Finite Group Databasebeta

← #190342

Subgroups — #190342

Order 512.

828 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#34231C211yesmaximal, characteristic
2#21425C211yesmaximal, characteristic
3#17404C211yesmaximal, characteristic
4#8775C214yesmaximal, complemented
5#8774C214yesmaximal, complemented
6#8071C211yesmaximal, characteristic
7#8068C211yesmaximal, characteristic
8#7976C211yesmaximal, characteristic
9#7972C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2369—21no—
16#2365—21no—
17#2362—22no—
18#2356C2214yes—
19#2347C412yes—
20#2346C412yes—
21#2332—22no—
22#2320C412yes—
23C4×#125—22no—
24#2283C2211yescharacteristic
25#2276C412yes—
26C4×#117C2211yescharacteristic
27#1997—21no—
28#1966—21no—
29#1901—22no—
30C4×#101C2212yes—
31C4×#101C2212yes—
32#1472—22no—
33#1470—22no—
34#1467C2214yes—
35#1300C2212yes—
36#1300C2214yes—
37#1300C2212yes—
38#1288C2212yes—
39#1288C2212yes—
40#1288C2214yes—
41C22×M16—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125D412yes—
53C2×C22⋊Q8D412yes—
54C2×C22⋊Q8D412yes—
55C2×C4×Q8—22no—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2312yes—
58C2×#117—21no—
59C2×#117C2×C412yes—
60C22×C42C2311yescharacteristic
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432C2312yes—
68#432C2312yes—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#423—21no—
74#423—21no—
75C2×C4⋊C8—21no—
76C2×#101—22no—
77C2×#101—22no—
78#407—21no—
79#407—21no—
80C2×C22⋊C8—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16C2311yescharacteristic
86C4×M16C2311yescharacteristic
87C4×M16C2312yes—
88C2×#97—21no—
89C2×#97—21no—
90C2×C4×C8—21no—
91#391—22no—
92#391—24no—
93#389—22no—
94#389—22no—
95#389—22no—
96#388—24no—
97#388—24no—
98#387—24no—
99#387—24no—
100#387—24no—
101#383C2×C414yes—
102#383C2312yes—
103#382—21no—
104#382C2×C414yes—
105#381—21no—
106#380C2×C411yescharacteristic
107#380C2×C411yescharacteristic
108C4×C4⋊C4—24no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C414yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4C2×C412yes—
121#376C2×C412yes—
122#376C2×C414yes—
123#376C2312yes—
124C43—21no—
125#343—22no—
126#343—22no—
127#324—28no—
128C22×Q8C22⋊C412yes—
129C22×Q8C2×D412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—42no—
137C2×M16—42no—
138C2×M16—41no—
139C2×M16—41no—
140C22×C8—41no—
141C22×C8—41no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150#125—22no—
151#125—22no—
152#125—22no—
153#125—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—48no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—44no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C414yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4C22⋊C414yes—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—24no—
199C2×C42C22×C414yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42C2411yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218#101—44no—
219#101—44no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—41no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226#97—21no—
227#97—21no—
228#97—22no—
229#97—22no—
230#97—22no—
231#97—41no—
232#97—41no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—22no—
236C4×C8—41no—
237C4×C8—41no—
238#95—22no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95C22×C412yes—
247#95C22×C412yes—
248#95C22×C412yes—
249#95C22×C412yes—
250C24C2×C22⋊C411yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—44no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C2×C22⋊C414yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C414yes—
271C22×C4#11711yescharacteristic
272C22×C4#11711yescharacteristic
273C22×C4#11711yescharacteristic
274C22×C4#11711yescharacteristic
275C22×C4#12311yescharacteristic
276C22×C4#12311yescharacteristic
277C22×C4#12311yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4#12311yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4#12311yescharacteristic
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286M16—22no—
287M16—22no—
288M16—41no—
289M16—41no—
290M16—44no—
291M16—42no—
292M16—42no—
293M16—82no—
294C2×C8—44no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—82no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—48no—
318C4⋊C4—84no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—44no—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C42—24no—
325C42—24no—
326C42—44no—
327C42—44no—
328C42—44no—
329C42—48no—
330C42—81no—
331C42—81no—
332C42—84no—
333C42#11711yescharacteristic
334C42#11711yescharacteristic
335C42#11711yescharacteristic
336C42#11711yescharacteristic
337C42#12311yescharacteristic
338C42#12311yescharacteristic
339C42#12311yescharacteristic
340C42#12311yescharacteristic
341C42#12311yescharacteristic
342C42#12311yescharacteristic
343C42#12311yescharacteristic
344C42#12311yescharacteristic
345C42C2×#4111yescharacteristic
346C42C2×#4111yescharacteristic
347C42C2×#4111yescharacteristic
348C42C2×#4111yescharacteristic
349C23—21no—
350C23—21no—
351C23—41no—
352C23—41no—
353C23#38111yescharacteristic
354C23#38111yescharacteristic
355C23C22×C22⋊C411yescharacteristic
356Q8—44no—
357Q8—44no—
358Q8—82no—
359Q8—82no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
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—42no—
397C2×C4#38111yescharacteristic
398C2×C4#38111yescharacteristic
399C2×C4#38111yescharacteristic
400C2×C4#38111yescharacteristic
401C2×C4#38111yescharacteristic
402C2×C4#38111yescharacteristic
403C2×C4C2×#11711yescharacteristic
404C2×C4C2×#11711yescharacteristic
405C2×C4C2×#12311yescharacteristic
406C2×C4C2×#12311yescharacteristic
407C2×C4C2×#12311yescharacteristic
408C2×C4C2×#12311yescharacteristic
409C8—42no—
410C8—42no—
411C8—82no—
412C8—82no—
413C8—82no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—81no—
423C22—81no—
424C22C2×#38111yescharacteristic
425C22#290111yescharacteristic
426C22#290111yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—44no—
438C4—44no—
439C4—44no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4#182712yes—
444C4#183012yes—
445C2—21no—
446C2—21no—
447C2—41no—
448C2—41no—
449C2#2070611yescharacteristic
450C2C2×#182711yescharacteristic
451C2C2×#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. #8775C2semidirect
5. #8774C2semidirect

Maximal subgroups

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

subgroupindex
#342312
#214252
#174042
#87752
#87742
#80712
#80682
#79762
#79722