Finite Group Databasebeta

← #200523

Subgroups — #200523

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#17746C212yesmaximal
5#9400C214yesmaximal, complemented
6#9392C214yesmaximal, complemented
7#8098C212yesmaximal
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2984—24no—
11#2983—22no—
12#2983C2212yes—
13#2982C2212yes—
14Q8×M16—24no—
15#2920—22no—
16#2920C2212yes—
17#2920C2212yes—
18#2567—21no—
19#2567—21no—
20#2567—22no—
21#2567C414yes—
22#2567C2212yes—
23#2567C2212yes—
24C2×#395C414yes—
25C2×#391—21no—
26C2×#389—22no—
27C2×#384—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1821—21no—
30#1801C2212yes—
31#1801C2212yes—
32#1784—21no—
33C4×#101—21no—
34C4×#101—21no—
35#1558—24no—
36#1548—24no—
37#1548C2218yes—
38#1303C2218yes—
39C23×Q8C2×C412yes—
40C23×Q8D411yescharacteristic
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×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D412yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68#446—216no—
69#446—28no—
70C8×Q8—28no—
71#432—24no—
72#432—24no—
73#432—22no—
74#432—22no—
75#432—22no—
76#432D414yes—
77#431—24no—
78#431—24no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431C2314yes—
83#423—24no—
84#423D414yes—
85#423C2314yes—
86C2×#101—22no—
87C2×#101—22no—
88#407—22no—
89#407—22no—
90C4×M16—24no—
91C4×M16—22no—
92C4×M16—22no—
93#395—24no—
94#395—24no—
95#395C2×C418yes—
96#395C2×C418yes—
97#391—24no—
98#391—22no—
99#391—22no—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#384—24no—
105#384—22no—
106#384—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2312yes—
111C2×#95—24no—
112C2×#95C2×C412yes—
113#343—22no—
114#343—22no—
115#333—216no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—42no—
123C22×Q8—84no—
124C22×Q8C22⋊C412yes—
125C22×Q8C22×C414yes—
126C22×Q8C2×D411yescharacteristic
127C22×Q8C2×D411yescharacteristic
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C2×M16—24no—
136C2×M16—24no—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—24no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—88no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—44no—
191C2×C42C22⋊C412yes—
192C2×C42C22⋊C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D412yes—
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—28no—
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—24no—
208C4⋊C8C2×D418yes—
209#101—44no—
210#101—44no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—42no—
214C22⋊C8—42no—
215#97—28no—
216#97—24no—
217#97—42no—
218C4×C8—28no—
219C4×C8—24no—
220C4×C8—42no—
221#95—28no—
222#95—28no—
223#95—22no—
224#95C22×C414yes—
225C24C2×C22⋊C411yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—28no—
234C2×Q8—42no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—44no—
238C2×Q8—88no—
239C2×Q8—84no—
240C2×Q8—84no—
241C2×Q8—88no—
242C2×Q8#13714yes—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—48no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C4×D412yes—
270C22×C4C4×D412yes—
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12112yes—
274C22×C4#12112yes—
275C22×C4#12311yescharacteristic
276C22×C4#12311yescharacteristic
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—48no—
281M16—44no—
282M16—44no—
283C2×C8—28no—
284C2×C8—44no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—88no—
301C4⋊C4—88no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—44no—
311C42—44no—
312C42C4×D412yes—
313C42C4×D412yes—
314C42#12012yes—
315C42#12112yes—
316C42#12112yes—
317C42#12312yes—
318C42C22×D412yes—
319C42C2×#4112yes—
320C23—21no—
321C23—21no—
322C23—22no—
323C23—22no—
324C23#38611yescharacteristic
325C23#38611yescharacteristic
326C23C22×C22⋊C411yescharacteristic
327Q8—28no—
328Q8—48no—
329Q8—44no—
330Q8—88no—
331Q8—88no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—28no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—416no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—82no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4#38611yescharacteristic
367C2×C4#38612yes—
368C2×C4#38612yes—
369C2×C4#38611yescharacteristic
370C2×C4#41914yes—
371C2×C4#45614yes—
372C2×C4C2×C4×D412yes—
373C2×C4C2×#12011yescharacteristic
374C2×C4C2×#12112yes—
375C2×C4C2×#12311yescharacteristic
376C2×C4C2×#13712yes—
377C2×C4#57512yes—
378C8—48no—
379C8—84no—
380C22—24no—
381C22—24no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22#188612yes—
389C22#191412yes—
390C22C2×#38611yescharacteristic
391C22C2×#41911yescharacteristic
392C22#289611yescharacteristic
393C4—22no—
394C4—22no—
395C4—21no—
396C4—21no—
397C4—22no—
398C4—24no—
399C4—48no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—84no—
404C4#188612yes—
405C4C2×#45612yes—
406C4#305614yes—
407C2—21no—
408C2—21no—
409C2—22no—
410C2—22no—
411C2C2×#188611yescharacteristic
412C2#2097211yescharacteristic
413C2C2×#191411yescharacteristic
subgroupG/H
center, socleC22C2×#386
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
5. #9400C2semidirect
6. #9392C2semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#25672
#207272
#177462
#94002
#93922
#80982