Finite Group Databasebeta

← #31889100

Subgroups — #31889100

Order 18176.

788 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556047C211yesmaximal, characteristic, complemented
2#22586942C212yesmaximal, complemented
3#22586783C212yesmaximal
4#22586684C211yesmaximal, characteristic, complemented
5C2×#554×C71C211yesmaximal, characteristic
6#17986056C2212yes—
7C2×#14556047—21no—
8C4×#11765849C2212yes—
9#17985768—21no—
10#17985741—24no—
11D4⋊(C71⋊C8)—24no—
12#17985725—21no—
13#17985725—21no—
14#554×C71—24no—
15#554×C71—24no—
16C2×C4⋊Q8×C71C411yescharacteristic
17C2×#125×C71C411yescharacteristic
18C2×#125×C71C2211yescharacteristic
19C2×C22⋊Q8×C71—21no—
20C2×C22⋊Q8×C71—21no—
21C2×C22⋊Q8×C71—21no—
22C2×C4×Q8×C71C2211yescharacteristic
23C2×C4×D4×C71C411yescharacteristic
24C2×C4×D4×C71C411yescharacteristic
25C2×C4×D4×C71C2211yescharacteristic
26C22×C4⋊C4×C71—21no—
27C2×#11765849—22no—
28C4⋊(C71⋊C8)—22no—
29#14556047—24no—
30C22⋊(C71⋊C8)—22no—
31#14556039—22no—
32#14556039—22no—
33C4×C71⋊C8—22no—
34C22×Q8×C71—21no—
35C22×Q8×C71D411yescharacteristic
36C22×D4×C71C2×C411yescharacteristic
37C23×C4×C71—21no—
38C23×C4×C71—21no—
39C23×C4×C71—21no—
40C4⋊Q8×C71—22no—
41C4⋊Q8×C71—22no—
42#125×C71—22no—
43#125×C71—22no—
44#125×C71—22no—
45#125×C71D414yes—
46C22⋊Q8×C71—28no—
47C22⋊Q8×C71—28no—
48C22⋊Q8×C71—28no—
49C4×Q8×C71—22no—
50C4×Q8×C71—22no—
51C4×D4×C71—22no—
52C4×D4×C71—22no—
53C4×D4×C71—22no—
54C4×D4×C71—22no—
55C4×D4×C71—22no—
56C4×D4×C71D414yes—
57C2×C4⋊C4×C71—28no—
58C2×C4⋊C4×C71—21no—
59C2×C4⋊C4×C71—21no—
60C2×C4⋊C4×C71—21no—
61C2×C4⋊C4×C71—21no—
62C2×C4⋊C4×C71—21no—
63C2×C4⋊C4×C71C2×C411yescharacteristic
64C2×C4⋊C4×C71C2×C411yescharacteristic
65C2×C4⋊C4×C71C2×C411yescharacteristic
66C2×C4⋊C4×C71D411yescharacteristic
67C2×C4⋊C4×C71D411yescharacteristic
68C2×C4⋊C4×C71D411yescharacteristic
69C2×C22⋊C4×C71—21no—
70C2×C22⋊C4×C71—21no—
71C2×C22⋊C4×C71—21no—
72C2×C42×C71C2×C411yescharacteristic
73C2×C42×C71C2×C411yescharacteristic
74C2×C42×C71C2311yescharacteristic
75#11765849—24no—
76#11765849—42no—
77C2×C71⋊C8—42no—
78C2×C71⋊C8—42no—
79C24×C71—21no—
80C2×Q8×C71—24no—
81C2×Q8×C71—22no—
82C2×Q8×C71—22no—
83C2×Q8×C71—41no—
84C2×Q8×C71—82no—
85C2×D4×C71—24no—
86C2×D4×C71—21no—
87C2×D4×C71C22⋊C412yes—
88C22×C4×C71—24no—
89C22×C4×C71—21no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71—24no—
93C22×C4×C71—24no—
94C22×C4×C71—42no—
95C22×C4×C71—42no—
96C22×C4×C71—82no—
97C22×C4×C71—82no—
98C22×C4×C71C22⋊C411yescharacteristic
99C22×C4×C71C22⋊C411yescharacteristic
100C22×C4×C71C22⋊C411yescharacteristic
101C22×C4×C71C22⋊C411yescharacteristic
102C22×C4×C71C22×C411yescharacteristic
103C22×C4×C71C2×D411yescharacteristic
104C22×C4×C71C2×D411yescharacteristic
105C4⋊C4×C71—24no—
