Finite Group Databasebeta

← #31896286

Subgroups — #31896286

Order 18176.

720 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588320C211yesmaximal, characteristic, complemented
2C4×C22⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
3C2×#17985725C212yesmaximal, complemented
4C2×#17985723C212yesmaximal
5C2×#388×C71C211yesmaximal, characteristic
6C2×C4⋊(C71⋊C8)—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8C2×C22⋊(C71⋊C8)C2212yes—
9C2×C4×C71⋊C8—21no—
10#17985754—21no—
11#17985754—21no—
12#17985725—24no—
13#17985723—24no—
14C22×C4⋊C4×C71C2211yescharacteristic
15C22×C22⋊C4×C71C411yescharacteristic
16C22×C22⋊C4×C71C411yescharacteristic
17C22×C22⋊C4×C71C2211yescharacteristic
18C22×C42×C71C2211yescharacteristic
19#388×C71—21no—
20#388×C71—21no—
21#388×C71—21no—
22#388×C71—21no—
23C2×#95×C71C411yescharacteristic
24C2×#95×C71C411yescharacteristic
25C22×C71⋊C8—22no—
26C22×C71⋊C8—22no—
27C4⋊(C71⋊C8)—42no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)D414yes—
32C4×C71⋊C8—42no—
33C25×C71C2×C411yescharacteristic
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71C2×C411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—22no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—22no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—22no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—22no—
53C2×C22⋊C4×C71—42no—
54C2×C22⋊C4×C71C814yescomplemented
55C2×C42×C71—42no—
56C2×C42×C71D411yescharacteristic
57C2×C42×C71D411yescharacteristic
58C2×C42×C71D411yescharacteristic
59C2×C42×C71D411yescharacteristic
60#95×C71—21no—
61#95×C71—21no—
62#95×C71—21no—
63#95×C71—21no—
64C2×C71⋊C8—24no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—42no—
67C2×C71⋊C8—42no—
68C2×C71⋊C8—42no—
69C2×C71⋊C8—42no—
70C24×C71—21no—
71C24×C71—21no—
72C24×C71—22no—
73C24×C71—22no—
74C24×C71—41no—
75C24×C71—41no—
76C24×C71—41no—
77C24×C71—41no—
78C24×C71C2×C812yes—
79C24×C71C22×C411yescharacteristic
80C22×C4×C71—21no—
81C22×C4×C71—22no—
82C22×C4×C71—21no—
83C22×C4×C71—22no—
84C22×C4×C71—22no—
85C22×C4×C71—22no—
86C22×C4×C71—21no—
87C22×C4×C71—21no—
88C22×C4×C71—22no—
89C22×C4×C71—22no—
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×C71—41no—
98C22×C4×C71—41no—
99C22×C4×C71C22⋊C411yescharacteristic
100C22×C4×C71C22⋊C411yescharacteristic
101C22×C4×C71C22⋊C411yescharacteristic
102C22×C4×C71C22⋊C411yescharacteristic
103C22×C4×C71C2×C812yes—
104C22×C4×C71C2×C812yes—
105C22×C4×C71M1612yes—
106C22×C4×C71C2×D412yes—
107C22×C4×C71C2×D411yescharacteristic
108C22×C4×C71C2×D411yescharacteristic
109C22×C4×C71#4112yes—
110C4⋊C4×C71—42no—
111C4⋊C4×C71—41no—
112C4⋊C4×C71—41no—
113C22⋊C4×C71—24no—
114C22⋊C4×C71—24no—
115C22⋊C4×C71—42no—
116C22⋊C4×C71—42no—
117C22⋊C4×C71—42no—
118C22⋊C4×C71—41no—
119C22⋊C4×C71—41no—
120C42×C71—22no—
121C42×C71—22no—
122C42×C71—41no—
123C42×C71—41no—
124C23×C71—21no—
125C23×C71—21no—
126C23×C71—21no—
127C23×C71—21no—
128C23×C71—22no—
129C23×C71—24no—
130C23×C71—42no—
131C23×C71—42no—
132C23×C71—42no—
133C23×C71—42no—
134C23×C71—41no—
