Finite Group Databasebeta

← #192972

Subgroups — #192972

Order 512.

804 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60249C211yesmaximal, characteristic
2#21970C211yesmaximal, characteristic
3#17793C211yesmaximal, characteristic
4#17491C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8909C214yesmaximal, complemented
7#8907C214yesmaximal, complemented
8#8078C211yesmaximal, characteristic
9#7885C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2986—22no—
13#2985—22no—
14#2985—22no—
15Q8×M16—22no—
16Q8×M16C2214yes—
17#2921—22no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#1950—22no—
25#1950C2214yes—
26#1811—22no—
27#1568—22no—
28#1567C2212yes—
29#1566C2212yes—
30#1564—22no—
31#1497—22no—
32#1493C412yes—
33#1493C412yes—
34#1493C412yes—
35#1493C412yes—
36#1487C2214yes—
37#1485—22no—
38C2×Q8⋊C8C2212yes—
39C2×Q8⋊C8C2212yes—
40#1301C2214yes—
41#1278C2212yes—
42#1278C2212yes—
43C22×M16—21no—
44C23×C8—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533C2×C412yes—
50#533C2×C412yes—
51#533D412yes—
52#533D412yes—
53C2×C4⋊Q8Q811yescharacteristic
54C2×C4⋊Q8Q811yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8Q812yes—
58C2×C4×Q8—21no—
59C2×C4×Q8C2312yes—
60C2×#117—21no—
61C22×C42C2311yescharacteristic
62#446—24no—
63#446—28no—
64#446—24no—
65#446—28no—
66C8×Q8—24no—
67C8×Q8—24no—
68#433—28no—
69#433—28no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432—21no—
77#432D412yes—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431D412yes—
85#431C2312yes—
86C2×#108—24no—
87#423—24no—
88#423—21no—
89#423—21no—
90#423—21no—
91#423C2×C412yes—
92#423C2×C412yes—
93#423D412yes—
94#423C2312yes—
95#423C2312yes—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2312yes—
98#407—21no—
99#407—21no—
100#407—21no—
101#407D412yes—
102C4×M16—24no—
103C4×M16—21no—
104C4×M16—21no—
105C4×M16—21no—
106C4×M16C2312yes—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2×C411yescharacteristic
111C2×C4×C8C2×C411yescharacteristic
112C2×C4×C8C2×C411yescharacteristic
113C2×C4×C8C2×C411yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#341—24no—
117#335—24no—
118#334—24no—
119Q8⋊C8—28no—
120Q8⋊C8C2×C414yes—
121Q8⋊C8C2×C414yes—
122Q8⋊C8C2×C414yes—
123Q8⋊C8C2×C414yes—
124C22×Q8—21no—
125C22×Q8C2×Q812yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×Q811yescharacteristic
128C23×C4#4111yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—41no—
134C2×M16—82no—
135C2×M16C2×D414yes—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—44no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8C4⋊C412yes—
149C4⋊Q8C4⋊C412yes—
150C4⋊Q8C4⋊C412yes—
151C4⋊Q8C4⋊C412yes—
152#125—22no—
153#125—22no—
154#125—22no—
155#125—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8C4⋊C414yes—
163C22⋊Q8C4⋊C414yes—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8C22×C414yes—
171C4×Q8C2×D414yes—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C2×Q812yes—
182C2×C22⋊C4—22no—
183C2×C42—22no—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q811yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196#108—84no—
197C4⋊C8—28no—
198C4⋊C8—28no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—41no—
209C4⋊C8C22×C414yes—
210C4⋊C8C2×D414yes—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—41no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219#97—28no—
220#97—22no—
221#97—22no—
222#97—41no—
223#97—41no—
224C4×C8—28no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—22no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—41no—
235C4×C8C22×C411yescharacteristic
236C4×C8C22×C411yescharacteristic
237C4×C8C22×C411yescharacteristic
238C4×C8C22×C411yescharacteristic
239C24C22⋊Q811yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—44no—
244C2×Q8—82no—
245C2×Q8C2×C4⋊C414yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C2×#4111yescharacteristic
270M16—24no—
271M16—24no—
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276M16—82no—
277M16—82no—
278M16—82no—
279C2×C8—24no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—44no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4C2×C4⋊C414yes—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C42—24no—
310C42—24no—
311C42C2×C4⋊C411yescharacteristic
312C42C2×C4⋊C411yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42#12011yescharacteristic
318C42#12011yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42#12311yescharacteristic
322C42#12311yescharacteristic
323C42C23×C411yescharacteristic
324C42C22×D411yescharacteristic
325C42C22×D411yescharacteristic
326C42C2×#4111yescharacteristic
327C23—22no—
328C23—41no—
329C23—41no—
330C23#38511yescharacteristic
331C23#38511yescharacteristic
332C23C2×C22⋊Q811yescharacteristic
333Q8—28no—
334Q8—44no—
335Q8—82no—
336Q8—82no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#38511yescharacteristic
361C2×C4#38511yescharacteristic
362C2×C4#38511yescharacteristic
363C2×C4#38511yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4C22×C4⋊C411yescharacteristic
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C2×C4×D411yescharacteristic
369C2×C4C2×#12011yescharacteristic
370C2×C4C2×C22⋊Q811yescharacteristic
371C2×C4C2×#12311yescharacteristic
372C2×C4#57712yes—
373C2×C4#57812yes—
374C8—22no—
375C8—22no—
376C8—41no—
377C8—41no—
378C8—44no—
379C8—44no—
380C8—44no—
381C8—82no—
382C8—82no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22C2×#38511yescharacteristic
393C22#308212yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—44no—
403C4—44no—
404C4#186112yes—
405C4#291612yes—
406C4#294911yescharacteristic
407C4#295011yescharacteristic
408C4#301312yes—
409C2—22no—
410C2—41no—
411C2—41no—
412C2#2085211yescharacteristic
413C2C2×#186111yescharacteristic
414C2#2088011yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C22⋊Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8909C2semidirect
7. #8907C2semidirect

Maximal subgroups

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

subgroupindex
#602492
#219702
#177932
#174912
C2×#14932
#89092
#89072
#80782
#78852