Finite Group Databasebeta

← #117463

Subgroups — #117463

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21815C211yesmaximal, characteristic
2#21187C211yesmaximal, characteristic
3#17817C211yesmaximal, characteristic, complemented
4#17714C211yesmaximal, characteristic
5#8097C211yesmaximal, characteristic, complemented
6#7890C211yesmaximal, characteristic, complemented
7#7885C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#408—22no—
14C22×C4×C8C2211yescharacteristic
15C4×#117C2211yescharacteristic
16#1980—22no—
17#1962—21no—
18#1962—21no—
19#1954—22no—
20#1954—22no—
21#1789—22no—
22#1574—22no—
23#1573—22no—
24#1567—21no—
25#1566—21no—
26#1551—22no—
27#1548C411yescharacteristic, complemented
28#1548C411yescharacteristic, complemented
29#1545C411yescharacteristic, complemented
30#1545C411yescharacteristic, complemented
31#1543—22no—
32#1303—22no—
33#1279—22no—
34#1278—22no—
35C22×#41D412yes—
36C22×M16—21no—
37C23×C8—21no—
38#532—21no—
39#532—21no—
40#532—21no—
41#532—21no—
42#532—21no—
43#532—21no—
44#532—21no—
45#532—21no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#127C2×C411yescharacteristic
48C2×#124Q812yes—
49C2×#121—21no—
50C2×#121—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×#117—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C411yescharacteristic
66#431C2×C411yescharacteristic
67#431D412yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423C2×C411yescharacteristic
73#423C2×C411yescharacteristic
74#423D411yescharacteristic
75#423D411yescharacteristic
76C2×C4⋊C8—21no—
77C2×C4⋊C8D412yes—
78C2×C4⋊C8D412yes—
79C2×#104—24no—
80C2×#104—24no—
81#408—48no—
82#407—21no—
83#407—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—28no—
86C4×M16—21no—
87C4×M16—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8D412yes—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D412yes—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99#376—21no—
100C43—21no—
101#335—24no—
102#334—24no—
103#333—22no—
104#333—22no—
105#331—22no—
106#331—22no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—22no—
111C2×#41—44no—
112C2×#41—44no—
113C22×Q8C4⋊C412yes—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4C4⋊C412yes—
118C23×C4—22no—
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—44no—
127C2×M16—41no—
128C2×M16—41no—
129C22×C8—24no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—42no—
134C22×C8—41no—
135C22×C8—41no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140#127—21no—
141#127—21no—
142#127—21no—
143#127—21no—
144#124—22no—
145#124—22no—
146#124—22no—
147#124—22no—
148#121—41no—
149#121—41no—
150#121—41no—
151#121—41no—
152#121—41no—
153#121—41no—
154#121—41no—
155#121—41no—
156C4×D4—41no—
157C4×D4—41no—
158C4×D4—41no—
159C4×D4—41no—
160C4×D4—41no—
161C4×D4—41no—
162C4×D4—41no—
163C4×D4—41no—
164#117—24no—
165#117—42no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—41no—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×Q811yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8C22⋊C412yes—
201C4⋊C8C22⋊C412yes—
202#104—84no—
203#104—84no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—22no—
207C22⋊C8—416no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212#97—41no—
213#97—41no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—41no—
220C4×C8—41no—
221C4×C8—41no—
222C4×C8—41no—
223#95—22no—
224#95—41no—
225C24—22no—
226C24#12311yescharacteristic
227#41—44no—
228#41—44no—
229#41—84no—
230#41—84no—
231#41—82no—
232#41—82no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—48no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—42no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4C2×C4⋊C411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4#12011yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4C22⋊Q811yescharacteristic
270C22×C4C22⋊Q811yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4#12311yescharacteristic
274C22×C4#12411yescharacteristic
275C22×C4#12411yescharacteristic
276C22×C4#12611yescharacteristic
277C22×C4#12611yescharacteristic
278M16—42no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—82no—
283M16—82no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—44no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—82no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C42—44no—
310C42—42no—
311C42—42no—
312C42—82no—
313C42C2×C22⋊C411yescharacteristic
314C42#11711yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42#12011yescharacteristic
318C42#12111yescharacteristic
319C42#12111yescharacteristic
320C42#12111yescharacteristic
321C42#12111yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42C22⋊Q811yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42C22⋊Q811yescharacteristic
326C42#12311yescharacteristic
327C42#12411yescharacteristic
328C42#12711yescharacteristic
329C23—22no—
330C23—22no—
331C23—41no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—42no—
336C23#38511yescharacteristic
337C23#39411yescharacteristic
338C23#39711yescharacteristic
339Q8—44no—
340Q8—82no—
341D4—44no—
342D4—82no—
343D4—82no—
344D4—84no—
345D4—84no—
346D4—82no—
347D4—82no—
348D4—81no—
349D4—81no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4#38011yescharacteristic
379C2×C4#38511yescharacteristic
380C2×C4#38611yescharacteristic
381C2×C4#38811yescharacteristic
382C2×C4#38811yescharacteristic
383C2×C4#39011yescharacteristic
384C2×C4#39211yescharacteristic
385C2×C4#39311yescharacteristic
386C2×C4#39411yescharacteristic
387C2×C4#39611yescharacteristic
388C2×C4#39611yescharacteristic
389C2×C4#39911yescharacteristic
390C2×C4#44312yes—
391C2×C4#44412yes—
392C2×C4#45312yes—
393C2×C4#45512yes—
394C8—44no—
395C8—42no—
396C8—42no—
397C8—81no—
398C8—81no—
399C8—82no—
400C8—81no—
401C8—81no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—44no—
410C22—88no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22#143911yescharacteristic
415C22#186612yes—
416C22#203712yes—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—84no—
425C4—82no—
426C4#185612yes—
427C4#188112yes—
428C4#192912yes—
429C4#197411yescharacteristic
430C4#197511yescharacteristic
431C4#200211yescharacteristic
432C4#200411yescharacteristic
433C2—22no—
434C2—22no—
435C2—41no—
436C2—162no—
437C2#1106511yescharacteristic
438C2#1112511yescharacteristic
439C2#1113511yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17817C2semidirect
5. #8097C2semidirect
6. #7890C2semidirect
7. #7885C2semidirect
27. #1548C4semidirect
28. #1548C4semidirect
29. #1545C4semidirect
30. #1545C4semidirect

Maximal subgroups

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

subgroupindex
#218152
#211872
#178172
#177142
#80972
#78902
#78852