Finite Group Databasebeta

← #362864

Subgroups — #362864

Order 512.

1311 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2#20637C211yesmaximal, characteristic
3C4×#410C211yesmaximal, characteristic
4#11348C214yesmaximal, complemented
5#10852C212yesmaximal, complemented
6#10836C212yesmaximal, complemented
7#10758C214yesmaximal
8C2×C4×#41C2212yes—
9C2×C4×#41C2211yescharacteristic
10C2×#423C2212yes—
11C2×#413C2212yes—
12C2×#410C2212yes—
13#2351—21no—
14Q8×C4⋊C4—21no—
15#2349—22no—
16#2349—21no—
17D4×C4⋊C4—21no—
18D4×C4⋊C4—22no—
19#2296C414yescomplemented
20#2296C2212yes—
21#2285C412yescomplemented
22#2285C2211yescharacteristic
23C4×#117C2212yes—
24C2×C4×C4⋊C4C412yescomplemented
25C2×C4×C4⋊C4C2211yescharacteristic
26#1801—21no—
27#1801—21no—
28#1800—21no—
29#1800—21no—
30C4×#99—22no—
31C4×#99—22no—
32#1742C2212yes—
33#1740C2212yes—
34C2×#339C2218yes—
35C2×H(3,ℤ/4)C2218yes—
36C22×#41C2311yescharacteristic
37C22×M16C2314yes—
38C4×#41—22no—
39C4×#41—22no—
40C4×#41—22no—
41C4×#41—22no—
42C4×#41—48no—
43C2×C4×Q8—22no—
44C2×C4×Q8D411yescharacteristic
45C2×C4×Q8D411yescharacteristic
46C2×C4×D4—24no—
47C2×C4×D4—22no—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×#117C2×C414yes—
53C2×#117C2×C412yes—
54C2×#117C2314yes—
55C2×#117C2312yes—
56C2×#117C2311yescharacteristic
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C42C2×C412yes—
60C22×C42C2×C414yes—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63#423—22no—
64#423—22no—
65#423—22no—
66#423—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69#413—28no—
70C2×#101—22no—
71C2×#100—22no—
72#410—24no—
73#410Q818yes—
74C2×#99—24no—
75#389—24no—
76#389—24no—
77#389—22no—
78#389—22no—
79#385—24no—
80#385—24no—
81#385—24no—
82#384—22no—
83#384—21no—
84#382—22no—
85#382—21no—
86#380—22no—
87#380—22no—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2×C411yescharacteristic
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4C2×C414yes—
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C414yes—
102C4×C4⋊C4C2×C412yes—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105#376—22no—
106C2×#95—22no—
107C2×#95—21no—
108C43—22no—
109#343—22no—
110#343—22no—
111#342—24no—
112#339—48no—
113H(3,ℤ/4)—48no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41—24no—
117C2×#41C2×Q814yes—
118C2×#41#4114yes—
119C22×Q8C2×D411yescharacteristic
120C22×D4C2×D412yes—
121C22×D4C2×D411yescharacteristic
122C23×C4—24no—
123C23×C4—22no—
124C23×C4C22⋊C412yes—
125C23×C4C22⋊C412yes—
126C23×C4C22×C412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C2×M16—24no—
131C2×M16—28no—
132C2×M16—24no—
133C2×M16—28no—
134C22×C8—24no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—48no—
140C4×D4—24no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—416no—
149C4×D4—48no—
150#117—22no—
151#117—22no—
152#117—28no—
153#117—24no—
154#117—24no—
155#117—24no—
156#117—24no—
157#117—24no—
158#117—48no—
159#117C2×Q818yes—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C2×D412yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42—28no—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—48no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42C22⋊C412yes—
194C2×C42C22⋊C412yes—
195C2×C42C22×C412yes—
196C2×C42C22×C414yes—
197C2×C42C22×C412yes—
198C2×C42C22×C412yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C4⋊C8—42no—
208C4⋊C8—42no—
209C4⋊C8—42no—
