Finite Group Databasebeta

← #200598

Subgroups — #200598

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60284C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#20728C211yesmaximal, characteristic
4#17742C212yesmaximal, complemented
5#9403C214yesmaximal, complemented
6#9396C214yesmaximal, complemented
7#8094C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2991—24no—
11#2982C2212yes—
12#2975—22no—
13#2975C2212yes—
14#2971—24no—
15#2920—22no—
16#2920C2212yes—
17#2920C2212yes—
18#2857—22no—
19#2628C414yes—
20#2568C2212yes—
21#2567—21no—
22#2563—21no—
23#2542C2212yes—
24#2487C414yes—
25#2409—22no—
26#2317—21no—
27#2311—21no—
28#2285C2211yescharacteristic
29#1822—21no—
30#1801C2212yes—
31#1800C2212yes—
32#1785—21no—
33C4×#101—21no—
34C4×#100—21no—
35#1556—24no—
36#1549C2218yes—
37#1546—24no—
38#1303C2218yes—
39C22×#41D411yescharacteristic
40C23×Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2312yes—
49C2×C22⋊Q8—24no—
50C4×#41—22no—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×#117—21no—
61C2×#117C2×C412yes—
62C2×#117D411yescharacteristic
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C412yes—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67C8×Q8—28no—
68#436—216no—
69#436—28no—
70#432—24no—
71#432—22no—
72#432—22no—
73#432—22no—
74#432D414yes—
75#431—24no—
76#431—24no—
77#431—22no—
78#431—22no—
79#431—22no—
80#431C2314yes—
81#423—24no—
82#423D414yes—
83#423C2314yes—
84C2×#101—22no—
85C2×#100—22no—
86#407—24no—
87#407—22no—
88#407—22no—
89C4×M16—24no—
90C4×M16—22no—
91C4×M16—22no—
92#400C2×C418yes—
93#399C2×C418yes—
94#395—24no—
95#393—24no—
96#391—22no—
97#391—24no—
98#390—22no—
99#389—24no—
100#389—24no—
101#386—24no—
102#385—24no—
103#384—22no—
104#384—21no—
105#382—22no—
106#382C2×C412yes—
107#380—24no—
108#380—24no—
109#380—21no—
110#380C2×C412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2312yes—
113#343—22no—
114#343—22no—
115#332—216no—
116C2×#41—24no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—42no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—84no—
125C22×Q8C2×D411yescharacteristic
126C22×D4C22⋊C412yes—
127C22×D4C2×D411yescharacteristic
128C23×C4—22no—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C2×M16—28no—
136C2×M16—24no—
137C2×M16—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C22⋊Q8—88no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161#117—22no—
162#117—22no—
163#117—24no—
164#117—24no—
165#117—44no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C414yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C22⋊C4—22no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22×C412yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C22⋊C412yes—
193C2×C42C22⋊C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4112yes—
201C2×C42#4112yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—28no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8C2×D418yes—
208#101—44no—
209#100—44no—
210C22⋊C8—28no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215#97—24no—
216#97—42no—
217C4×C8—28no—
218C4×C8—24no—
219C4×C8—42no—
220#95—22no—
221#95—28no—
222#95—28no—
223#95C22×C414yes—
224C24C2×C22⋊C412yes—
225C24C2×C22⋊C411yescharacteristic
226#41—48no—
227#41—48no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—48no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—88no—
235C2×Q8—84no—
236C2×Q8—88no—
237C2×Q8#13714yes—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—42no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—84no—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C4×D412yes—
269C22×C4C4×D412yes—
270C22×C4#12012yes—
271C22×C4#12112yes—
272C22×C4#12112yes—
273C22×C4#12312yes—
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277M16—44no—
278M16—44no—
279M16—84no—
280M16—88no—
281C2×C8—28no—
282C2×C8—44no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—28no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—88no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—44no—
301C22⋊C4—88no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—44no—
309C42C4×D412yes—
310C42C4×D412yes—
311C42#12011yescharacteristic
312C42#12011yescharacteristic
313C42#12112yes—
314C42#12112yes—
315C42#12311yescharacteristic
316C42#12311yescharacteristic
317C42C22×D412yes—
318C42C2×#4112yes—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23#38612yes—
329C23C22×C22⋊C411yescharacteristic
330Q8—28no—
331Q8—84no—
332Q8—88no—
333Q8—84no—
334D4—48no—
335D4—44no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—48no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—88no—
366C2×C4#38611yescharacteristic
367C2×C4#38611yescharacteristic
368C2×C4#38611yescharacteristic
369C2×C4#38612yes—
370C2×C4#38611yescharacteristic
371C2×C4#41914yes—
372C2×C4C2×C4×D412yes—
373C2×C4C2×#12011yescharacteristic
374C2×C4C2×#12112yes—
375C2×C4C2×#12311yescharacteristic
376C2×C4C2×#13712yes—
377C2×C4#57512yes—
378C8—48no—
379C8—84no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—42no—
386C22—42no—
387C22—44no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22C2×#38611yescharacteristic
393C22C2×#41911yescharacteristic
394C22#289611yescharacteristic
395C4—21no—
396C4—22no—
397C4—21no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—24no—
402C4—42no—
403C4—42no—
404C4—84no—
405C4—84no—
406C4#191512yes—
407C4#302112yes—
408C4#305614yes—
409C2—21no—
410C2—21no—
411C2—42no—
412C2—82no—
413C2#2095611yescharacteristic
414C2#2097211yescharacteristic
415C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #17742C2semidirect
5. #9403C2semidirect
6. #9396C2semidirect
7. #8094C2semidirect

Maximal subgroups

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

subgroupindex
#602842
#352792
#207282
#177422
#94032
#93962
#80942