Finite Group Databasebeta

← #190199

Subgroups — #190199

Order 512.

1171 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#8760C218yesmaximal, complemented
5#8014C212yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×C4×#41—21no—
8C2×C4×#41C2211yescharacteristic
9C2×#431C2212yes—
10C2×#410—21no—
11#2351C412yescomplemented
12Q8×C4⋊C4—21no—
13#2349C412yescomplemented
14#2349C2214yes—
15D4×C4⋊C4—21no—
16D4×C4⋊C4—22no—
17#2296—22no—
18#2296—21no—
19#2285C412yescomplemented
20#2285C2212yes—
21#2285C2211yescharacteristic
22C2×C4×C4⋊C4C412yescomplemented
23C2×C4×C4⋊C4C2211yescharacteristic
24#1799—22no—
25#1799C2214yes—
26#1740—21no—
27#1461—21no—
28#1460—21no—
29#1459C2214yes—
30#1458—22no—
31#1294C2218yes—
32#1289C2218yes—
33C22×#41D411yescharacteristic
34C4×#41—22no—
35C4×#41—22no—
36C4×#41—22no—
37C4×#41—22no—
38C4×#41—44no—
39C4×#41—44no—
40C2×C4×Q8—22no—
41C2×C4×Q8C2×C411yescharacteristic
42C2×C4×Q8D411yescharacteristic
43C2×C4×D4—22no—
44C2×C4×D4—24no—
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4D412yes—
47C2×C4×D4D411yescharacteristic
48C2×C4×D4C2312yes—
49C2×#117—22no—
50C2×#117—21no—
51C2×#117—22no—
52C2×#117D411yescharacteristic
53C2×#117D412yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C42—22no—
57C22×C42—21no—
58C22×C42C2×C412yes—
59C22×C42C2311yescharacteristic
60#431—22no—
61#431—22no—
62#431—22no—
63#431—22no—
64C2×C4⋊C8—22no—
65#410—44no—
66C2×#99—24no—
67C2×C22⋊C8C2314yes—
68C2×C4×C8—22no—
69#389—22no—
70#389—24no—
71#389—22no—
72#389—24no—
73#385—24no—
74#385—24no—
75#385—24no—
76#384—22no—
77#384—21no—
78#384—22no—
79#382—22no—
80#382C2×C412yes—
81#382C2×C414yes—
82#380—22no—
83#380—21no—
84#380—22no—
85#380C2×C414yes—
86#380C2314yes—
87#380C2312yes—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C414yes—
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C411yescharacteristic
101C2×#95C2×C414yes—
102C2×#95C2312yes—
103#342—24no—
104#324—24no—
105#323—24no—
106C2×#41—22no—
107C2×#41—22no—
108C2×#41—24no—
109C2×#41D814yes—
110C2×#41SD1614yes—
111C22×Q8C22⋊C411yescharacteristic
112C22×D4C22⋊C411yescharacteristic
113C22×D4C2×D412yes—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22×C412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—24no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—48no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136C4×D4—48no—
137C4×D4—48no—
138C4×D4—48no—
139#117—22no—
140#117—22no—
141#117—24no—
142#117—24no—
143#117—48no—
144#117—42no—
145#117—42no—
146#117—44no—
147#117—44no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C2×D412yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C2×D412yes—
171C2×C22⋊C4C2×D412yes—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—48no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42#4112yes—
193C2×C42#4112yes—
194C2×C42#4112yes—
195C2×C42#4112yes—
196C4⋊C8—42no—
197C4⋊C8—42no—
198#99—48no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C4×C8—42no—
202C4×C8—42no—
203#95—24no—
204#95—24no—
205#95—22no—
206#95—22no—
207#95—24no—
208#95—42no—
209#95C22×C412yes—
210#95C22×C414yes—
211#95C22×C412yes—
212C24C2×C22⋊C412yes—
213C24C22×D411yescharacteristic
214#41—28no—
215#41—28no—
216#41—28no—
217#41—28no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8#10214yes—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4#10214yes—
226C2×D4C2×D814yes—
227C2×D4C2×SD1614yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—48no—
245C22×C4—44no—
246C22×C4—81no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4—82no—
250C22×C4#10218yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4#12312yes—
259C22×C4#12312yes—
260C22×C4#12312yes—
261C22×C4#12312yes—
262C22×C4C2×D812yes—
263C22×C4C2×SD1612yes—
264C22×C4#13714yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C2×C8—44no—
270C2×C8—84no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—84no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—48no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—42no—
302C42—44no—
303C42—42no—
304C42—48no—
305C42—44no—
306C42—84no—
307C42#11711yescharacteristic
308C42#11711yescharacteristic
309C42#11712yes—
310C42#11712yes—
311C42#11711yescharacteristic
312C42#11711yescharacteristic
313C42#12312yes—
314C42#12312yes—
315C42#12312yes—
316C42#12312yes—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23C2×#10214yes—
324C23C22×C22⋊C411yescharacteristic
325C23C2×#12312yes—
326C23C22×D811yescharacteristic
327C23C22×SD1611yescharacteristic
328C23C2×#13712yes—
329Q8—28no—
330D4—28no—
331D4—28no—
332D4—28no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—816no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4#38111yescharacteristic
369C2×C4#38112yes—
370C2×C4#38111yescharacteristic
371C2×C4#38112yes—
372C2×C4#38111yescharacteristic
373C2×C4#38111yescharacteristic
374C2×C4C2×#10214yes—
375C2×C4C2×#10212yes—
376C2×C4C2×#10212yes—
377C2×C4#41912yes—
378C2×C4#41912yes—
379C2×C4#48014yes—
380C2×C4#48014yes—
381C2×C4#48314yes—
382C2×C4#48314yes—
383C2×C4C2×#11711yescharacteristic
384C2×C4C2×#11711yescharacteristic
385C2×C4C2×#12312yes—
386C8—84no—
387C22—21no—
388C22—22no—
389C22—21no—
390C22—22no—
391C22—24no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—44no—
396C22—44no—
397C22—42no—
398C22—42no—
399C22C2×#38111yescharacteristic
400C22C22×#10211yescharacteristic
401C22C2×#41911yescharacteristic
402C22C2×#48012yes—
403C22C2×#48312yes—
404C4—24no—
405C4—21no—
406C4—22no—
407C4—22no—
408C4—21no—
409C4—21no—
410C4—44no—
411C4—42no—
412C4—42no—
413C4—84no—
414C4—82no—
415C4—164no—
416C4#182312yes—
417C4#182312yes—
418C4#182314yes—
419C4#182712yes—
420C2—21no—
421C2—22no—
422C2—21no—
423C2—22no—
424C2—44no—
425C2C2×#182311yescharacteristic
426C2C2×#182711yescharacteristic
427C2#2071011yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8760C2semidirect
5. #8014C2semidirect
6. #7981C2semidirect
11. #2351C4semidirect
13. #2349C4semidirect
19. #2285C4semidirect
22. 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
#206322
#173492
#87602
#80142
#79812