Finite Group Databasebeta

← #31887975

Subgroups — #31887975

Order 18176.

710 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588102C211yesmaximal, characteristic, complemented
2C4×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
3C2×#17985723C212yesmaximal
4#22586183C212yesmaximal, complemented
5C2×#388×C71C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)C2212yes—
7C2×C22⋊(C71⋊C8)C2212yes—
8C2×#14556039—21no—
9C2×C4×C71⋊C8—21no—
10#17985754—21no—
11#17985754—21no—
12#17985723C418yescomplemented
13C22×C4⋊C4×C71C411yescharacteristic
14C22×C22⋊C4×C71C411yescharacteristic
15C22×C22⋊C4×C71C411yescharacteristic
16C22×C22⋊C4×C71C2211yescharacteristic
17C22×C42×C71C2211yescharacteristic
18#388×C71—21no—
19#388×C71—21no—
20#388×C71—21no—
21#388×C71—21no—
22C2×#95×C71C411yescharacteristic
23C2×#95×C71C2211yescharacteristic
24C22×C71⋊C8—22no—
25C22×C71⋊C8—22no—
26C22⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)C2×C414yes—
29C22⋊(C71⋊C8)C2×C414yes—
30#14556039—42no—
31C4×C71⋊C8—42no—
32C25×C71C2×C411yescharacteristic
33C23×C4×C71C2×C411yescharacteristic
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71C2311yescharacteristic
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—42no—
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—42no—
50C2×C22⋊C4×C71—42no—
51C2×C22⋊C4×C71C2×C414yes—
52C2×C42×C71—42no—
53C2×C42×C71D411yescharacteristic
54C2×C42×C71D411yescharacteristic
55C2×C42×C71D411yescharacteristic
56C2×C42×C71D411yescharacteristic
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61C2×C71⋊C8—24no—
62C2×C71⋊C8—24no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—42no—
67C24×C71—21no—
68C24×C71—21no—
69C24×C71—22no—
70C24×C71—22no—
71C24×C71—41no—
72C24×C71—41no—
73C24×C71—41no—
74C24×C71—41no—
75C24×C71C4212yes—
76C24×C71C22×C411yescharacteristic
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—22no—
80C22×C4×C71—22no—
81C22×C4×C71—22no—
82C22×C4×C71—22no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—22no—
86C22×C4×C71—22no—
87C22×C4×C71—41no—
88C22×C4×C71—41no—
89C22×C4×C71—41no—
90C22×C4×C71—41no—
91C22×C4×C71—41no—
92C22×C4×C71—41no—
93C22×C4×C71—41no—
94C22×C4×C71—41no—
95C22×C4×C71—41no—
96C22×C4×C71—41no—
97C22×C4×C71C4212yes—
98C22×C4×C71C22⋊C411yescharacteristic
99C22×C4×C71C22⋊C411yescharacteristic
100C22×C4×C71C22⋊C411yescharacteristic
101C22×C4×C71C22⋊C411yescharacteristic
102C22×C4×C71C22×C412yes—
103C22×C4×C71C2×D411yescharacteristic
104C22×C4×C71C2×D411yescharacteristic
105C22×C4×C71#4112yes—
106C4⋊C4×C71—42no—
107C4⋊C4×C71—42no—
108C22⋊C4×C71—24no—
109C22⋊C4×C71—24no—
110C22⋊C4×C71—42no—
111C22⋊C4×C71—42no—
112C22⋊C4×C71—42no—
113C22⋊C4×C71—42no—
114C42×C71—22no—
115C42×C71—22no—
116C42×C71—41no—
117C42×C71—41no—
118C23×C71—21no—
119C23×C71—21no—
120C23×C71—21no—
121C23×C71—21no—
122C23×C71—22no—
123C23×C71—24no—
124C23×C71—42no—
125C23×C71—42no—
126C23×C71—42no—
127C23×C71—42no—
128C23×C71—41no—
