Finite Group Databasebeta

← #375303

Subgroups — #375303

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3147C211yesmaximal, characteristic
2C2×#1976C214yesmaximal, complemented
3#21212C212yesmaximal, complemented
4#21209C212yesmaximal, complemented
5C4×#427C211yesmaximal, characteristic, complemented
6#20437C211yesmaximal, characteristic, complemented
7#11026C214yesmaximal, complemented
8C22×#135C2212yes—
9C22×#135C2212yes—
10C2×#532C2212yes—
11#3147—22no—
12#3147—22no—
13#3147—22no—
14#3147—22no—
15C2×#495C412yescomplemented
16C2×#495C412yescomplemented
17C2×#494C411yescharacteristic, complemented
18C2×#493C411yescharacteristic, complemented
19C2×#492C412yescomplemented
20C2×#427C2212yes—
21C2×#423C2211yescharacteristic
22C2×#421C2218yes—
23C2×#410C2218yes—
24C2×#405C2212yes—
25C22×C4×C8C2211yescharacteristic
26#2285C2211yescharacteristic
27C4×#117C2212yes—
28#1976—216no—
29#1964—28no—
30#1963—24no—
31#1962—24no—
32C4×#107—22no—
33C4×#106—22no—
34#1770—22no—
35C8×C4⋊C4—22no—
36#1735C2212yes—
37#1729C2212yes—
38C22×#41C2314yes—
39C2×#135—24no—
40C2×#135—24no—
41C2×#135—22no—
42C2×#135—22no—
43C2×#135—22no—
44C2×#135—22no—
45C2×#135—28no—
46C2×#135—28no—
47C2×#135—44no—
48C2×#135—44no—
49C22×Q16C2×C412yes—
50C22×Q16C2×C412yes—
51C22×SD16C2×C414yes—
52C22×SD16C2×C414yes—
53C22×D8C2×C412yes—
54C22×D8C2×C412yes—
55C22×M16C2312yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58#532—24no—
59#532—24no—
60#532—22no—
61#532—22no—
62#532—22no—
63#532—22no—
64C2×C4⋊Q8C2×C412yes—
65C2×#127C2×C412yes—
66C2×#124C2×C414yes—
67C2×#121—24no—
68C2×C4×D4—24no—
69C2×#117C2314yes—
70C2×#117C2312yes—
71C22×C42C2311yescharacteristic
72#495—24no—
73#495—24no—
74#495—24no—
75#495—24no—
76#494—22no—
77#494—22no—
78#493—22no—
79#493—22no—
80#492—24no—
81#492—24no—
82#427—24no—
83#427D418yescomplemented
84C2×#107—22no—
85C2×#106—22no—
86#423—22no—
87#423D412yes—
88#423D412yes—
89C2×C4⋊C8—22no—
90#421—216no—
91#421—48no—
92C2×#104—216no—
93#410—216no—
94#410—48no—
95C2×#99—216no—
96#405—28no—
97C2×#97—22no—
98C2×C4×C8—22no—
99C2×C4×C8C2×C411yescharacteristic
100C2×C4×C8C2×C411yescharacteristic
101C2×C4×C8C2×C411yescharacteristic
102C2×C4×C8C2×C411yescharacteristic
103C2×C4×C8D412yes—
104C2×C4×C8D412yes—
105C2×C4×C8D412yes—
106C2×C4×C8D412yes—
107#384—21no—
108#382—21no—
109#380D414yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—24no—
112C4×C22⋊C4—24no—
113#376—22no—
114C43—22no—
115#340—24no—
116#336—24no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—28no—
121C2×#41—24no—
122C2×#41—28no—
123C2×#41—28no—
124C2×#41—28no—
125C22×Q8C22×C414yes—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4C22×C414yes—
129C23×C4—24no—
130C23×C4C2×D412yes—
131C23×C4C2411yescharacteristic
132#135—48no—
133#135—48no—
134#135—416no—
135#135—48no—
136#135—48no—
137#135—44no—
138#135—44no—
139#135—44no—
140#135—44no—
141C2×Q16—24no—
142C2×Q16—22no—
143C2×Q16—22no—
144C2×Q16—28no—
145C2×Q16—44no—
146C2×SD16—28no—
147C2×SD16—24no—
148C2×SD16—24no—
149C2×SD16—216no—
150C2×SD16—48no—
151C2×D8—24no—
152C2×D8—22no—
153C2×D8—22no—
154C2×D8—28no—
155C2×D8—44no—
156C2×M16—22no—
157C2×M16—24no—
158C2×M16—24no—
159C2×M16C2×D414yes—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166C22×C8C22×C411yescharacteristic
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D414yes—
174C22×C8#4114yes—
175C4⋊Q8—24no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178#127—24no—
179#127—22no—
180#127—22no—
181#124—28no—
182#124—24no—
183#124—24no—
184#121—44no—
185#121—44no—
186#121—44no—
187#121—44no—
188C4×D4—44no—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
192#117—24no—
193#117—24no—
194#117—24no—
195#117—24no—
196#117—24no—
197#117C2×D418yes—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4C2×D414yes—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—44no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42#4112yes—
217#107—28no—
218#106—28no—
219C4⋊C8—24no—
220C4⋊C8—24no—
221#104—432no—
222#99—432no—
223#97—44no—
224C4×C8—44no—
225C4×C8C22⋊C412yes—
226C4×C8C22⋊C412yes—
227C4×C8C22⋊C412yes—
228C4×C8C22⋊C412yes—
