Finite Group Databasebeta

← #99585

Subgroups — #99585

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22168C211yesmaximal, characteristic
2C4×#481C211yesmaximal, characteristic
3#11346C211yesmaximal, characteristic, complemented
4#8707C211yesmaximal, characteristic, complemented
5#7565C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic, complemented
7#7016C211yesmaximal, characteristic, complemented
8C2×#481C411yescharacteristic
9C2×C8×D4—21no—
10C4×#121C2211yescharacteristic
11C2×C4×C4⋊C4C2211yescharacteristic
12#1926—21no—
13#1923—21no—
14#1919—21no—
15#1916—21no—
16#1874C411yescharacteristic
17#1869—21no—
18#1869—21no—
19#1853C411yescharacteristic
20#1850—21no—
21#1824C411yescharacteristic
22#1797C2211yescharacteristic
23C4×#106—21no—
24C4×#102—21no—
25C4×C22⋊C8C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27C2×#336—21no—
28C2×#336—21no—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1279—21no—
32#1279—21no—
33#1271—21no—
34#1270—21no—
35C23×C8—21no—
36C23×C8—21no—
37C23×C8D411yescharacteristic
38C23×C8D411yescharacteristic
39C2×#121C2×C411yescharacteristic
40C2×C4×D4—21no—
41C2×C4×D4D411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#481—22no—
47#481C814yescomplemented
48C8×D4—42no—
49C8×D4—42no—
50C2×#106—21no—
51C2×#106—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8D411yescharacteristic
54C2×C4⋊C8D411yescharacteristic
55C2×#102—21no—
56C2×#102—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8C2×C411yescharacteristic
63C2×C22⋊C8C2×C411yescharacteristic
64C2×C4×C8—21no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67#390D411yescharacteristic
68#389D411yescharacteristic
69#388—21no—
70#386—21no—
71#380C2×C411yescharacteristic
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4D411yescharacteristic
77C4×C22⋊C4—21no—
78C2×#95—21no—
79#340—21no—
80#340—21no—
81#340—21no—
82#340—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336—21no—
91#336—21no—
92#336—42no—
93#336—42no—
94C22×D4—21no—
95C22×D4C22⋊C411yescharacteristic
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—42no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—81no—
125C22×C8—81no—
126C22×C8SD1612yes—
127#121—21no—
128#121—21no—
129#121C2×C812yes—
130#121M1612yes—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C2×C812yescomplemented
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C42—21no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168#106—24no—
169#106—42no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—42no—
173#102—24no—
174#102—42no—
175C22⋊C8—21no—
176C22⋊C8—22no—
177C22⋊C8—21no—
178C22⋊C8—21no—
179C22⋊C8—42no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—81no—
184C22⋊C8—81no—
185C22⋊C8C2×C812yescomplemented
186C4×C8—41no—
187C4×C8—41no—
188C4×C8—41no—
189C4×C8—41no—
190C4×C8—42no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—41no—
195#95—41no—
196C24—21no—
197C24C4×D411yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4C22⋊C812yes—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4C22×C811yescharacteristic
248C22×C4C2×M1611yescharacteristic
249C22×C4#13111yescharacteristic
250C22×C4#13111yescharacteristic
251C22×C4C2×SD1611yescharacteristic
252C22×C4#13511yescharacteristic
253C22×C4AGL(1,ℤ/8)11yescharacteristic
254C22×C4#13711yescharacteristic
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—81no—
280C2×C8—84no—
281C2×C8—81no—
282C2×C8—81no—
283C2×C8—81no—
284C2×C8—81no—
285C2×C8—81no—
286C2×C8—81no—
287C2×C8—81no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4C22⋊C812yes—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—81no—
319C42—81no—
320C42—81no—
321C42—81no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23#38611yescharacteristic
334C23C8×D411yescharacteristic
335C23#43511yescharacteristic
336C23C4×SD1611yescharacteristic
337C23#44011yescharacteristic
338C23#44311yescharacteristic
339C23#44411yescharacteristic
340D4—42no—
341D4—42no—
342D4—42no—
343D4—42no—
344D4—42no—
345D4—81no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
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—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4C2×C22⋊C811yescharacteristic
400C2×C4#40811yescharacteristic
401C2×C4C8×D411yescharacteristic
402C2×C4#43511yescharacteristic
403C2×C4#45011yescharacteristic
404C2×C4#45211yescharacteristic
405C2×C4#45311yescharacteristic
406C2×C4#45411yescharacteristic
407C2×C4#45811yescharacteristic
408C2×C4#45811yescharacteristic
409C2×C4#46111yescharacteristic
410C2×C4#46311yescharacteristic
411C8—42no—
412C8—42no—
413C8—81no—
414C8—82no—
415C8—82no—
416C8—81no—
417C8—81no—
418C8—81no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22C8×SD1611yescharacteristic
439C22SD16⋊C811yescharacteristic
440C22#158211yescharacteristic
441C22#158511yescharacteristic
442C22#186911yescharacteristic
443C22#187611yescharacteristic
444C22#187711yescharacteristic
445C22#188111yescharacteristic
446C22#188311yescharacteristic
447C22#189011yescharacteristic
448C22#189111yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—41no—
456C4—41no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4—82no—
461C4—82no—
462C4#161311yescharacteristic
463C4#161511yescharacteristic
464C4#161611yescharacteristic
465C4#161811yescharacteristic
466C2—21no—
467C2—21no—
468C2—21no—
469C2—21no—
470C2—82no—
471C2#894511yescharacteristic
472C2#895411yescharacteristic
473C2#896611yescharacteristic
474C2#903811yescharacteristic
475C2#904511yescharacteristic
476C2#904711yescharacteristic
477C2#905211yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11346C2semidirect
4. #8707C2semidirect
6. #7029C2semidirect
7. #7016C2semidirect
47. #481C8semidirect
149. C2×C4⋊C4C2×C8semidirect
185. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#221682
C4×#4812
#113462
#87072
#75652
#70292
#70162