210C4⋊C8—42no—
211#101—44no—
212#100—44no—
213#99—216no—
214#95—21no—
215#95—22no—
216#95—21no—
217#95—24no—
218#95—22no—
219#95—24no—
220#95—22no—
221#95—42no—
222#95—42no—
223C24C2×C22⋊C412yes—
224C24C22×D411yescharacteristic
225#41—28no—
226#41—28no—
227#41—28no—
228#41—28no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8#12412yes—
232C2×Q8C4⋊Q812yes—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4C22⋊Q818yes—
238C2×D4#12412yes—
239C2×D4C4⋊Q812yes—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—81no—
258C22×C4—82no—
259C22×C4—84no—
260C22×C4—82no—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C4×Q818yes—
269C22×C4#12312yes—
270C22×C4#12312yes—
271C22×C4#12312yes—
272C22×C4#12312yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×Q812yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4112yes—
278C22×C4C2×#4112yes—
279C22×C4H(5,𝔽2)14yes—
280M16—44no—
281M16—44no—
282M16—48no—
283C2×C8—44no—
284C2×C8—48no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C4⋊C4—48no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—82no—
299C4⋊C4—84no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—28no—
304C22⋊C4—48no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—24no—
314C42—24no—
315C42—28no—
316C42—28no—
317C42—48no—
318C42—44no—
319C42—44no—
320C42—84no—
321C42—84no—
322C42#11711yescharacteristic
323C42#11711yescharacteristic
324C42#11712yes—
325C42#11711yescharacteristic
326C42#11711yescharacteristic
327C42#11712yes—
328C42#12312yes—
329C42#12312yes—
330C42#12312yes—
331C42#12312yes—
332C23—22no—
333C23—22no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23#39114yes—
339C23C22×C22⋊C411yescharacteristic
340C23C2×C22⋊Q812yes—
341C23C2×#12312yes—
342C23#53512yes—
343Q8—24no—
344Q8—24no—
345D4—216no—
346D4—24no—
347D4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—88no—
379C2×C4—816no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4#38111yescharacteristic
383C2×C4#38111yescharacteristic
384C2×C4#38111yescharacteristic
385C2×C4#38112yes—
386C2×C4#38112yes—
387C2×C4#38111yescharacteristic
388C2×C4#39114yes—
389C2×C4C2×#11711yescharacteristic
390C2×C4C2×#11711yescharacteristic
391C2×C4C2×C4×Q812yes—
392C2×C4#51812yes—
393C2×C4C2×#12312yes—
394C2×C4C2×#12411yescharacteristic
395C2×C4C2×C4⋊Q811yescharacteristic
396C2×C4#53512yes—
397C2×C4#55314yes—
398C2×C4#55314yes—
399C2×C4#55414yes—
400C2×C4#55414yes—
401C8—84no—
402C22—21no—
403C22—22no—
404C22—21no—
405C22—22no—
406C22—24no—
407C22—42no—
408C22—42no—
409C22—44no—
410C22—44no—
411C22—42no—
412C22—42no—
413C22C2×#38111yescharacteristic
414C22C2×#39111yescharacteristic
415C22#230911yescharacteristic
416C22#248714yes—
417C4—24no—
418C4—21no—
419C4—22no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—44no—
424C4—42no—
425C4—42no—
426C4—82no—
427C4—84no—
428C4—88no—
429C4#182712yes—
430C4#234512yes—
431C4#234512yes—
432C4#250212yes—
433C4#250412yes—
434C2—21no—
435C2—24no—
436C2—21no—
437C2—44no—
438C2#1392411yescharacteristic
439C2C2×#182711yescharacteristic
440C2#2070811yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2285 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11348C2semidirect
5. #10852C2semidirect
6. #10836C2semidirect
19. #2296C4semidirect
21. #2285C4semidirect
24. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
#206372
C4×#4102
#113482
#108522
#108362
#107582