229C4×C8C22⋊C412yes—
230C4×C8C22⋊C412yes—
231C4×C8C22⋊C412yes—
232C4×C8C22⋊C412yes—
233#95—24no—
234#95—44no—
235C24—24no—
236C24C22×D411yescharacteristic
237#41—48no—
238#41—44no—
239#41—44no—
240#41—44no—
241#41—44no—
242#41—48no—
243#41—48no—
244#41—48no—
245#41—48no—
246#41—48no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—28no—
250C2×Q8—28no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—28no—
254C2×D4—28no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—88no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4#12112yes—
273C22×C4#12112yes—
274C22×C4#12712yes—
275C22×C4#12712yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D412yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4112yes—
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4H(5,𝔽2)12yes—
285C22×C4H(5,𝔽2)12yes—
286Q16—48no—
287Q16—44no—
288Q16—44no—
289SD16—416no—
290SD16—48no—
291SD16—48no—
292D8—48no—
293D8—44no—
294D8—44no—
295M16—28no—
296M16—44no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8C2×C22⋊C412yes—
306C2×C8C2×C22⋊C412yes—
307C2×C8C2×C22⋊C411yescharacteristic
308C2×C8C2×C22⋊C411yescharacteristic
309C2×C8C2×C22⋊C411yescharacteristic
310C2×C8C2×C22⋊C411yescharacteristic
311C2×C8C4×D414yes—
312C2×C8C4×D414yes—
313C2×C8#12114yes—
314C2×C8#12114yes—
315C2×C8#12412yes—
316C2×C8#12412yes—
317C2×C8#12712yes—
318C2×C8#12712yes—
319C4⋊C4—28no—
320C4⋊C4—48no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—44no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C42—28no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42—88no—
335C42C2×C22⋊C412yes—
336C42C2×C22⋊C412yes—
337C42C4×D411yescharacteristic
338C42C4×D411yescharacteristic
339C42C4×D411yescharacteristic
340C42C4×D411yescharacteristic
341C42#12112yes—
342C42#12112yes—
343C42#12412yes—
344C42#12412yes—
345C23—42no—
346C23—44no—
347C23—48no—
348C23—88no—
349C23C2×C4×D411yescharacteristic
350C23C2×#12111yescharacteristic
351C23C2×#12711yescharacteristic
352C23D4212yes—
353C23#55012yes—
354Q8—44no—
355Q8—48no—
356Q8—44no—
357D4—44no—
358D4—48no—
359D4—44no—
360D4—88no—
361D4—88no—
362D4—84no—
363D4—84no—
364D4—88no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—88no—
377C2×C4—84no—
378C2×C4—88no—
379C2×C4—84no—
380C2×C4#39012yes—
381C2×C4#39012yes—
382C2×C4#39012yes—
383C2×C4#39012yes—
384C2×C4C22×C22⋊C411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4C2×C4×D412yes—
387C2×C4#51812yes—
388C2×C4C2×#12112yes—
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×#12411yescharacteristic
391C2×C4C2×#12411yescharacteristic
392C2×C4C2×#12711yescharacteristic
393C2×C4#53511yescharacteristic
394C2×C4#53511yescharacteristic
395C2×C4#53512yes—
396C2×C4D4212yes—
397C2×C4#55012yes—
398C2×C4#55012yes—
399C2×C4#55012yes—
400C2×C4#55912yes—
401C2×C4#55912yes—
402C8—22no—
403C8—22no—
404C8—44no—
405C8#39012yes—
406C8#39012yes—
407C8#39012yes—
408C8#39012yes—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—42no—
413C22—42no—
414C22—84no—
415C22—88no—
416C22—168no—
417C22C2×#39011yescharacteristic
418C22#236612yes—
419C22#254112yes—
420C22#254111yescharacteristic
421C22#254811yescharacteristic
422C4—22no—
423C4—22no—
424C4—22no—
425C4—84no—
426C4—88no—
427C4—84no—
428C4#197614yescomplemented
429C4C2×#39011yescharacteristic
430C4#230911yescharacteristic
431C4#236612yes—
432C4#254111yescharacteristic
433C4#254812yes—
434C4#254811yescharacteristic
435C2—21no—
436C2—22no—
437C2—22no—
438C2—21no—
439C2—164no—
440C2#1415411yescharacteristic
441C2C2×#197612yes—
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#390
prime radicalC2×#3147C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1976C2semidirect
3. #21212C2semidirect
4. #21209C2semidirect
5. C4×#427C2semidirect
6. #20437C2semidirect
7. #11026C2semidirect
15. C2×#495C4semidirect
16. C2×#495C4semidirect
17. C2×#494C4semidirect
18. C2×#493C4semidirect
19. C2×#492C4semidirect
83. #427D4semidirect
428. C4#1976semidirect

Maximal subgroups

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

subgroupindex
C2×#31472
C2×#19762
#212122
#212092
C4×#4272
#204372
#110262