Finite Group Databasebeta

← #190291

Subgroups — #190291

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34192C211yesmaximal, characteristic
2C2×#1801C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#8769C212yesmaximal, complemented
5#8768C212yesmaximal, complemented
6#8767C212yesmaximal, complemented
7#8766C212yesmaximal, complemented
8#8093C212yesmaximal
9#7960C212yesmaximal
10C22×C4×Q8C2211yescharacteristic
11#2920C2212yes—
12C2×#423C2212yes—
13C22×#101—21no—
14#2356C412yes—
15#2356C412yes—
16#2356C2211yescharacteristic
17#2356C2211yescharacteristic
18#2351C2211yescharacteristic
19#2351C2211yescharacteristic
20#2345—22no—
21#2345—21no—
22Q8×C22⋊C4—21no—
23#2320—21no—
24#2320—22no—
25#2283C412yes—
26#2283C412yes—
27#2283C2211yescharacteristic
28#2283C2211yescharacteristic
29#2276—21no—
30#1801—21no—
31#1801—21no—
32#1801C2212yes—
33#1801C2212yes—
34C2×#343—21no—
35#1467—22no—
36#1463—22no—
37#1461C2212yes—
38#1461C2212yes—
39#1303C2214yes—
40#1303C2214yes—
41#1286C2214yes—
42#1286C2214yes—
43C23×Q8D411yescharacteristic
44C22×M16—22no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D412yes—
55C2×C4×Q8C2311yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C4⋊C4D411yescharacteristic
59C22×C42C2×C412yes—
60C22×C42C2311yescharacteristic
61#432—22no—
62#432—22no—
63#431—22no—
64#431C2314yes—
65#423—22no—
66#423—22no—
67#423—22no—
68#423—22no—
69#423C2314yes—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×#101—22no—
73C2×#101—22no—
74C2×#101—44no—
75#407—22no—
76C4×M16—22no—
77#391—22no—
78#391—22no—
79#391—21no—
80#391—21no—
81#389—21no—
82#389—21no—
83#389—22no—
84#389—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#383—22no—
92#383—21no—
93#383C2×C411yescharacteristic
94#383C2×C411yescharacteristic
95#383C2×C412yes—
96#383C2×C412yes—
97#382—22no—
98#382—21no—
99#382C2×C412yes—
100#382C2×C412yes—
101#382C2311yescharacteristic
102#382C2311yescharacteristic
103#381C2×C411yescharacteristic
104#381C2×C412yes—
105#381C2×C411yescharacteristic
106#380—22no—
107#380—22no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124#376—22no—
125#376—21no—
126#376C2×C412yes—
127#376C2×C412yes—
128#376C2×C412yes—
129#343—22no—
130#343—22no—
131#324—24no—
132#324—24no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—42no—
138C22×Q8C22⋊C412yes—
139C22×Q8C2×D411yescharacteristic
140C22×Q8C2×D411yescharacteristic
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—42no—
148C2×M16—42no—
149C2×M16—44no—
150C2×M16—44no—
151C22×C8—42no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—44no—
165C4×Q8—42no—
166C4×Q8—42no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C22×C412yes—
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—42no—
216C4⋊C8—42no—
217C4⋊C8—42no—
218#101—44no—
219#101—44no—
220C22⋊C8—24no—
221C22⋊C8—42no—
222#97—42no—
223C4×C8—24no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95C22×C412yes—
243#95C22×C412yes—
244#95C22×C411yescharacteristic
245#95C22×C411yescharacteristic
246#95C22×C411yescharacteristic
247#95C22×C411yescharacteristic
248C24C2×C22⋊C411yescharacteristic
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—42no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8AGL(1,ℤ/8)12yes—
263C2×Q8#13712yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—81no—
281C22×C4—82no—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4#11711yescharacteristic
290C22×C4#11711yescharacteristic
291C22×C4#11711yescharacteristic
292C22×C4#11711yescharacteristic
293C22×C4#12312yes—
294C22×C4#12312yes—
295C22×C4#12312yes—
296C22×C4#12312yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300M16—44no—
301M16—44no—
302M16—84no—
303M16—84no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—84no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—44no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—81no—
320C4⋊C4—81no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C22⋊C4—42no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—44no—
335C42—41no—
336C42—41no—
337C42—42no—
338C42—44no—
339C42—82no—
340C42—82no—
341C42—82no—
342C42#11712yes—
343C42#11712yes—
344C42#12312yes—
345C42#12312yes—
346C42#12312yes—
347C42#12312yes—
348C42C2×#4112yes—
349C42C2×#4112yes—
350C23—22no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23#38111yescharacteristic
355C23#38111yescharacteristic
356C23C22×C22⋊C411yescharacteristic
357Q8—24no—
358Q8—24no—
359Q8—48no—
360Q8—42no—
361Q8—42no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#38111yescharacteristic
410C2×C4#38111yescharacteristic
411C2×C4#38112yes—
412C2×C4#38112yes—
413C2×C4#41714yes—
414C2×C4C2×#11711yescharacteristic
415C2×C4C2×#11711yescharacteristic
416C2×C4C2×#12312yes—
417C2×C4C2×#12312yes—
418C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
419C2×C4C2×#13711yescharacteristic
420C2×C4#57512yes—
421C8—84no—
422C8—84no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22#183012yes—
433C22C2×#38111yescharacteristic
434C22C2×#41711yescharacteristic
435C22#289611yescharacteristic
436C4—22no—
437C4—24no—
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—81no—
448C4—82no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4#309412yes—
454C4#309512yes—
455C2—21no—
456C2—21no—
457C2—22no—
458C2—41no—
459C2#2070211yescharacteristic
460C2#2070811yescharacteristic
461C2C2×#183011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
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. #8769C2semidirect
5. #8768C2semidirect
6. #8767C2semidirect
7. #8766C2semidirect

Maximal subgroups

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

subgroupindex
#341922
C2×#18012
#173652
#87692
#87682
#87672
#87662
#80932
#79602