Finite Group Databasebeta

← #190330

Subgroups — #190330

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34190C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3C2×#1466C211yesmaximal, characteristic
4#8773C214yesmaximal, complemented
5#8772C214yesmaximal, complemented
6#8069C212yesmaximal, complemented
7#7977C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9#2920C2212yes—
10C2×#432C2212yes—
11#2357C412yes—
12#2353C2212yes—
13#2353C2212yes—
14#2349C412yes—
15#2344—21no—
16#2342—22no—
17D4×C22⋊C4—21no—
18#2319—21no—
19#2318—22no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#2280C412yes—
23C2×#381C412yes—
24#2275—21no—
25#1821—21no—
26#1800—22no—
27#1783—21no—
28C4×#100C2214yes—
29#1466—21no—
30#1466—21no—
31#1466C2214yes—
32C2×#323—22no—
33#1300C2214yescomplemented
34#1300C2214yescomplemented
35#1288C2218yescomplemented
36C23×D4D411yescharacteristic
37C24×C4C2×C412yes—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4D412yes—
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4C2312yes—
49C2×#117—22no—
50C22×C4⋊C4D411yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4D412yes—
53C22×C42C2311yescharacteristic
54#432—22no—
55#432—22no—
56#432—22no—
57#432—22no—
58#432—22no—
59#432C2314yes—
60#431—22no—
61#431—22no—
62#423—22no—
63C2×C4⋊C8—22no—
64C2×#100—24no—
65#407—22no—
66C2×C22⋊C8—22no—
67C4×M16C2314yes—
68C2×#97—22no—
69#390—22no—
70#389—22no—
71#388—22no—
72#388—22no—
73#388—22no—
74#388—22no—
75#387—22no—
76#387—22no—
77#386—22no—
78#386—22no—
79#385—22no—
80#385—22no—
81#383—21no—
82#383C2312yes—
83#382—21no—
84#382C2×C412yes—
85#381—22no—
86#381—22no—
87#381C2×C412yes—
88#381C2×C412yes—
89#381C2×C411yescharacteristic
90#381C2×C412yes—
91#381C2×C411yescharacteristic
92#380—22no—
93#380—21no—
94#380—21no—
95#380C2×C412yes—
96#380C2×C412yes—
97#379—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4C2×C412yes—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4C2×C412yes—
107C4×C22⋊C4C2×C412yes—
108#376—21no—
109#376C2311yescharacteristic
110#376C2311yescharacteristic
111C2×#95—22no—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114#343—24no—
115#323—24no—
116#323—24no—
117C25C22⋊C412yes—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—44no—
121C22×D4C22⋊C411yescharacteristic
122C22×D4C22⋊C411yescharacteristic
123C22×D4C2×D412yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—84no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C22×C8—42no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—48no—
154#117—84no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C2×D412yes—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—84no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C2×D412yes—
179C2×C22⋊C4C2×D412yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4112yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4112yes—
192C2×C42C2411yescharacteristic
193C4⋊C8—24no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196C4⋊C8—42no—
197#100—48no—
198C22⋊C8—24no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202#97—24no—
203#97—24no—
204#97—42no—
205C4×C8—24no—
206#95—21no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—21no—
217#95—22no—
218#95C22×C412yes—
219#95C22×C412yes—
220#95C22×C412yes—
221#95C22×C412yes—
222C24—22no—
223C24—22no—
224C24—24no—
225C24—42no—
226C24—44no—
227C24—82no—
228C24C2×C22⋊C411yescharacteristic
229C24C2×C22⋊C412yes—
230C24C2×C22⋊C412yes—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—48no—
239C2×D4—48no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4—81no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4#11711yescharacteristic
278C22×C4#11711yescharacteristic
279C22×C4#11711yescharacteristic
280C22×C4#11711yescharacteristic
281C22×C4#12312yes—
282C22×C4#12312yes—
283C22×C4#12312yes—
284C22×C4#12312yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288M16—48no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—84no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—44no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C42—44no—
313C42—81no—
314C42—81no—
315C42—82no—
316C42—82no—
317C42#11712yes—
318C42#11712yes—
319C42#12312yes—
320C42#12312yes—
321C42#12312yes—
322C42#12312yes—
323C42C2×#4112yes—
324C42C2×#4112yes—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—41no—
333C23—44no—
334C23—44no—
335C23—42no—
336C23—42no—
337C23—44no—
338C23—42no—
339C23—42no—
340C23—44no—
341C23—82no—
342C23—82no—
343C23—88no—
344C23—88no—
345C23#38111yescharacteristic
346C23#38111yescharacteristic
347C23#42114yes—
348C23C22×C22⋊C411yescharacteristic
349D4—28no—
350D4—48no—
351D4—44no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—164no—
393C2×C4—164no—
394C2×C4#38112yes—
395C2×C4#38112yes—
396C2×C4#38111yescharacteristic
397C2×C4#38111yescharacteristic
398C2×C4C2×#11711yescharacteristic
399C2×C4C2×#11711yescharacteristic
400C2×C4C2×#12312yes—
401C2×C4C2×#12312yes—
402C8—84no—
403C8—84no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—24no—
410C22—21no—
411C22—21no—
412C22—44no—
413C22—44no—
414C22—44no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—88no—
421C22—82no—
422C22—82no—
423C22—88no—
424C22#182912yes—
425C22C2×#38111yescharacteristic
426C22C2×#42112yes—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4—82no—
441C2—22no—
442C2—22no—
443C2—41no—
444C2—44no—
445C2—82no—
446C2#2070511yescharacteristic
447C2#2070811yescharacteristic
448C2C2×#182911yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8773C2semidirect
5. #8772C2semidirect
6. #8069C2semidirect
7. #7977C2semidirect
33. #1300C22semidirect
34. #1300C22semidirect
35. #1288C22semidirect

Maximal subgroups

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

subgroupindex
#341902
#207262
C2×#14662
#87732
#87722
#80692
#79772