Finite Group Databasebeta

← #200575

Subgroups — #200575

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#20637C211yesmaximal, characteristic
4C2×#1546C212yesmaximal, complemented
5#9402C214yesmaximal, complemented
6#9401C214yesmaximal, complemented
7#8095C212yesmaximal, complemented
8C2×#536—21no—
9C22×C4⋊Q8C2211yescharacteristic
10C2×C4×#41C2211yescharacteristic
11#2989—24no—
12#2983C2212yes—
13#2975C2212yes—
14#2973—22no—
15#2965—24no—
16#2920—22no—
17C2×#431C2212yes—
18C2×#423C2212yes—
19C2×#413—21no—
20#2857—22no—
21#2621C414yes—
22#2547—21no—
23#2547C2212yes—
24#2546—21no—
25#2546C2212yes—
26#2486—22no—
27#2486C414yes—
28#2310—21no—
29#2285C2211yescharacteristic
30#1801—21no—
31#1801C2212yes—
32#1800—21no—
33#1800C2212yes—
34#1742—21no—
35#1546—24no—
36#1546C2218yes—
37C2×#332—24no—
38#1303C2218yes—
39C22×#41D411yescharacteristic
40C23×Q8—21no—
41C22×M16—22no—
42#536—48no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2312yes—
52C2×#124—22no—
53C2×#123—24no—
54C2×C22⋊Q8—24no—
55C4×#41—22no—
56C4×#41—22no—
57C4×#41—22no—
58C4×#41—22no—
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4C2×C412yes—
62C2×C4×D4D412yes—
63C2×C4×D4D411yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×#117—21no—
66C2×#117C2×C412yes—
67C2×#117D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C412yes—
70C22×C42D412yes—
71C22×C42C2311yescharacteristic
72#446—28no—
73#436—28no—
74#435—216no—
75#432—24no—
76#432—24no—
77#431—24no—
78#431—22no—
79#431—22no—
80#431D414yes—
81#431C2314yes—
82#423—24no—
83#423—22no—
84#423—22no—
85#423—22no—
86#423C2314yes—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8D414yes—
90#413—44no—
91C2×#101—22no—
92C2×#100—22no—
93#407—24no—
94C2×C22⋊C8—22no—
95C4×M16—24no—
96C2×C4×C8—22no—
97#398—24no—
98#398C2×C418yes—
99#397—24no—
100#397C2×C418yes—
101#391—22no—
102#391—22no—
103#391—24no—
104#389—24no—
105#389—24no—
106#384—21no—
107#382C2×C412yes—
108#381—24no—
109#380—24no—
110#380—24no—
111#380—21no—
112#380C2×C412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2312yes—
115#343—22no—
116#343—22no—
117#332—216no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—24no—
121C2×#41—22no—
122C2×#41—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—44no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8C2×D411yescharacteristic
131C22×D4C22⋊C412yes—
132C22×D4C2×D411yescharacteristic
133C23×C4—22no—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×M16—28no—
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—44no—
144C2×M16—44no—
145C22×C8—24no—
146C22×C8—42no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—48no—
152C4⋊Q8—41no—
153C4⋊Q8—41no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158#124—48no—
159#123—416no—
160C22⋊Q8—416no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173#117—22no—
174#117—22no—
175#117—24no—
176#117—24no—
177#117—44no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22×C412yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—28no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221C4⋊C8C2×D418yes—
222#101—44no—
223#100—44no—
224C22⋊C8—28no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227#97—28no—
228C4×C8—24no—
229C4×C8—24no—
230#95—22no—
231#95—28no—
232#95—28no—
233#95C22×C414yes—
234C24C2×C22⋊C412yes—
235C24C2×C22⋊C411yescharacteristic
236#41—44no—
237#41—44no—
238#41—44no—
239#41—44no—
240C2×Q8—22no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—21no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—84no—
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—22no—
262C2×D4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C4×D412yes—
287C22×C4C4×D412yes—
288C22×C4#12012yes—
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4#12312yes—
292C22×C4AGL(1,ℤ/8)12yes—
293C22×C4AGL(1,ℤ/8)14yes—
294C22×C4#13712yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298M16—44no—
299M16—44no—
300M16—88no—
301M16—84no—
302C2×C8—28no—
303C2×C8—44no—
304C2×C8—84no—
305C2×C8—84no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—28no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—416no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—416no—
324C22⋊C4—44no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—44no—
332C42C4×D412yes—
333C42C4×D412yes—
334C42#12011yescharacteristic
335C42#12011yescharacteristic
336C42#12112yes—
337C42#12112yes—
338C42#12311yescharacteristic
339C42#12311yescharacteristic
340C42C22×D412yes—
341C42C2×#4112yes—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23#38612yes—
350C23#41712yes—
351C23#41812yes—
352C23C22×C22⋊C411yescharacteristic
353Q8—42no—
354Q8—42no—
355Q8—84no—
356Q8—84no—
357Q8—84no—
358Q8—84no—
359D4—48no—
360D4—42no—
361D4—42no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—88no—
390C2×C4#38611yescharacteristic
391C2×C4#38612yes—
392C2×C4#38611yescharacteristic
393C2×C4#38611yescharacteristic
394C2×C4#38611yescharacteristic
395C2×C4#41712yes—
396C2×C4#41812yes—
397C2×C4#46814yes—
398C2×C4#46814yes—
399C2×C4#46914yes—
400C2×C4#46914yes—
401C2×C4C2×C4×D412yes—
402C2×C4C2×#12011yescharacteristic
403C2×C4C2×#12112yes—
404C2×C4C2×#12311yescharacteristic
405C2×C4C2×AGL(1,ℤ/8)12yes—
406C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C8—48no—
409C8—84no—
410C22—21no—
411C22—22no—
412C22—21no—
413C22—22no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22#189514yes—
422C22C2×#38611yescharacteristic
423C22C2×#41811yescharacteristic
424C22C2×#41711yescharacteristic
425C4—21no—
426C4—22no—
427C4—21no—
428C4—22no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—84no—
433C4—84no—
434C4#189512yes—
435C4#189512yes—
436C4#191512yes—
437C4#302212yes—
438C4C2×#46812yes—
439C4C2×#46912yes—
440C2—21no—
441C2—21no—
442C2—24no—
443C2—82no—
444C2#2095511yescharacteristic
445C2C2×#189511yescharacteristic
446C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1546C2semidirect
5. #9402C2semidirect
6. #9401C2semidirect
7. #8095C2semidirect

Maximal subgroups

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

subgroupindex
#602772
#353162
#206372
C2×#15462
#94022
#94012
#80952