135C23×C71—41no—
136C23×C71—41no—
137C23×C71—41no—
138C23×C71—41no—
139C23×C71—41no—
140C23×C71—42no—
141C23×C71—82no—
142C23×C71—82no—
143C23×C71—82no—
144C23×C71—82no—
145C23×C71C2×C22⋊C411yescharacteristic
146C23×C71C4×D412yes—
147C23×C71C22×C811yescharacteristic
148C23×C71C2×M1611yescharacteristic
149C23×C71#13112yes—
150C2×C4×C71—21no—
151C2×C4×C71—21no—
152C2×C4×C71—21no—
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—21no—
160C2×C4×C71—22no—
161C2×C4×C71—21no—
162C2×C4×C71—21no—
163C2×C4×C71—21no—
164C2×C4×C71—24no—
165C2×C4×C71—42no—
166C2×C4×C71—42no—
167C2×C4×C71—42no—
168C2×C4×C71—42no—
169C2×C4×C71—44no—
170C2×C4×C71—41no—
171C2×C4×C71—41no—
172C2×C4×C71—42no—
173C2×C4×C71—41no—
174C2×C4×C71—41no—
175C2×C4×C71—41no—
176C2×C4×C71—41no—
177C2×C4×C71—41no—
178C2×C4×C71—41no—
179C2×C4×C71—82no—
180C2×C4×C71—82no—
181C2×C4×C71C22⋊C812yes—
182C2×C4×C71C22⋊C812yes—
183C2×C4×C71#9912yes—
184C2×C4×C71#12112yes—
185C2×C4×C71#12411yescharacteristic
186C2×C4×C71#12711yescharacteristic
187C71⋊C8—44no—
188C71⋊C8—82no—
189C22×C71—22no—
190C22×C71—22no—
191C22×C71—21no—
192C22×C71—21no—
193C22×C71—21no—
194C22×C71—21no—
195C22×C71—22no—
196C22×C71—22no—
197C22×C71—42no—
198C22×C71—42no—
199C22×C71—42no—
200C22×C71—84no—
201C22×C71—84no—
202C22×C71—81no—
203C22×C71—81no—
204C22×C71—81no—
205C22×C71—81no—
206C22×C71#39011yescharacteristic
207C22×C71C2×C22⋊C811yescharacteristic
208C22×C71#40811yescharacteristic
209C22×C71C2×#9911yescharacteristic
210C22×C71#41011yescharacteristic
211C22×C71#41311yescharacteristic
212C22×C71#41311yescharacteristic
213C22×C71C8×D412yes—
214C22×C71#43612yes—
215C4×C71—22no—
216C4×C71—21no—
217C4×C71—21no—
218C4×C71—22no—
219C4×C71—22no—
220C4×C71—22no—
221C4×C71—42no—
222C4×C71—41no—
223C4×C71—81no—
224C4×C71—81no—
225#9651—711nomaximal, Sylow
226C2×C71—22no—
227C2×C71—22no—
228C2×C71—82no—
229C2×C71#145811yescharacteristic
230C2×C71#145911yescharacteristic
231C2×C71#146811yescharacteristic
232C2×C71#147311yescharacteristic
233C2×C71#195411yescharacteristic
234C2×C71#196411yescharacteristic
235C2×C71#196711yescharacteristic
236C2×#388D7111yescharacteristic
237#1797—711no—
238C4×C22⋊C8—711no—
239C2×#324—712no—
240C2×#323—712no—
241C71#965111yescharacteristic, complemented, Sylow
242C22×C4⋊C4C2×D7111yescharacteristic
243C22×C22⋊C4Dic7111yescharacteristic
244C22×C22⋊C4Dic7111yescharacteristic
245C22×C22⋊C4C2×D7111yescharacteristic
246C22×C42C2×D7111yescharacteristic
247C2×C4⋊C8—1421no—
248C2×C22⋊C8—712no—
249C2×C22⋊C8—712no—
250C2×C4×C8—1421no—
251#388—21no—
252#388—21no—
253#388—21no—
254#388—21no—
255C2×#95Dic7111yescharacteristic
256C2×#95Dic7111yescharacteristic
257#336—1421no—
258#336—1421no—
259#324—1424no—
260#323—1424no—
261C25C2×Dic7111yescharacteristic
262C23×C4C2×Dic7111yescharacteristic
263C23×C4C2×Dic7111yescharacteristic
264C23×C4C2×Dic7111yescharacteristic
265C23×C4C2×Dic7111yescharacteristic
266C23×C4C2×Dic7111yescharacteristic
267C23×C4C22×D7111yescharacteristic
268C22×C8—1422no—
