic71—24no—
93C4⋊Dic71—22no—
94C4⋊Dic71C22×C414yes—
95Dic71⋊C4—28no—
96Dic71⋊C4—28no—
97C4×Dic71—28no—
98C4×Dic71—28no—
99C24×C71C2×D411yescharacteristic
100C22×C4×C71—24no—
101C22×C4×C71—24no—
102C22×C4×C71—24no—
103C22×C4×C71—24no—
104C22×C4×C71C22⋊C414yes—
105C22×C4×C71C22⋊C414yes—
106C22×C4×C71C22×C412yes—
107C22×C4×C71C2×D411yescharacteristic
108C22×C4×C71C2×D411yescharacteristic
109C22×C4×C71C2×D411yescharacteristic
110C22×C4×C71C2×D411yescharacteristic
111C22×C4×C71C2×D411yescharacteristic
112C22×C4×C71C2×D412yes—
113C22×C4×C71C2411yescharacteristic
114C4⋊C4×C71—28no—
115C22⋊C4×C71—44no—
116C42×C71—44no—
117C42×C71C22⋊C414yes—
118C42×C71C22⋊C414yes—
119C42×C71C22×C412yes—
120C42×C71C22×C412yes—
121C42×C71C2×D412yes—
122C42×C71C2×D412yes—
123C23×C71—22no—
124C23×C71—22no—
125C23×C71C2×C22⋊C412yes—
126C23×C71#13514yes—
127C23×C71C22×D411yescharacteristic
128C2×C4×C71—22no—
129C2×C4×C71—22no—
130C2×C4×C71—22no—
131C2×C4×C71—22no—
132C2×C4×C71—28no—
133C2×C4×C71—24no—
134C2×C4×C71—24no—
135C2×C4×C71—28no—
136C2×C4×C71—44no—
137C2×C4×C71—44no—
138C2×C4×C71—44no—
139C2×C4×C71C2×C22⋊C412yes—
140C2×C4×C71C2×C22⋊C412yes—
141C2×C4×C71C2×C22⋊C412yes—
142C2×C4×C71C2×C22⋊C412yes—
143C2×C4×C71C2×C22⋊C412yes—
144C2×C4×C71#13714yes—
145C2×C4×C71C23×C411yescharacteristic
146C2×C4×C71C22×D411yescharacteristic
147C2×C4×C71C2×#4114yes—
148C2×Dic71—216no—
149C2×Dic71—24no—
150C2×Dic71—28no—
151C2×Dic71—24no—
152C2×Dic71—44no—
153C2×Dic71—48no—
154Dic142—216no—
155Dic142—24no—
156Dic142—28no—
157Dic142#11718yes—
158C71⋊C8—44no—
159C22×C71—28no—
160C22×C71—22no—
161C22×C71—22no—
162C22×C71#41614yes—
163C22×C71C22×C22⋊C411yescharacteristic
164C22×C71#52512yes—
165C22×C71C2×#13512yes—
166C22×C71C2×#13712yes—
167C4×C71—22no—
168C4×C71—22no—
169C4×C71—22no—
170C4×C71—22no—
171C4×C71—84no—
172C4×C71#41914yes—
173C4×C71C2×#11712yes—
174C4×C71#52512yes—
175Dic71—28no—
176Dic71—44no—
177#20735—711nomaximal, Sylow
178C2×C71—24no—
179C2×C71#228011yescharacteristic
180C2×C71C2×#41611yescharacteristic
181C2×C71C2×#41911yescharacteristic
182C2×C71#310814yes—
183C22×C4×Q8—711no—
184C2×#431—711no—
185#2283D7111yescharacteristic
186#1823—712no—
187#1805—714no—
188#1787—712no—
189C4×#103—714no—
190C71#2073511yescharacteristic, complemented, Sylow
191C23×Q8—711no—
192C2×C4×Q8—718no—
193C2×C4×Q8—712no—
194C2×C4×Q8—712no—
195C2×C4×Q8—1424no—
196C2×C4×Q8—1421no—
197C2×C4×Q8—1421no—
198C22×C4⋊C4—711no—
199C22×C4⋊C4—1421no—
200C22×C42—1421no—
201C22×C42C2×D7111yescharacteristic
202#431—1424no—
203C2×C4⋊C8—712no—
204C2×#103—718no—
205C2×C22⋊C8—712no—
206C2×C4×C8—712no—
207#383—21no—
208#382C2×D7112yes—
209#381—21no—
210#380C2×D7112yes—
211C4×C4⋊C4C2×D7112yes—
212C4×C22⋊C4—21no—
213#376—21no—
214#340—718no—
215C22×Q8—714no—
216C22×Q8—712no—
217C22×Q8—1428no—
218C22×Q8—1421no—
219C22×Q8—1422no—
220C23×C4—1421no—
221C23×C4—1421no—
222C23×C4C22⋊D7111yescharacteristic
223C23×C4C22⋊D7111yescharacteristic
224C23×C4C22×D7111yescharacteristic
225C22×C8—714no—
226C4×Q8—718no—
227C4×Q8—718no—
228C4×Q8—1428no—
229C4×Q8—1428no—
230C4×Q8—1424no—
231C4×Q8—1424no—
232C2×C4⋊C4—714no—
233C2×C4⋊C4—712no—
