Finite Group Databasebeta

← #192207

Subgroups — #192207

Order 512.

887 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#21994C212yesmaximal, complemented
3#21849C211yesmaximal, characteristic
4#17363C211yesmaximal, characteristic
5#8860C214yesmaximal, complemented
6#8859C214yesmaximal, complemented
7#8058C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2989—22no—
10#2989—22no—
11#2983—21no—
12#2975C2212yes—
13#2973—21no—
14#2973C2212yes—
15#2965—24no—
16#2920—21no—
17#2920C2212yes—
18C2×#432C2212yes—
19C2×#431—21no—
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#1984—21no—
23#1954—21no—
24#1950—24no—
25#1870C2212yes—
26#1870C2212yes—
27#1855C2218yes—
28#1849—22no—
29#1838—22no—
30#1797—21no—
31#1797—21no—
32#1467C412yes—
33#1466C412yes—
34#1464C412yes—
35#1464C412yes—
36#1459C2212yes—
37#1459C2212yes—
38#1298C2218yes—
39C22×#41Q811yescharacteristic
40C22×M16C2312yes—
41C4×#41—22no—
42C4×#41—22no—
43C4×#41—22no—
44C4×#41—22no—
45C2×C4×Q8C2×C411yescharacteristic
46C2×C4×Q8D411yescharacteristic
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4Q811yescharacteristic
50C2×C4×D4Q811yescharacteristic
51C2×C4×D4C2311yescharacteristic
52C2×C4×D4C2311yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117Q811yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59#446—24no—
60#446—24no—
61#436—24no—
62#436—24no—
63#435—24no—
64#435—24no—
65#435—24no—
66#435—24no—
67#432—22no—
68#432—22no—
69#432—22no—
70#432—22no—
71#432—22no—
72#432—22no—
73#432C2×C414yes—
74#432C2×C414yes—
75#431—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83C2×#108—28no—
84#423—21no—
85#423—21no—
86#423—22no—
87#423—22no—
88#423D412yes—
89#423D412yes—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96#407—22no—
97#407C2314yes—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8C2312yes—
102C2×C22⋊C8C2314yes—
103C4×M16—22no—
104C4×M16D414yes—
105C2×#97—22no—
106C2×C4×C8—22no—
107C2×C4×C8—21no—
108#341—28no—
109#336—22no—
110#336—22no—
111#324C2×C414yes—
112#324C2×C414yes—
113#323C2×C414yes—
114#323C2×C414yes—
115C2×#41—22no—
116C2×#41—22no—
117C2×#41—42no—
118C2×#41—42no—
119C22×Q8C4⋊C411yescharacteristic
120C22×D4C4⋊C411yescharacteristic
121C22×D4C2×Q811yescharacteristic
122C22×D4C2×Q811yescharacteristic
123C23×C4C4⋊C411yescharacteristic
124C23×C4C4⋊C411yescharacteristic
125C23×C4C4⋊C411yescharacteristic
126C23×C4C4⋊C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×Q811yescharacteristic
130C23×C4C2×Q811yescharacteristic
131C23×C4C2411yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—24no—
135C2×M16—82no—
136C2×M16C2×D414yes—
137C2×M16C2×D414yes—
138C22×C8—24no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163#117—22no—
164#117—22no—
165C2×C4⋊C4C4⋊C411yescharacteristic
166C2×C4⋊C4C4⋊C411yescharacteristic
167C2×C4⋊C4C4⋊C411yescharacteristic
168C2×C4⋊C4C4⋊C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C22⋊C4C4⋊C411yescharacteristic
172C2×C22⋊C4C4⋊C411yescharacteristic
173C2×C22⋊C4C22×C411yescharacteristic
174C2×C22⋊C4C22×C411yescharacteristic
175C2×C22⋊C4C2×Q811yescharacteristic
176C2×C22⋊C4C2×Q811yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42C4⋊C411yescharacteristic
186C2×C42C4⋊C411yescharacteristic
187C2×C42C4⋊C411yescharacteristic
188C2×C42C4⋊C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42#4112yes—
198C2×C42#4112yes—
199#108—88no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—41no—
211C4⋊C8—42no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—82no—
222C22⋊C8—82no—
223C22⋊C8C22×C418yes—
224#97—24no—
225#97—24no—
226#97—24no—
227#97—24no—
228C4×C8—24no—
229C4×C8—24no—
230C4×C8—42no—
231C24C2×C4⋊C411yescharacteristic
232C24C2×C4⋊C411yescharacteristic
233C24C22×Q811yescharacteristic
234#41—48no—
235#41—48no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4C2×C4⋊C411yescharacteristic
276C22×C4C2×C4⋊C411yescharacteristic
277C22×C4C2×C4⋊C411yescharacteristic
278C22×C4C2×C4⋊C411yescharacteristic
279C22×C4C2×C4⋊C411yescharacteristic
280C22×C4C2×C4⋊C411yescharacteristic
281C22×C4C2×C4⋊C411yescharacteristic
282C22×C4C2×C4⋊C411yescharacteristic
283C22×C4C2×C4⋊C411yescharacteristic
284C22×C4C2×C4⋊C411yescharacteristic
285C22×C4C22⋊Q812yes—
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4C22⋊Q812yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D412yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4112yes—
293M16—44no—
294M16—44no—
295M16—48no—
296M16—82no—
297M16—82no—
298C2×C8—28no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—44no—
302C2×C8—44no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—82no—
310C2×C8—82no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42C4×D412yes—
336C42C4×D412yes—
337C42C4×D412yes—
338C42C4×D412yes—
339C42#12011yescharacteristic
340C42#12012yes—
341C42#12011yescharacteristic
342C42#12312yes—
343C42#12311yescharacteristic
344C42#12311yescharacteristic
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23#43012yes—
359C23C22×C4⋊C411yescharacteristic
360C23C2×C22⋊Q812yes—
361Q8—44no—
362D4—44no—
363D4—44no—
364D4—44no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4#38511yescharacteristic
409C2×C4#38511yescharacteristic
410C2×C4#38511yescharacteristic
411C2×C4#38512yes—
412C2×C4#38512yes—
413C2×C4#38511yescharacteristic
414C2×C4#43012yes—
415C2×C4C2×C4×D412yes—
416C2×C4C2×#12011yescharacteristic
417C2×C4C2×#12311yescharacteristic
418C8—48no—
419C8—44no—
420C8—44no—
421C8—82no—
422C8—82no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—42no—
430C22—42no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22C2×#38511yescharacteristic
441C22C2×#43011yescharacteristic
442C22C2×#43011yescharacteristic
443C4—21no—
444C4—21no—
445C4—22no—
446C4—22no—
447C4—21no—
448C4—22no—
449C4—22no—
450C4—21no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4#185812yes—
457C4#302711yescharacteristic
458C4#302811yescharacteristic
459C2—21no—
460C2—21no—
461C2—42no—
462C2—42no—
463C2—81no—
464C2#2085111yescharacteristic
465C2C2×#185811yescharacteristic
466C2#2089011yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC22×#41Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21994C2semidirect
5. #8860C2semidirect
6. #8859C2semidirect
7. #8058C2semidirect

Maximal subgroups

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

subgroupindex
#602772
#219942
#218492
#173632
#88602
#88592
#80582