129C23×C71—41no—
130C23×C71—41no—
131C23×C71—41no—
132C23×C71—41no—
133C23×C71—41no—
134C23×C71—42no—
135C23×C71—82no—
136C23×C71—82no—
137C23×C71—82no—
138C23×C71—82no—
139C23×C71C2×C4211yescharacteristic
140C23×C71C2×C22⋊C411yescharacteristic
141C23×C71#11711yescharacteristic
142C23×C71#13112yes—
143C23×C71#13112yes—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—21no—
149C2×C4×C71—21no—
150C2×C4×C71—21no—
151C2×C4×C71—21no—
152C2×C4×C71—22no—
153C2×C4×C71—21no—
154C2×C4×C71—21no—
155C2×C4×C71—21no—
156C2×C4×C71—21no—
157C2×C4×C71—21no—
158C2×C4×C71—21no—
159C2×C4×C71—42no—
160C2×C4×C71—42no—
161C2×C4×C71—44no—
162C2×C4×C71—44no—
163C2×C4×C71—42no—
164C2×C4×C71—41no—
165C2×C4×C71—41no—
166C2×C4×C71—42no—
167C2×C4×C71—41no—
168C2×C4×C71—41no—
169C2×C4×C71—41no—
170C2×C4×C71—41no—
171C2×C4×C71—41no—
172C2×C4×C71—41no—
173C2×C4×C71—82no—
174C2×C4×C71—82no—
175C2×C4×C71#9912yes—
176C2×C4×C71#10012yes—
177C2×C4×C71C4×D412yes—
178C2×C4×C71C4×D412yes—
179C71⋊C8—44no—
180C71⋊C8—44no—
181C22×C71—22no—
182C22×C71—22no—
183C22×C71—22no—
184C22×C71—22no—
185C22×C71—21no—
186C22×C71—21no—
187C22×C71—21no—
188C22×C71—21no—
189C22×C71—42no—
190C22×C71—42no—
191C22×C71—42no—
192C22×C71—84no—
193C22×C71—84no—
194C22×C71—81no—
195C22×C71—81no—
196C22×C71—81no—
197C22×C71—81no—
198C22×C71C4×C22⋊C411yescharacteristic
199C22×C71#40512yes—
200C22×C71#40811yescharacteristic
201C22×C71#40811yescharacteristic
202C22×C71C2×#9911yescharacteristic
203C22×C71#41011yescharacteristic
204C22×C71C2×#10011yescharacteristic
205C22×C71#41311yescharacteristic
206C22×C71#43312yes—
207C4×C71—22no—
208C4×C71—22no—
209C4×C71—22no—
210C4×C71—22no—
211C4×C71—41no—
212C4×C71—41no—
213C4×C71—42no—
214C4×C71—81no—
215C4×C71—81no—
216#8155—711nomaximal, Sylow
217C2×C71—22no—
218C2×C71—22no—
219C2×C71—82no—
220C2×C71#146811yescharacteristic
221C2×C71#146911yescharacteristic
222C2×C71#147111yescharacteristic
223C2×C71#147311yescharacteristic
224C2×C71#175011yescharacteristic
225C2×C71C4×#9911yescharacteristic
226C2×C71C4×#10011yescharacteristic
227C2×#388D7111yescharacteristic
228#1749—711no—
229C4×C22⋊C8—711no—
230C2×#323—712no—
231#1280—712no—
232C71#815511yescharacteristic, complemented, Sylow
233C22×C4⋊C4Dic7111yescharacteristic
234C22×C22⋊C4Dic7111yescharacteristic
235C22×C22⋊C4Dic7111yescharacteristic
236C22×C22⋊C4C2×D7111yescharacteristic
237C22×C42C2×D7111yescharacteristic
238C2×C22⋊C8—712no—
239C2×C22⋊C8—712no—
240C2×#97—1421no—
241C2×C4×C8—1421no—
242#388—21no—
243#388—21no—
244#388—21no—
245#388—21no—
246C2×#95Dic7111yescharacteristic
247C2×#95C2×D7111yescharacteristic
248#336—1421no—
249#336—1421no—
250#323—718no—
251C25C2×Dic7111yescharacteristic
252C23×C4C2×Dic7111yescharacteristic
253C23×C4C2×Dic7111yescharacteristic
254C23×C4C2×Dic7111yescharacteristic
255C23×C4C2×Dic7111yescharacteristic
256C23×C4C2×Dic7111yescharacteristic
257C23×C4C22×D7111yescharacteristic
258C22×C8—1422no—