106C4⋊C4×C71—24no—
107C4⋊C4×C71—24no—
108C4⋊C4×C71—21no—
109C4⋊C4×C71—21no—
110C4⋊C4×C71—21no—
111C4⋊C4×C71—22no—
112C4⋊C4×C71—24no—
113C4⋊C4×C71—24no—
114C4⋊C4×C71—22no—
115C4⋊C4×C71—22no—
116C4⋊C4×C71C22⋊C412yes—
117C4⋊C4×C71C22⋊C412yes—
118C4⋊C4×C71C22⋊C412yes—
119C22⋊C4×C71—24no—
120C22⋊C4×C71—24no—
121C22⋊C4×C71—24no—
122C42×C71—21no—
123C42×C71—21no—
124C42×C71—21no—
125C42×C71—21no—
126C42×C71C2×D412yes—
127C42×C71C2×D412yes—
128C23×C71—24no—
129C23×C71—22no—
130C23×C71—42no—
131C23×C71—41no—
132C23×C71—41no—
133C23×C71C2×C22⋊C411yescharacteristic
134Q8×C71—44no—
135Q8×C71—84no—
136D4×C71—24no—
137D4×C71—22no—
138D4×C71#10114yescomplemented
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—22no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—22no—
148C2×C4×C71—22no—
149C2×C4×C71—22no—
150C2×C4×C71—21no—
151C2×C4×C71—21no—
152C2×C4×C71—41no—
153C2×C4×C71—48no—
154C2×C4×C71—42no—
155C2×C4×C71—42no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—41no—
159C2×C4×C71—41no—
160C2×C4×C71—84no—
161C2×C4×C71—84no—
162C2×C4×C71#10412yes—
163C2×C4×C71C2×C22⋊C411yescharacteristic
164C2×C4×C71C2×C22⋊C411yescharacteristic
165C2×C4×C71#12012yes—
166C2×C4×C71#12012yes—
167C71⋊C8—44no—
168C71⋊C8—82no—
169C22×C71—21no—
170C22×C71—21no—
171C22×C71—24no—
172C22×C71—44no—
173C22×C71—44no—
174C22×C71—42no—
175C22×C71—42no—
176C22×C71#37911yescharacteristic
177C22×C71C2×#10411yescharacteristic
178C22×C71#42111yescharacteristic
179C4×C71—22no—
180C4×C71—21no—
181C4×C71—22no—
182C4×C71—22no—
183C4×C71—42no—
184C4×C71—42no—
185C4×C71—41no—
186C4×C71—41no—
187C4×C71—41no—
188C4×C71—42no—
189C4×C71C2×#10112yes—
190#8372—711nomaximal, Sylow
191C2×C71—21no—
192C2×C71—21no—
193C2×C71—44no—
194C2×C71#178411yescharacteristic
195C2×C71#178911yescharacteristic
196C2×C71#179511yescharacteristic
197C2×#554D7111yescharacteristic
198C4×#101—711no—
199#1528—712no—
200#1490—712no—
201#1467—711no—
202C71#837211yescharacteristic, complemented, Sylow
203#554—24no—
204#554—24no—
205C2×C4⋊Q8Dic7111yescharacteristic
206C2×#125Dic7111yescharacteristic
207C2×#125C2×D7111yescharacteristic
208C2×C22⋊Q8—21no—
209C2×C22⋊Q8—21no—
210C2×C22⋊Q8—21no—
211C2×C4×Q8C2×D7111yescharacteristic
212C2×C4×D4Dic7111yescharacteristic
213C2×C4×D4Dic7111yescharacteristic
214C2×C4×D4C2×D7111yescharacteristic
215C22×C4⋊C4—21no—
216#432—712no—
217C2×#101—1421no—
218C4×M16—712no—
219#343—1421no—
220#330—1424no—
221D4⋊C8—1424no—
222#324—1421no—
223#324—1421no—
224C22×Q8—21no—
225C22×Q8C22⋊D7111yescharacteristic
226C22×D4C2×Dic7111yescharacteristic
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C2×M16—1422no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233#125—22no—
234#125—22no—
235#125—22no—
