Finite Group Databasebeta

← #32000979

Subgroups — #32000979

Order 18176.

906 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986048C212yesmaximal
2C2×#17986023C211yesmaximal, characteristic
3C2×C4×#11765849C211yesmaximal, characteristic
4#22589116C218yesmaximal, complemented
5#22588067C212yesmaximal
6C2×#423×C71C211yesmaximal, characteristic
7C22×#11765849C2212yes—
8C22×#11765849C2211yescharacteristic
9C22×#11765849C2211yescharacteristic
10#17986048—28no—
11C2×#14556064C2218yes—
12#17986023—21no—
13#17986023—21no—
14#17986023C2214yes—
15C2×C4⋊(C71⋊C8)—22no—
16C4×#11765849—22no—
17C4×#11765849C2214yes—
18C2×#14556039—21no—
19C2×C4×C71⋊C8—21no—
20#17985764C2218yes—
21C22×M16×C71C2212yes—
22C22×C42×C71C2211yescharacteristic
23#423×C71C412yes—
24#423×C71C414yes—
25#423×C71C412yes—
26C2×C4⋊C8×C71C2214yes—
27C2×#11765849—21no—
28C2×#11765849—21no—
29C2×#11765849—22no—
30C2×#11765849—24no—
31C2×#11765849—22no—
32C2×#11765849—22no—
33C2×#11765849—24no—
34C2×#11765849D414yes—
35C2×#11765849Q814yes—
36C2×#11765849C2314yes—
37C2×#11765849C2312yes—
38C2×#11765849C2312yes—
39C22×C71⋊C8—21no—
40C22×C71⋊C8—21no—
41C22×C71⋊C8—22no—
42#14556064—216no—
43C4⋊(C71⋊C8)—24no—
44C4⋊(C71⋊C8)—24no—
45#14556039—24no—
46C4×C71⋊C8—24no—
47C23×C4×C71Q812yes—
48C23×C4×C71C2311yescharacteristic
49C2×M16×C71—24no—
50C2×M16×C71—24no—
51C2×M16×C71C2×C414yes—
52C2×M16×C71C2×C414yes—
53C22×C8×C71C2314yes—
54C2×C42×C71C2×C411yescharacteristic
55C2×C42×C71C2×C412yes—
56C2×C42×C71C2×C411yescharacteristic
57C2×C42×C71D411yescharacteristic
58C2×C42×C71D411yescharacteristic
59C2×C42×C71D412yes—
60C2×C42×C71Q811yescharacteristic
61C2×C42×C71Q811yescharacteristic
62C2×C42×C71C2312yes—
63C4⋊C8×C71C2×C418yes—
64C4⋊C8×C71C2×C418yes—
65#11765849—24no—
66#11765849—24no—
67#11765849—22no—
68#11765849—22no—
69#11765849—24no—
70#11765849—24no—
71#11765849—44no—
72#11765849C2×D414yes—
73#11765849C2×Q814yes—
74C2×C71⋊C8—24no—
75C2×C71⋊C8—24no—
76C2×C71⋊C8—22no—
77C2×C71⋊C8—22no—
78C2×C71⋊C8—24no—
79C2×C71⋊C8—44no—
80C24×C71C2×Q811yescharacteristic
81C22×C4×C71—44no—
82C22×C4×C71C4⋊C412yes—
83C22×C4×C71C4⋊C412yes—
84C22×C4×C71C4⋊C412yes—
85C22×C4×C71C4⋊C412yes—
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C22×C411yescharacteristic
88C22×C4×C71C2×D412yes—
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91C22×C4×C71C2×Q812yes—
92C22×C4×C71C2×Q812yes—
93C22×C4×C71C2×Q812yes—
94C22×C4×C71C2×Q811yescharacteristic
95C22×C4×C71#4112yes—
96C22×C4×C71#4112yes—
97C22×C4×C71C2411yescharacteristic
98M16×C71—28no—
99M16×C71—28no—
100C2×C8×C71—24no—
101C2×C8×C71—24no—
102C2×C8×C71C22×C418yes—
103C42×C71C4⋊C411yescharacteristic
104C42×C71C4⋊C411yescharacteristic
105C42×C71C4⋊C411yescharacteristic
106C42×C71C4⋊C412yes—
107C42×C71C4⋊C412yes—
108C42×C71C4⋊C411yescharacteristic
109C42×C71C22×C412yes—
110C42×C71C22×C412yes—
111C42×C71C2×D412yes—
112C42×C71C2×D412yes—
113C23×C71—42no—
114C23×C71C2×C4⋊C411yescharacteristic
115C23×C71C2×C4⋊C411yescharacteristic
116C23×C71#12512yes—
117C23×C71C4⋊Q812yes—