269C22×C8—1422no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4C71⋊C814yescomplemented
285C2×C42—42no—
286C2×C42C22⋊D7111yescharacteristic
287C2×C42C22⋊D7111yescharacteristic
288C2×C42C22⋊D7111yescharacteristic
289C2×C42C22⋊D7111yescharacteristic
290C4⋊C8—2842no—
291C22⋊C8—714no—
292C22⋊C8—1422no—
293C22⋊C8—1422no—
294C22⋊C8—1422no—
295C4×C8—2842no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—22no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24C2×C71⋊C812yes—
309C24C22×Dic7111yescharacteristic
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4C22⋊Dic7111yescharacteristic
330C22×C4C22⋊Dic7111yescharacteristic
331C22×C4C22⋊Dic7111yescharacteristic
332C22×C4C22⋊Dic7111yescharacteristic
333C22×C4C2×C71⋊C812yes—
334C22×C4C2×C71⋊C812yes—
335C22×C4#1176584912yes—
336C22×C4D4×D7112yes—
337C22×C4C2×C22⋊D7111yescharacteristic
338C22×C4C2×C22⋊D7111yescharacteristic
339C22×C4#1176586412yes—
340C2×C8—2842no—
341C2×C8—2842no—
342C2×C8—2842no—
343C2×C8—2842no—
344C2×C8—2842no—
345C2×C8—1424no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C42—22no—
357C42—22no—
358C42—41no—
359C42—41no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—22no—
365C23—24no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23C2×C22⋊Dic7111yescharacteristic
382C23D4×Dic7112yes—
383C23C22×C71⋊C811yescharacteristic
384C23C2×#1176584911yescharacteristic
385C23#1455613512yes—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4C22⋊(C71⋊C8)12yes—
418C2×C4C22⋊(C71⋊C8)12yes—
419C2×C4#1455604312yes—
420C2×C4#1455609912yes—
421C2×C4#1455611511yescharacteristic
422C2×C4#1455612411yescharacteristic
423C8—2844no—
424C8—5682no—
425C22—22no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—84no—
437C22—84no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22#1798590211yescharacteristic
443C22C2×C22⋊(C71⋊C8)11yescharacteristic
444C22#1798596611yescharacteristic
445C22C2×#1455604311yescharacteristic
446C22#1798597311yescharacteristic
447C22#1798598311yescharacteristic
448C22#1798598311yescharacteristic
449C22D4×C71⋊C812yes—
450C22#1798607712yes—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—21no—
456C4—21no—
457C4—41no—
458C4—42no—
459C4—81no—
460C4—81no—
461C2—22no—
462C2—22no—
463C2—82no—
464C2#2258665411yescharacteristic
465C2#2258665811yescharacteristic
466C2#2258668811yescharacteristic
467C2#2258670911yescharacteristic
468C2#2258913511yescharacteristic
469C2#2258917711yescharacteristic
470C2#2258919011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC23×C71C22×C8
FrattiniC23×C4C22×D71
FittingC2×#388×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC25×C71C2×C4
cosocleC23×C4×C71C23
hypercenterC2×#388D71
hypocenterC71#9651
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. #22588320C2semidirect
2. C4×C22⋊(C71⋊C8)C2semidirect
3. C2×#17985725C2semidirect
54. C2×C22⋊C4×C71C8semidirect
241. C71#9651semidirect
284. C2×C22⋊C4C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225883202
C4×C22⋊(C71⋊C8)2
C2×#179857252
C2×#179857232
C2×#388×C712
#965171