234C2×C4⋊C4—1428no—
235C2×C4⋊C4—1421no—
236C2×C4⋊C4—1422no—
237C2×C4⋊C4—1422no—
238C2×C4⋊C4—1422no—
239C2×C4⋊C4C22×D7114yes—
240C2×C22⋊C4—22no—
241C2×C42—22no—
242C2×C42—1428no—
243C2×C42—1422no—
244C2×C42—1422no—
245C2×C42C4×D7114yes—
246C2×C42D28414yes—
247C2×C42C22⋊D7111yescharacteristic
248C2×C42C22⋊D7111yescharacteristic
249C2×C42C22×D7111yescharacteristic
250C2×C42C22×D7111yescharacteristic
251C4⋊C8—1424no—
252#103—14216no—
253C22⋊C8—1424no—
254C4×C8—1424no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259C24C2×C22⋊D7111yescharacteristic
260C2×Q8—714no—
261C2×Q8—718no—
262C2×Q8—14216no—
263C2×Q8—1428no—
264C2×Q8—14216no—
265C2×Q8—1422no—
266C2×Q8—1424no—
267C2×Q8—1424no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—2844no—
273C22×C4—1422no—
274C22×C4—1424no—
275C22×C4—1428no—
276C22×C4—1422no—
277C22×C4—1424no—
278C22×C4(C2×D71)⋊C414yes—
279C22×C4(C2×D71)⋊C414yes—
280C22×C4C2×C4×D7112yes—
281C22×C4C2×C22⋊D7111yescharacteristic
282C22×C4C2×C22⋊D7111yescharacteristic
283C22×C4C2×C22⋊D7111yescharacteristic
284C22×C4C2×C22⋊D7111yescharacteristic
285C22×C4C2×C22⋊D7111yescharacteristic
286C22×C4C2×D28412yes—
287C22×C4C23×D7111yescharacteristic
288C2×C8—1424no—
289C2×C8—1428no—
290C4⋊C4—28no—
291C4⋊C4—714no—
292C4⋊C4—1422no—
293C4⋊C4—1428no—
294C4⋊C4—1428no—
295C4⋊C4—1424no—
296C22⋊C4—44no—
297C42—44no—
298C42—1428no—
299C42—1428no—
300C42(C2×D71)⋊C414yes—
301C42(C2×D71)⋊C414yes—
302C42C2×C4×D7112yes—
303C42C2×C4×D7112yes—
304C42C2×D28412yes—
305C42C2×D28412yes—
306C23—22no—
307C23—22no—
308C23C2×(C2×D71)⋊C412yes—
309C23#1455614814yes—
310C23C22×C22⋊D7111yescharacteristic
311Q8—718no—
312Q8—14216no—
313Q8—1424no—
314Q8—1428no—
315C2×C4—28no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—28no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—2848no—
327C2×C4—2844no—
328C2×C4—1424no—
329C2×C4—14216no—
330C2×C4—1424no—
331C2×C4—1428no—
332C2×C4C2×(C2×D71)⋊C412yes—
333C2×C4C2×(C2×D71)⋊C412yes—
334C2×C4C2×(C2×D71)⋊C412yes—
335C2×C4C2×(C2×D71)⋊C412yes—
336C2×C4C2×(C2×D71)⋊C412yes—
337C2×C4#1455615814yes—
338C2×C4C22×C4×D7111yescharacteristic
339C2×C4C22×D28411yescharacteristic
340C2×C4#41×D7114yes—
341C8—2844no—
342C22—28no—
343C22—22no—
344C22—22no—
345C22#1798599314yes—
346C22C22×(C2×D71)⋊C411yescharacteristic
347C22#1798650612yes—
348C22C2×#1455614812yes—
349C22C2×#1455615812yes—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—84no—
355C4—2844no—
356C4—1428no—
357C4#1798600714yes—
358C4#117×D7112yes—
359C4#1798651112yes—
360C2—24no—
361C2#2259071911yescharacteristic
362C2C2×#1798599311yescharacteristic
363C2C2×#1798600711yescharacteristic
364C2#2259841614yes—
subgroupG/H
centerC22×C4C2×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2283C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenter#2283D71
hypocenterC71#20735
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
4. #22588352C2semidirect
6. C4×#14556051C2semidirect
12. C2×C4×Dic142C4semidirect
190. C71#20735semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1422
C2×#179860522
#225884242
#225883522
#225882742
C4×#145560512
C71×#22832
#2073571