259C22×C8—1422no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—42no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4C4×D7114yes—
273C2×C42—42no—
274C2×C42C22⋊D7111yescharacteristic
275C2×C42C22⋊D7111yescharacteristic
276C2×C42C22⋊D7111yescharacteristic
277C2×C42C22⋊D7111yescharacteristic
278C22⋊C8—714no—
279C22⋊C8—714no—
280C22⋊C8—1422no—
281C22⋊C8—1422no—
282#97—2842no—
283C4×C8—2842no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288C24—21no—
289C24—21no—
290C24—22no—
291C24—22no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24C4×Dic7112yes—
297C24C22×Dic7111yescharacteristic
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4C4×Dic7112yes—
319C22×C4C22⋊Dic7111yescharacteristic
320C22×C4C22⋊Dic7111yescharacteristic
321C22×C4C22⋊Dic7111yescharacteristic
322C22×C4C22⋊Dic7111yescharacteristic
323C22×C4C2×C4×D7112yes—
324C22×C4C2×C22⋊D7111yescharacteristic
325C22×C4C2×C22⋊D7111yescharacteristic
326C22×C4#1176586512yes—
327C2×C8—2842no—
328C2×C8—2842no—
329C2×C8—2842no—
330C2×C8—2842no—
331C2×C8—1424no—
332C2×C8—1424no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42—41no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—22no—
350C23—24no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23C2×C4×Dic7111yescharacteristic
367C23C2×C22⋊Dic7111yescharacteristic
368C23#1455608711yescharacteristic
369C23#1455613512yes—
370C23#1455613512yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#1455604312yes—
403C2×C4#1455604512yes—
404C2×C4C4×C22⋊D7112yes—
405C2×C4C4×C22⋊D7112yes—
406C8—2844no—
407C8—2844no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—84no—
420C22—84no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22C4×C22⋊Dic7111yescharacteristic
426C22#1798595512yes—
427C22#1798596611yescharacteristic
428C22#1798596611yescharacteristic
429C22C2×#1455604311yescharacteristic
430C22#1798597311yescharacteristic
431C22C2×#1455604511yescharacteristic
432C22#1798598311yescharacteristic
433C22#1798606012yes—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—41no—
439C4—41no—
440C4—42no—
441C4—81no—
442C4—81no—
443C2—22no—
444C2—22no—
445C2—82no—
446C2#2258668811yescharacteristic
447C2#2258669211yescharacteristic
448C2#2258670011yescharacteristic
449C2#2258670911yescharacteristic
450C2#2258810611yescharacteristic
451C2C4×#1455604311yescharacteristic
452C2C4×#1455604511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC23×C71C2×C42
FrattiniC23×C4C22×D71
FittingC2×#388×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC25×C71C2×C4
cosocleC23×C4×C71C23
hypercenterC2×#388D71
hypocenterC71#8155
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
1. #22588102C2semidirect
2. C4×C22⋊(C71⋊C8)C2semidirect
4. #22586183C2semidirect
12. #17985723C4semidirect
232. C71#8155semidirect

Maximal subgroups

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

subgroupindex
#225881022
C4×C22⋊(C71⋊C8)2
C2×#179857232
#225861832
C2×#388×C712
#815571