Finite Group Databasebeta

← #190267

Subgroups — #190267

Order 512.

864 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34192C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×#1461C211yesmaximal, characteristic
4#8769C212yesmaximal, complemented
5#8768C212yesmaximal, complemented
6#8767C212yesmaximal, complemented
7#8766C212yesmaximal, complemented
8#8096C212yesmaximal
9#7956C212yesmaximal
10C22×C4×Q8C2211yescharacteristic
11#2920C2212yes—
12C2×#431C2212yes—
13#2356C412yes—
14#2356C412yes—
15#2356C2211yescharacteristic
16#2356C2211yescharacteristic
17#2351C2211yescharacteristic
18#2351C2211yescharacteristic
19#2345—21no—
20#2345—22no—
21Q8×C22⋊C4—21no—
22#2320—21no—
23#2320—22no—
24#2283C412yes—
25#2283C412yes—
26#2283C2211yescharacteristic
27#2283C2211yescharacteristic
28#2276—21no—
29#1821—21no—
30#1801C2212yes—
31#1801C2212yes—
32#1784—21no—
33C4×#101—21no—
34C4×#101—21no—
35#1461—22no—
36#1461C2212yes—
37#1461C2212yes—
38C2×#324—22no—
39#1303C2214yes—
40#1303C2214yes—
41#1286C2214yes—
42#1286C2214yes—
43C23×Q8D411yescharacteristic
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×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8C2311yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C42C2×C412yes—
59C22×C42C2311yescharacteristic
60#432—22no—
61#432—22no—
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431C2314yes—
68#423C2314yes—
69C2×C4⋊C8—22no—
70C2×#101—22no—
71C2×#101—22no—
72#407—22no—
73C2×C22⋊C8—22no—
74C4×M16—22no—
75C2×C4×C8—22no—
76#391—22no—
77#391—22no—
78#391—21no—
79#391—21no—
80#389—21no—
81#389—21no—
82#389—22no—
83#389—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#383—22no—
91#383—21no—
92#383C2×C411yescharacteristic
93#383C2×C411yescharacteristic
94#383C2×C412yes—
95#383C2×C412yes—
96#382—22no—
97#382—21no—
98#382C2×C412yes—
99#382C2×C412yes—
100#382C2311yescharacteristic
101#382C2311yescharacteristic
102#381C2×C411yescharacteristic
103#381C2×C412yes—
104#381C2×C411yescharacteristic
105#380—22no—
106#380—22no—
107#380—21no—
108#380—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123#376—22no—
124#376—21no—
125#376C2×C412yes—
126#376C2×C412yes—
127#376C2×C412yes—
128#343—22no—
129#343—22no—
130#324—24no—
131#324—24no—
132C22×Q8—24no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8C22⋊C412yes—
137C22×Q8C2×D411yescharacteristic
138C22×Q8C2×D411yescharacteristic
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C22×C8—42no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—24no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—44no—
159C4×Q8—44no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C22⋊C412yes—
194C2×C42C22⋊C412yes—
195C2×C42C22×C412yes—
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C2×C42#4112yes—
204C2×C42C2411yescharacteristic
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—24no—
208C4⋊C8—42no—
209#101—44no—
210#101—44no—
211C22⋊C8—24no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215#97—42no—
216C4×C8—24no—
217C4×C8—42no—
218C4×C8—42no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95C22×C412yes—
238#95C22×C412yes—
239#95C22×C411yescharacteristic
240#95C22×C411yescharacteristic
241#95C22×C411yescharacteristic
242#95C22×C411yescharacteristic
243C24C2×C22⋊C411yescharacteristic
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—42no—
252C2×Q8—48no—
253C2×Q8—48no—
254C2×Q8#13514yes—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—81no—
271C22×C4—82no—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4#11711yescharacteristic
280C22×C4#11711yescharacteristic
281C22×C4#11711yescharacteristic
282C22×C4#11711yescharacteristic
283C22×C4#12312yes—
284C22×C4#12312yes—
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290M16—44no—
291M16—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—84no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—44no—
322C42—42no—
323C42—42no—
324C42—44no—
325C42—82no—
326C42—82no—
327C42—82no—
328C42#11712yes—
329C42#11712yes—
330C42#12312yes—
331C42#12312yes—
332C42#12312yes—
333C42#12312yes—
334C42C2×#4112yes—
335C42C2×#4112yes—
336C23—22no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23#38111yescharacteristic
341C23#38111yescharacteristic
342C23C22×C22⋊C411yescharacteristic
343Q8—28no—
344Q8—48no—
345Q8—44no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4#38111yescharacteristic
392C2×C4#38111yescharacteristic
393C2×C4#38112yes—
394C2×C4#38112yes—
395C2×C4#41614yes—
396C2×C4C2×#11711yescharacteristic
397C2×C4C2×#11711yescharacteristic
398C2×C4C2×#12312yes—
399C2×C4C2×#12312yes—
400C2×C4C2×#13512yes—
401C2×C4#57512yes—
402C8—84no—
403C8—84no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—42no—
412C22—42no—
413C22#183012yes—
414C22C2×#38111yescharacteristic
415C22C2×#41611yescharacteristic
416C22#289611yescharacteristic
417C4—22no—
418C4—24no—
419C4—22no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—81no—
429C4—82no—
430C4—81no—
431C4—82no—
432C4—81no—
433C4—81no—
434C4#309114yes—
435C2—21no—
436C2—21no—
437C2—22no—
438C2—41no—
439C2#2069911yescharacteristic
440C2#2070811yescharacteristic
441C2C2×#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
#207272
C2×#14612
#87692
#87682
#87672
#87662
#80962
#79562