Finite Group Databasebeta

← #304543

Subgroups — #304543

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic
4#29219C211yesmaximal, characteristic
5#10350C214yesmaximal
6#9896C211yesmaximal, characteristic
7#9870C211yesmaximal, characteristic
8#8987C214yesmaximal
9C4×Q8⋊C8C211yesmaximal, characteristic
10C2×Q82—21no—
11C2×#462—21no—
12C2×#462—21no—
13C2×#462C2212yes—
14#2868—21no—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2567—22no—
18#2567—21no—
19#2551—22no—
20#2551—21no—
21#2551—22no—
22#2551C2212yes—
23Q8×C4⋊C4—21no—
24Q8×C4⋊C4C2212yes—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8C2211yescharacteristic
27C42×Q8C2211yescharacteristic
28#2099—22no—
29#1990—22no—
30#1990—21no—
31#1990C2214yes—
32#1988C2214yes—
33#1981—21no—
34#1981—21no—
35#1981C2212yes—
36#1842C2211yescharacteristic
37#1834—22no—
38#1834—21no—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42#1296C2214yes—
43#1279C2214yes—
44#1279C2212yes—
45C22×Q16—22no—
46Q82—44no—
47Q82—44no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8D412yes—
59C2×C4⋊Q8C2312yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—21no—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66#462—28no—
67#462—44no—
68#462—44no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—22no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8C2312yes—
80#395—24no—
81#395—24no—
82#395—24no—
83#391—22no—
84#391—21no—
85#389—22no—
86#389—24no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—24no—
91#389—21no—
92#389D412yes—
93#384—22no—
94#384—22no—
95#384—22no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384D411yescharacteristic
103#384D412yes—
104#384C2312yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4C2312yes—
114C4×C4⋊C4C2311yescharacteristic
115C43—21no—
116C43C2311yescharacteristic
117#340—24no—
118#340—28no—
119#340—24no—
120#336—21no—
121Q8⋊C8—24no—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8C2×D411yescharacteristic
127C2×Q16—44no—
128C2×Q16—44no—
129C2×Q16—44no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—48no—
137C4⋊Q8—44no—
138C4⋊Q8—416no—
139C4⋊Q8—44no—
140C4⋊Q8—44no—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147C4⋊Q8—82no—
148C4⋊Q8—84no—
149C4⋊Q8—84no—
150C4⋊Q8Q1614yescomplemented
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—48no—
155C4×Q8—44no—
156C4×Q8—44no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8D814yes—
160C4×Q8SD1614yes—
161C4×Q8#4114yes—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D414yes—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—24no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2411yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—24no—
207C4⋊C8—22no—
208C4⋊C8—42no—
209C4⋊C8—42no—
210#103—44no—
211#103—44no—
212#103—44no—
213C4×C8—24no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—42no—
217C4×C8—42no—
218#95—24no—
219#95—24no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95—21no—
224#95—22no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—24no—
229C2×Q8—48no—
230C2×Q8—42no—
231C2×Q8—44no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8#12412yes—
244C2×Q8C2×D812yes—
245C2×Q8C2×SD1612yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—42no—
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261Q16—88no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C4⋊C4—21no—
272C4⋊C4—24no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—48no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—82no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4#12412yes—
301C4⋊C4C2×D812yes—
302C4⋊C4C2×SD1612yes—
303C4⋊C4C2×Q1614yes—
304C42—24no—
305C42—22no—
306C42—24no—
307C42—24no—
308C42—22no—
309C42—22no—
310C42—48no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42C2×D812yes—
317C42C2×SD1612yes—
318C42C2×Q1612yes—
319C42AGL(1,ℤ/8)12yes—
320C42#13712yes—
321C42#13712yes—
322C42C2×#4112yes—
323C422−1+412yes—
324C23C2×#12011yescharacteristic
325Q8—22no—
326Q8—22no—
327Q8—22no—
328Q8—48no—
329Q8—82no—
330Q8—82no—
331Q8—82no—
332Q8—82no—
333Q8—164no—
334Q8—168no—
335Q8#48614yes—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4#44712yes—
367C2×C4#44712yes—
368C2×C4#44812yes—
369C2×C4#44812yes—
370C2×C4#45112yes—
371C2×C4#45112yes—
372C2×C4#45512yes—
373C2×C4#45612yes—
374C2×C4C2×#12411yescharacteristic
375C2×C4#53611yescharacteristic
376C2×C4#54812yes—
377C2×C4#54812yes—
378C2×C4C22×D811yescharacteristic
379C2×C4C22×SD1611yescharacteristic
380C2×C4C22×Q1611yescharacteristic
381C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
382C2×C4C2×#13711yescharacteristic
383C2×C4C2×#13711yescharacteristic
384C2×C4#57712yes—
385C8—44no—
386C8—82no—
387C8—82no—
388C22#243611yescharacteristic
389C22C2×#44711yescharacteristic
390C22C2×#44811yescharacteristic
391C22C2×#45111yescharacteristic
392C22#300911yescharacteristic
393C22C2×#45511yescharacteristic
394C22C2×#45611yescharacteristic
395C4—24no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—44no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—82no—
404C4—84no—
405C4—162no—
406C4#164514yes—
407C4#164914yes—
408C4C2×#48611yescharacteristic
409C4#314111yescharacteristic
410C4#322611yescharacteristic
411C4#322711yescharacteristic
412C4#336312yes—
413C4#336912yes—
414C2C2×#164911yescharacteristic
415C2C2×#164511yescharacteristic
416C2#1863011yescharacteristic
417C2#1865211yescharacteristic
418C2#2590311yescharacteristic
419C2#2592311yescharacteristic
420C2#2593611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
150. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#332892
#331632
#292192
#103502
#98962
#98702
#89872
C4×Q8⋊C82