Finite Group Databasebeta

← #193312

Subgroups — #193312

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34371C211yesmaximal, characteristic
2#21962C211yesmaximal, characteristic
3#21962C211yesmaximal, characteristic
4#21324C211yesmaximal, characteristic
5C22×#332C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8923C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7895C211yesmaximal, characteristic
11#7895C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2381—22no—
17#2365C2212yes—
18#2365C2212yes—
19#2345—21no—
20#2345—21no—
21Q8×C22⋊C4—21no—
22Q8×C22⋊C4—21no—
23#2331—22no—
24#2327—22no—
25#2313—22no—
26#2285C2211yescharacteristic
27#2283C2211yescharacteristic
28#2283C2211yescharacteristic
29#1993—22no—
30#1990C2212yes—
31#1990C2212yes—
32#1949—22no—
33#1949—22no—
34#1948C2212yes—
35#1948C2212yes—
36#1947C2212yes—
37#1947C2212yes—
38#1808—21no—
39#1808—21no—
40#1807—21no—
41#1807—21no—
42#1736—22no—
43C2×#332C418yescomplemented
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#332C2211yescharacteristic
47C2×#332C2211yescharacteristic
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C23×Q8—21no—
53C22×M16—21no—
54C23×C8—21no—
55C23×C8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yes—
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8Q814yes—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D412yes—
70C22×C42C2311yescharacteristic
71C2×#107—22no—
72C2×#107—22no—
73C2×#106—22no—
74C2×#106—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8C2×C414yes—
80C2×C4⋊C8C2×C414yes—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C4×M16—22no—
86C4×M16—22no—
87C2×#97—21no—
88C2×C4×C8C2312yes—
89#391—24no—
90#391—24no—
91#391—22no—
92#391—22no—
93#389—22no—
94#389—22no—
95#387—24no—
96#387—24no—
97#387—24no—
98#387—24no—
99#387—24no—
100#387—24no—
101#384C2312yes—
102#383—21no—
103#383—21no—
104#382—21no—
105#382D412yes—
106#382D412yes—
107#381—21no—
108#381—21no—
109#380—22no—
110#380—22no—
111#380—21no—
112#380—21no—
113#380D411yescharacteristic
114#380D411yescharacteristic
115#380D411yescharacteristic
116#380D411yescharacteristic
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2312yes—
122C4×C4⋊C4C2312yes—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127#376—21no—
128#376—21no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—24no—
134#340—24no—
135#332C2×C414yes—
136#332C2×C414yes—
137#332C2×C414yes—
138#332C2×C414yes—
139C22×Q8—24no—
140C22×Q8—22no—
141C22×Q8—24no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—82no—
145C22×Q8—82no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—24no—
159C22×C8—24no—
160C22×C8—82no—
161C22×C8—82no—
162C4⋊Q8—24no—
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C4⋊C418yes—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C2×Q812yes—
175C4⋊Q8C2×Q812yes—
176C4×Q8—82no—
177C4×Q8—82no—
178C4×Q8—82no—
179C4×Q8—82no—
180#117—44no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4C4⋊C418yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—41no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—41no—
210C2×C42C22×C412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218#107—82no—
219#107—82no—
220#106—82no—
221#106—82no—
222C4⋊C8—24no—
223C4⋊C8—24no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8C22×C412yes—
229C4⋊C8C22×C412yes—
230C4⋊C8C22×C412yes—
231C4⋊C8C22×C412yes—
232#97—42no—
233C4×C8—22no—
234C4×C8—22no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—41no—
250#95—41no—
251C24#12011yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—28no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—84no—
262C2×Q8—88no—
263C2×Q8—84no—
264C2×Q8—84no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—84no—
280C22×C4C2×C4⋊C412yes—
281C22×C4C4×D412yes—
282C22×C4C4×D412yes—
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4C22⋊Q812yes—
287C22×C4C22⋊Q812yes—
288C22×C4#12312yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—48no—
293C2×C8—24no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—84no—
305C2×C8—84no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—28no—
313C4⋊C4—28no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4C2×C4⋊C414yes—
330C4⋊C4C2×C4⋊C414yes—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—82no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—82no—
341C42—84no—
342C42—84no—
343C42C2×C4⋊C412yes—
344C42C4×D412yes—
345C42C4×D412yes—
346C42C22⋊Q812yes—
347C42C22⋊Q812yes—
348C42#12312yes—
349C42C23×C411yescharacteristic
350C42C22×Q811yescharacteristic
351C42C2×#4111yescharacteristic
352C42C2×#4111yescharacteristic
353C23—22no—
354C23—22no—
355C23#38512yes—
356C23#45412yes—
357C23#45612yes—
358C23C2×#12011yescharacteristic
359Q8—44no—
360Q8—44no—
361Q8—42no—
362Q8—42no—
363Q8—42no—
364Q8—42no—
365Q8—88no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—88no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#38512yes—
401C2×C4#38512yes—
402C2×C4#38512yes—
403C2×C4C22×C4⋊C411yescharacteristic
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×C4×D411yescharacteristic
406C2×C4C2×C22⋊Q811yescharacteristic
407C2×C4C2×C22⋊Q811yescharacteristic
408C2×C4C2×#12311yescharacteristic
409C2×C4#57612yes—
410C2×C4#57712yes—
411C2×C4#57712yes—
412C2×C4#57812yes—
413C8—44no—
414C8—42no—
415C8—42no—
416C8—42no—
417C8—42no—
418C22—24no—
419C22—24no—
420C22—22no—
421C22—22no—
422C22#185714yes—
423C22C2×#38511yescharacteristic
424C22#301212yes—
425C22#301312yes—
426C22C2×#45411yescharacteristic
427C22C2×#45611yescharacteristic
428C4—21no—
429C4—21no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—44no—
435C4—82no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4—162no—
440C4#294811yescharacteristic
441C4#294911yescharacteristic
442C4#294911yescharacteristic
443C4#295011yescharacteristic
444C4#308112yes—
445C4#308212yes—
446C2—24no—
447C2#1872612yes—
448C2#1873312yes—
449C2C2×#185711yescharacteristic
450C2#2087911yescharacteristic
451C2#2088011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8926C2semidirect
7. #8925C2semidirect
8. #8923C2semidirect
9. #8922C2semidirect
43. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#343712
#219622
#213242
C22×#3322
#89262
#89252
#89232
#89222
#78952