Finite Group Databasebeta

← #201418

Subgroups — #201418

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#21804C212yesmaximal, complemented
4#20632C211yesmaximal, characteristic
5#9471C218yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9#2989—24no—
10#2983—21no—
11#2975—21no—
12#2973C2214yes—
13#2965—24no—
14#2920—22no—
15C2×#431C2212yes—
16C2×#410—21no—
17#2857—22no—
18#2628C414yescomplemented
19#2568—21no—
20#2567C412yescomplemented
21#2563C412yescomplemented
22#2542—21no—
23#2487—22no—
24#2409C2214yes—
25#2317—21no—
26#2311—21no—
27#2285C2212yes—
28#2285C2211yescharacteristic
29#1981—24no—
30#1956—24no—
31#1799—22no—
32#1799C2214yes—
33#1787C2218yes—
34#1740—21no—
35#1289C2218yes—
36C22×#41D411yescharacteristic
37C23×Q8C2312yes—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×C22⋊Q8—24no—
47C4×#41—22no—
48C4×#41—22no—
49C4×#41—22no—
50C4×#41—22no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4D412yes—
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2312yes—
57C2×#117—21no—
58C2×#117—22no—
59C2×#117C2×C412yes—
60C2×#117D411yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65#446—28no—
66#436—28no—
67#435—216no—
68#432—24no—
69#432—24no—
70#431—24no—
71#431—22no—
72#431—22no—
73#431—22no—
74#431—22no—
75#423—24no—
76C2×C4⋊C8D414yes—
77C2×#103—216no—
78#410—44no—
79C2×#99—24no—
80#407—24no—
81C2×C22⋊C8C2314yes—
82C4×M16—24no—
83C2×C4×C8D414yes—
84#400—24no—
85#399—24no—
86#395C2×C418yes—
87#393C2×C418yes—
88#391—22no—
89#391—24no—
90#390—22no—
91#389—24no—
92#389—24no—
93#386—24no—
94#385—24no—
95#384—22no—
96#384—21no—
97#384—22no—
98#382—22no—
99#382—22no—
100#382C2×C412yes—
101#380—24no—
102#380—24no—
103#380—21no—
104#380—22no—
105#380C2314yes—
106#380C2312yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111#342—24no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41—24no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—84no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22×C414yes—
123C22×D4C22⋊C411yescharacteristic
124C22×D4C2×D412yes—
125C23×C4—22no—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C2×M16—28no—
133C22×C8C2×D418yes—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—41no—
139C4⋊Q8—42no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—42no—
144C22⋊Q8—88no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157#117—22no—
158#117—22no—
159#117—24no—
160#117—24no—
161#117—42no—
162#117—42no—
163#117—44no—
164#117—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22×C412yes—
189C2×C22⋊C4C2×D412yes—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C4⋊C8—28no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210#103—816no—
211#99—48no—
212C22⋊C8—28no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215#97—28no—
216C4×C8—24no—
217C4×C8—24no—
218#95—22no—
219#95—28no—
220#95—28no—
221#95—24no—
222#95—42no—
223#95C22×C414yes—
224C24C2×C22⋊C412yes—
225C24C22×D411yescharacteristic
226#41—44no—
227#41—44no—
228#41—44no—
229#41—44no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—28no—
234C2×Q8—48no—
235C2×Q8—416no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—84no—
239C2×Q8—88no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—84no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4#12112yes—
272C22×C4#12112yes—
273C22×C4#12112yes—
274C22×C4#12112yes—
275C22×C4AGL(1,ℤ/8)12yes—
276C22×C4#13712yes—
277C22×C4#13714yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D412yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4112yes—
282M16—88no—
283C2×C8—28no—
284C2×C8—48no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—48no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—84no—
300C4⋊C4—88no—
301C4⋊C4—84no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—88no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—44no—
317C42—84no—
318C42C4×D412yes—
319C42C4×D412yes—
320C42C4×D412yes—
321C42C4×D412yes—
322C42#12011yescharacteristic
323C42#12012yes—
324C42#12011yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12312yes—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—42no—
335C23C22×C22⋊C411yescharacteristic
336C23C2×#12112yes—
337C23C2×AGL(1,ℤ/8)11yescharacteristic
338C23C2×#13712yes—
339C23C2×#13711yescharacteristic
340Q8—416no—
341Q8—44no—
342Q8—84no—
343Q8—84no—
344D4—44no—
345D4—44no—
346D4—44no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—88no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4#38611yescharacteristic
378C2×C4#38612yes—
379C2×C4#38611yescharacteristic
380C2×C4#38611yescharacteristic
381C2×C4#38612yes—
382C2×C4#38611yescharacteristic
383C2×C4#41712yes—
384C2×C4#41712yes—
385C2×C4#41912yes—
386C2×C4#41912yes—
387C2×C4#46814yes—
388C2×C4#46814yes—
389C2×C4#47014yes—
390C2×C4#47014yes—
391C2×C4C2×C4×D412yes—
392C2×C4C2×#12011yescharacteristic
393C2×C4C2×#12311yescharacteristic
394C8—48no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—44no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—82no—
407C22C2×#38611yescharacteristic
408C22C2×#41911yescharacteristic
409C22C2×#41711yescharacteristic
410C22C2×#46812yes—
411C22C2×#47012yes—
412C4—21no—
413C4—22no—
414C4—21no—
415C4—22no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—84no—
420C4—84no—
421C4—164no—
422C4#189412yes—
423C4#189412yes—
424C4#189414yes—
425C4#190912yes—
426C4#302712yes—
427C2—21no—
428C2—22no—
429C2—21no—
430C2—22no—
431C2—82no—
432C2#2096111yescharacteristic
433C2C2×#189411yescharacteristic
434C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21804C2semidirect
5. #9471C2semidirect
6. #7981C2semidirect
18. #2628C4semidirect
20. #2567C4semidirect
21. #2563C4semidirect

Maximal subgroups

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

subgroupindex
#602772
#352792
#218042
#206322
#94712
#79812