236#125C22⋊D7114yes—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4C22⋊D7114yes—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C2×Dic7111yescharacteristic
255C2×C4⋊C4C2×Dic7111yescharacteristic
256C2×C4⋊C4C2×Dic7111yescharacteristic
257C2×C4⋊C4C22⋊D7111yescharacteristic
258C2×C4⋊C4C22⋊D7111yescharacteristic
259C2×C4⋊C4C22⋊D7111yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C42C2×Dic7111yescharacteristic
264C2×C42C2×Dic7111yescharacteristic
265C2×C42C22×D7111yescharacteristic
266C4⋊C8—1422no—
267#101—1424no—
268C22⋊C8—1422no—
269#97—1422no—
270#97—1422no—
271C4×C8—1422no—
272C24—21no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—82no—
278C2×D4—24no—
279C2×D4—21no—
280C2×D4C22⋊Dic7112yes—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C22⋊Dic7111yescharacteristic
292C22×C4C22⋊Dic7111yescharacteristic
293C22×C4C22⋊Dic7111yescharacteristic
294C22×C4C22⋊Dic7111yescharacteristic
295C22×C4C22×Dic7111yescharacteristic
296C22×C4C2×C22⋊D7111yescharacteristic
297C22×C4C2×C22⋊D7111yescharacteristic
298M16—2842no—
299M16—1424no—
300C2×C8—2842no—
301C2×C8—2842no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4C22⋊Dic7112yes—
314C4⋊C4C22⋊Dic7112yes—
315C4⋊C4C22⋊Dic7112yes—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42C2×C22⋊D7112yes—
324C42C2×C22⋊D7112yes—
325C23—24no—
326C23—22no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23C2×C22⋊Dic7111yescharacteristic
331Q8—44no—
332Q8—84no—
333D4—24no—
334D4—22no—
335D4#1455604714yescomplemented
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—48no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4#1455605512yes—
360C2×C4C2×C22⋊Dic7111yescharacteristic
361C2×C4C2×C22⋊Dic7111yescharacteristic
362C2×C4#1455609712yes—
363C2×C4#1455609712yes—
364C8—2844no—
365C8—5682no—
366C22—21no—
367C22—21no—
368C22—24no—
369C22—44no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22#1798585511yescharacteristic
374C22C2×#1455605511yescharacteristic
375C22#1798601611yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4C2×#1455604712yes—
387C2—21no—
388C2—21no—
389C2—44no—
390C2#2258826411yescharacteristic
391C2#2258828411yescharacteristic
392C2#2258831011yescharacteristic
subgroupG/H
centerC22#17985855
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#554×C71C2
socleC22×C71#379
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenterC2×#554D71
hypocenterC71#8372
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C4×#14556047C2semidirect
2. #22586942C2semidirect
4. #22586684C2semidirect
138. D4×C71#101semidirect
202. C71#8372semidirect
335. D4#14556047semidirect

Maximal subgroups

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

subgroupindex
C4×#145560472
#225869422
#225867832
#225866842
C2×#554×C712
#837271