118C23×C71C22×Q811yescharacteristic
119C2×C4×C71—24no—
120C2×C4×C71—24no—
121C2×C4×C71—24no—
122C2×C4×C71—42no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71C2×C4⋊C412yes—
126C2×C4×C71C2×C4⋊C412yes—
127C2×C4×C71C2×C4⋊C412yes—
128C2×C4×C71C2×C4⋊C411yescharacteristic
129C2×C4×C71C2×C4⋊C411yescharacteristic
130C2×C4×C71C2×C4⋊C412yes—
131C2×C4×C71C4×D412yes—
132C2×C4×C71C4×D412yes—
133C2×C4×C71C4×Q812yes—
134C2×C4×C71C4×Q812yes—
135C2×C4×C71#12112yes—
136C2×C4×C71#12112yes—
137C2×C4×C71C22⋊Q812yes—
138C2×C4×C71C22⋊Q812yes—
139C2×C4×C71#12512yes—
140C2×C4×C71C4⋊Q812yes—
141C2×C4×C71C23×C411yescharacteristic
142C2×C4×C71C22×D411yescharacteristic
143C2×C4×C71C22×D411yescharacteristic
144C2×C4×C71C22×Q811yescharacteristic
145C2×C4×C71C2×#4111yescharacteristic
146C2×C4×C71C2×#4111yescharacteristic
147C8×C71—28no—
148C71⋊C8—24no—
149C71⋊C8—24no—
150C71⋊C8—44no—
151C22×C71—22no—
152C22×C71—22no—
153C22×C71—22no—
154C22×C71—42no—
155C22×C71—42no—
156C22×C71#38912yes—
157C22×C71#38912yes—
158C22×C71C22×C4⋊C411yescharacteristic
159C22×C71C2×#12511yescharacteristic
160C22×C71C2×C4⋊Q811yescharacteristic
161C4×C71—22no—
162C4×C71—22no—
163C4×C71—22no—
164C4×C71#38912yes—
165C4×C71#38912yes—
166C4×C71#43014yescomplemented
167C4×C71C2×C4×D411yescharacteristic
168C4×C71C2×C4×Q811yescharacteristic
169C4×C71C2×#12111yescharacteristic
170C4×C71C2×C22⋊Q811yescharacteristic
171#21177—711nomaximal, Sylow
172C2×C71—21no—
173C2×C71—21no—
174C2×C71—22no—
175C2×C71—22no—
176C2×C71C2×#38911yescharacteristic
177C2×C71C2×#43012yes—
178C2×#430—712no—
179C2×#423—711no—
180C2×#423D7111yescharacteristic
181C2×C4×M16—711no—
182#1950—718no—
183#1738—712no—
184C71#2117711yescharacteristic, complemented, Sylow
185C22×M16—711no—
186C22×M16—711no—
187C22×M16—712no—
188C22×M16C2×D7112yes—
189C22×C42C2×D7111yescharacteristic
190#430—1428no—
191C2×#108—718no—
192#423—714no—
193#423—1421no—
194#423—1421no—
195#423Dic7112yes—
196#423Dic7114yes—
197#423Dic7112yes—
198C2×C4⋊C8—1422no—
199C2×C4⋊C8C2×D7114yes—
200C4×M16—714no—
201C4×M16—1422no—
202C2×#97—1421no—
203C2×C4×C8—1421no—
204#341—718no—
205C23×C4Dic14212yes—
206C23×C4C22×D7111yescharacteristic
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—714no—
210C2×M16—714no—
211C2×M16—712no—
212C2×M16—712no—
213C2×M16—714no—
214C2×M16—1424no—
215C2×M16—1424no—
216C2×M16—1422no—
217C2×M16—1421no—
218C2×M16—1421no—
219C2×M16—1422no—
220C2×M16—1422no—
221C2×M16C2×Dic7114yes—
222C2×M16C2×Dic7114yes—
223C22×C8—1422no—
224C22×C8—1421no—
225C22×C8—1421no—
226C22×C8C22×D7114yes—
227C2×C42Dic14211yescharacteristic
228C2×C42Dic14211yescharacteristic
229C2×C42D28411yescharacteristic
230C2×C42D28411yescharacteristic
231C2×C42D28412yes—
232C2×C42C2×Dic7111yescharacteristic
233C2×C42C2×Dic7112yes—
234C2×C42C2×Dic7111yescharacteristic
235C2×C42C22×D7112yes—
236#108—14216no—
237C4⋊C8—1424no—
238C4⋊C8—1424no—
239C4⋊C8C2×Dic7118yes—
240C4⋊C8C2×Dic7118yes—
241#97—1424no—
242C4×C8—1424no—
243C24C2×Dic14211yescharacteristic
244C22×C4—44no—
245C22×C4C4⋊Dic7112yes—
246C22×C4C4⋊Dic7112yes—
