Finite Group Databasebeta

← #98242

Subgroups — #98242

Order 512.

743 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Q8C211yesmaximal, characteristic
2#9896C211yesmaximal, characteristic
3#9694C211yesmaximal, characteristic
4#8586C211yesmaximal, characteristic
5#8564C211yesmaximal, characteristic
6#8564C211yesmaximal, characteristic
7#7570C211yesmaximal, characteristic
8C2×Q82—21no—
9#2868—21no—
10#2868C2211yescharacteristic
11#2567—22no—
12#2567—21no—
13#2551—22no—
14#2551—21no—
15#2551—22no—
16#2551C412yes—
17Q8×C4⋊C4—21no—
18Q8×C4⋊C4C412yes—
19C4×C4⋊Q8—21no—
20C4×C4⋊Q8C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22#1842C2211yescharacteristic
23#1842C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C2×#332—21no—
27C2×#332—22no—
28C2×Q8⋊C8—21no—
29C2×Q8⋊C8—21no—
30#1296—21no—
31#1296—22no—
32#1279—21no—
33#1279—21no—
34#1271—22no—
35Q82—42no—
36Q82—42no—
37Q82—42no—
38Q82—42no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—24no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D412yes—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4×C8—21no—
63C2×C4×C8—21no—
64#395—24no—
65#395—24no—
66#395—24no—
67#391—22no—
68#391—21no—
69#389—22no—
70#389—24no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—24no—
75#389—21no—
76#389D412yes—
77#384—22no—
78#384—22no—
79#384—22no—
80#384—21no—
81#384—21no—
82#384C2×C412yes—
83#384D411yescharacteristic
84#384D412yes—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C4⋊C4D411yescharacteristic
94C4×C4⋊C4D411yescharacteristic
95C43—21no—
96C43C2311yescharacteristic
97#336—21no—
98#336—21no—
99#332—44no—
100#332—42no—
101Q8⋊C8—42no—
102Q8⋊C8—42no—
103C22×Q8—24no—
104C22×Q8—21no—
105C22×Q8—22no—
106C22×Q8C22⋊C411yescharacteristic
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—44no—
114C4⋊Q8—44no—
115C4⋊Q8—42no—
116C4⋊Q8—42no—
117C4⋊Q8—48no—
118C4⋊Q8—48no—
119C4⋊Q8—44no—
120C4⋊Q8—44no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—82no—
126C4⋊Q8—84no—
127C4⋊Q8—82no—
128C4⋊Q8—82no—
129C4⋊Q8SD1612yes—
130C4⋊Q8Q1612yes—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—44no—
138C4×Q8—44no—
139C4×Q8—44no—
140C4×Q8—44no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8D812yes—
146C4×Q8SD1612yes—
147C4×Q8SD1612yes—
148C4×Q8Q1612yes—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C414yes—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—42no—
191C4⋊C8—42no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4×C8—41no—
201C4×C8—41no—
202C4×C8—41no—
203C4×C8—41no—
204#95—24no—
205#95—24no—
206#95—21no—
207#95—22no—
208#95—21no—
209#95—21no—
210#95—22no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—48no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8#10112yes—
228C2×Q8#10212yes—
229C2×Q8#10312yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8—81no—
249C2×C8—81no—
250C2×C8—81no—
251C2×C8—81no—
252C2×C8—81no—
253C2×C8—81no—
254C2×C8—81no—
255C2×C8—81no—
256C4⋊C4—24no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—82no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C4⋊C4#10112yes—
288C4⋊C4#10212yes—
289C4⋊C4#10312yes—
290C4⋊C4#10314yes—
291C42—24no—
292C42—21no—
293C42—21no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—41no—
303C42—48no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42C2×D811yescharacteristic
312C42C2×SD1611yescharacteristic
313C42C2×SD1611yescharacteristic
314C42C2×SD1611yescharacteristic
315C42C2×Q1611yescharacteristic
316C42C2×Q1611yescharacteristic
317C42AGL(1,ℤ/8)11yescharacteristic
318C42AGL(1,ℤ/8)11yescharacteristic
319C42#13711yescharacteristic
320C42#13711yescharacteristic
321C42#13711yescharacteristic
322C42#13711yescharacteristic
323C23#37911yescharacteristic
324Q8—22no—
325Q8—22no—
326Q8—22no—
327Q8—42no—
328Q8—42no—
329Q8—42no—
330Q8—42no—
331Q8—162no—
332Q8—162no—
333Q8—168no—
334Q8#33214yescomplemented
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4C2×#10111yescharacteristic
366C2×C4C2×#10111yescharacteristic
367C2×C4C2×#10211yescharacteristic
368C2×C4C2×#10311yescharacteristic
369C2×C4C2×#10311yescharacteristic
370C2×C4#41811yescharacteristic
371C2×C4#41911yescharacteristic
372C2×C4#41911yescharacteristic
373C2×C4#44711yescharacteristic
374C2×C4#44711yescharacteristic
375C2×C4#44811yescharacteristic
376C2×C4#44811yescharacteristic
377C2×C4#44811yescharacteristic
378C2×C4#44811yescharacteristic
379C2×C4#45011yescharacteristic
380C2×C4#45011yescharacteristic
381C2×C4#45111yescharacteristic
382C2×C4#45111yescharacteristic
383C2×C4#45111yescharacteristic
384C2×C4#45111yescharacteristic
385C2×C4#45311yescharacteristic
386C2×C4#45411yescharacteristic
387C2×C4#45511yescharacteristic
388C2×C4#45611yescharacteristic
389C8—82no—
390C8—82no—
391C8—161no—
392C8—161no—
393C22#178411yescharacteristic
394C22#178611yescharacteristic
395C22#178711yescharacteristic
396C22#178711yescharacteristic
397C22#179211yescharacteristic
398C22#179311yescharacteristic
399C22#179511yescharacteristic
400C4—24no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—44no—
412C4—84no—
413C4—82no—
414C4C2×#33211yescharacteristic
415C4#154611yescharacteristic
416C4#154911yescharacteristic
417C4#155611yescharacteristic
418C4#162311yescharacteristic
419C4#162711yescharacteristic
420C4#163311yescharacteristic
421C4#163711yescharacteristic
422C4#164511yescharacteristic
423C4#164511yescharacteristic
424C4#164911yescharacteristic
425C4#165111yescharacteristic
426C2#834011yescharacteristic
427C2#839611yescharacteristic
428C2#842011yescharacteristic
429C2#842411yescharacteristic
430C2#844711yescharacteristic
431C2#845111yescharacteristic
432C2#847311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
334. Q8#332semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#98962
#96942
#85862
#85642
#75702