Finite Group Databasebeta

← #117798

Subgroups — #117798

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21856C211yesmaximal, characteristic
2C2×#1837C211yesmaximal, characteristic, complemented
3#17481C211yesmaximal, characteristic
4#17473C211yesmaximal, characteristic
5#8097C211yesmaximal, characteristic, complemented
6#7934C211yesmaximal, characteristic, complemented
7#7890C211yesmaximal, characteristic, complemented
8#3444C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1994—21no—
16#1962—21no—
17#1871—22no—
18#1837—22no—
19#1837—22no—
20#1806—22no—
21C2×#341—22no—
22#1495—22no—
23#1494—22no—
24#1493—21no—
25#1493C411yescharacteristic, complemented
26#1493C411yescharacteristic, complemented
27#1492—21no—
28#1492C411yescharacteristic, complemented
29#1492C411yescharacteristic, complemented
30#1477—22no—
31#1476—22no—
32#1303—22no—
33#1283—22no—
34#1279—22no—
35C22×M16—21no—
36C22×M16D412yes—
37C23×C8—21no—
38#533C2×C411yescharacteristic
39#533D411yescharacteristic
40#533Q811yescharacteristic
41#532C2×C411yescharacteristic
42#532D411yescharacteristic
43#532Q811yescharacteristic
44#529—21no—
45#529—21no—
46#529—21no—
47#529—21no—
48#529—21no—
49#529—21no—
50#525—21no—
51#525—21no—
52#525—21no—
53C4×#41—21no—
54C4×#41—21no—
55C4×#41—21no—
56C2×#117—21no—
57C22×C42C2311yescharacteristic
58#432—21no—
59#432—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431D412yes—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423C2×C411yescharacteristic
74#423C2×C411yescharacteristic
75#423D412yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×#104—24no—
82#407—21no—
83C2×C22⋊C8—21no—
84C4×M16—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2×C411yescharacteristic
90C2×C4×C8C2×C411yescharacteristic
91C2×C4×C8D412yes—
92C2×C4×C8D412yes—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97#376—21no—
98C43—21no—
99#341—28no—
100#339—24no—
101Q8⋊C8—24no—
102Q8⋊C8—22no—
103Q8⋊C8—22no—
104D4⋊C8—24no—
105D4⋊C8—22no—
106D4⋊C8—22no—
107C2×#41—21no—
108C2×#41—21no—
109C2×#41—21no—
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—41no—
119C2×M16—44no—
120C2×M16—44no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8—41no—
129C22×C8—41no—
130C4⋊Q8C4⋊C411yescharacteristic
131C4⋊Q8C4⋊C411yescharacteristic
132C4⋊Q8C4⋊C411yescharacteristic
133#127C4⋊C411yescharacteristic
134#126—41no—
135#126—41no—
136#125—21no—
137#125—21no—
138#125—21no—
139#124—21no—
140#124—21no—
141#124—21no—
142#123—41no—
143#123—41no—
144#123—41no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151#121—21no—
152#121—21no—
153#121—21no—
154#121—21no—
155#121—21no—
156#121—21no—
157#120—41no—
158C4×Q8—21no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176#117—24no—
177#117—21no—
178#117—21no—
179#117—21no—
180#117—42no—
181#117—41no—
182#117—41no—
183#117—41no—
184C2×C4⋊C4—22no—
185C2×C22⋊C4—41no—
186C2×C42—22no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×Q811yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216#104—84no—
217C22⋊C8—22no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220C22⋊C8—41no—
221#97—41no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—41no—
228C4×C8—41no—
229C4×C8C22⋊C412yes—
230C4×C8C22⋊C412yes—
231#95—22no—
232#95—41no—
233C24#12411yescharacteristic
234#41—44no—
235#41—42no—
236#41—42no—
237#41—42no—
238#41—42no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4#11711yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4C22⋊Q811yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4#12311yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12411yescharacteristic
273C22×C4#12411yescharacteristic
274C22×C4#12611yescharacteristic
275M16—44no—
276M16—44no—
277M16—42no—
278M16—42no—
279M16—44no—
280M16—82no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—44no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—82no—
294C2×C8—82no—
295C2×C8—82no—
296C4⋊C4—24no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—44no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42—44no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—42no—
325C42—42no—
326C42—82no—
327C42C2×C22⋊C411yescharacteristic
328C42C2×C4⋊C411yescharacteristic
329C42C4×D411yescharacteristic
330C42C4×D411yescharacteristic
331C42#12011yescharacteristic
332C42#12111yescharacteristic
333C42#12111yescharacteristic
334C42#12111yescharacteristic
335C42#12111yescharacteristic
336C42C22⋊Q811yescharacteristic
337C42C22⋊Q811yescharacteristic
338C42C22⋊Q811yescharacteristic
339C42#12311yescharacteristic
340C42#12311yescharacteristic
341C42#12611yescharacteristic
342C42#12711yescharacteristic
343C23—22no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23#38811yescharacteristic
348C23#39311yescharacteristic
349C23#39411yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—42no—
353D4—42no—
354D4—42no—
355D4—42no—
356D4—82no—
357D4—82no—
358D4—82no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4#38011yescharacteristic
391C2×C4#38511yescharacteristic
392C2×C4#38511yescharacteristic
393C2×C4#38611yescharacteristic
394C2×C4#38811yescharacteristic
395C2×C4#39011yescharacteristic
396C2×C4#39211yescharacteristic
397C2×C4#39411yescharacteristic
398C2×C4#39611yescharacteristic
399C2×C4#39611yescharacteristic
400C2×C4#39711yescharacteristic
401C2×C4#39911yescharacteristic
402C2×C4#44312yes—
403C2×C4#44412yes—
404C2×C4#45412yes—
405C2×C4#45512yes—
406C8—42no—
407C8—42no—
408C8—82no—
409C8—81no—
410C8—81no—
411C8—82no—
412C8—82no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—44no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22#143911yescharacteristic
425C22#192912yes—
426C22#204012yes—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—84no—
435C4—81no—
436C4—81no—
437C4—81no—
438C4#181412yes—
439C4#186012yes—
440C4#186612yes—
441C4#197411yescharacteristic
442C4#197511yescharacteristic
443C4#200311yescharacteristic
444C4#200411yescharacteristic
445C2—22no—
446C2—22no—
447C2—41no—
448C2—161no—
449C2#1107911yescharacteristic
450C2#1110811yescharacteristic
451C2#1112511yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radical#3444C22
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
2. C2×#1837C2semidirect
5. #8097C2semidirect
6. #7934C2semidirect
7. #7890C2semidirect
25. #1493C4semidirect
26. #1493C4semidirect
28. #1492C4semidirect
29. #1492C4semidirect

Maximal subgroups

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

subgroupindex
#218562
C2×#18372
#174812
#174732
#80972
#79342
#78902