247C22×C4C4⋊Dic7112yes—
248C22×C4C4⋊Dic7112yes—
249C22×C4C22×Dic7111yescharacteristic
250C22×C4C22×Dic7111yescharacteristic
251C22×C4D4×D7112yes—
252C22×C4C2×D28411yescharacteristic
253C22×C4C2×D28411yescharacteristic
254C22×C4Q8×D7112yes—
255C22×C4C2×Dic14212yes—
256C22×C4C2×Dic14212yes—
257C22×C4C2×Dic14211yescharacteristic
258C22×C4#1176586412yes—
259C22×C4#1176586612yes—
260C22×C4C23×D7111yescharacteristic
261M16—28no—
262M16—28no—
263M16—2844no—
264M16—714no—
265M16—714no—
266M16—1424no—
267M16—1424no—
268M16—1422no—
269M16—1422no—
270M16—1424no—
271M16—1424no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—2844no—
275C2×C8—1424no—
276C2×C8—1422no—
277C2×C8—1422no—
278C2×C8—1424no—
279C2×C8—1424no—
280C2×C8C22×Dic7118yes—
281C42C4⋊Dic7111yescharacteristic
282C42C4⋊Dic7111yescharacteristic
283C42C4⋊Dic7111yescharacteristic
284C42C4⋊Dic7112yes—
285C42C4⋊Dic7112yes—
286C42C4⋊Dic7111yescharacteristic
287C42C22×Dic7112yes—
288C42C22×Dic7112yes—
289C42C2×D28412yes—
290C42C2×D28412yes—
291C23—42no—
292C23C2×C4⋊Dic7111yescharacteristic
293C23C2×C4⋊Dic7111yescharacteristic
294C23#1455611912yes—
295C23#1455612712yes—
296C23C22×Dic14211yescharacteristic
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4C2×C4⋊Dic7112yes—
304C2×C4C2×C4⋊Dic7112yes—
305C2×C4C2×C4⋊Dic7112yes—
306C2×C4C2×C4⋊Dic7112yes—
307C2×C4C2×C4⋊Dic7111yescharacteristic
308C2×C4C2×C4⋊Dic7111yescharacteristic
309C2×C4D4×Dic7112yes—
310C2×C4D4×Dic7112yes—
311C2×C4Q8×Dic7112yes—
312C2×C4Q8×Dic7112yes—
313C2×C4#1455610312yes—
314C2×C4#1455610312yes—
315C2×C4#1455610812yes—
316C2×C4#1455610812yes—
317C2×C4#1455611912yes—
318C2×C4#1455612712yes—
319C2×C4C23×Dic7111yescharacteristic
320C2×C4C2×D4×D7111yescharacteristic
321C2×C4C22×D28411yescharacteristic
322C2×C4C2×Q8×D7111yescharacteristic
323C2×C4C2×#1176586411yescharacteristic
324C2×C4C2×#1176586611yescharacteristic
325C8—28no—
326C8—2844no—
327C8—1424no—
328C8—1424no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22#1798589812yes—
335C22#1798589812yes—
336C22C22×C4⋊Dic7111yescharacteristic
337C22C2×#1455611911yescharacteristic
338C22C2×#1455612711yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4#1798589812yes—
343C4#1798589812yes—
344C4#1798604814yescomplemented
345C4C2×D4×Dic7111yescharacteristic
346C4C2×Q8×Dic7111yescharacteristic
347C4C2×#1455610311yescharacteristic
348C4C2×#1455610811yescharacteristic
349C2—22no—
350C2—22no—
351C2—21no—
352C2—21no—
353C2C2×#1798589811yescharacteristic
354C2C2×#1798604812yes—
subgroupG/H
centerC2×C4C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#423×C71C2
socleC22×C71C22×C4⋊C4
prime radicalC24×C71C2×Q8
cosocleC22×C4×C71C24
hypercenterC2×#423D71
hypocenterC71#21177
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
4. #22589116C2semidirect
166. C4×C71#430semidirect
184. C71#21177semidirect
344. C4#17986048semidirect

Maximal subgroups

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

subgroupindex
C2×#179860482
C2×#179860232
C2×C4×#117658492
#225891162
#225880672
C2×#423×C712
#2117771