Finite Group Databasebeta

← #192536

Subgroups — #192536

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2961C211yesmaximal, characteristic
2#21353C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#17562C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6#8881C214yesmaximal, complemented
7#8879C214yesmaximal, complemented
8#8093C211yesmaximal, characteristic
9#7919C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2961—22no—
12#2961—22no—
13#2961—22no—
14#2961C2214yes—
15C2×#446—22no—
16C2×#446—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#424—21no—
21C2×#423C2211yescharacteristic
22C2×#405—21no—
23C22×#97C2211yescharacteristic
24#2283C2211yescharacteristic
25#1940C2212yes—
26#1940C2212yes—
27#1850—21no—
28#1850—21no—
29#1768—22no—
30#1574—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1565—22no—
34#1515C414yes—
35#1511C414yes—
36#1503C2214yes—
37#1485—22no—
38#1481C2212yes—
39#1481C2212yes—
40#1479—22no—
41#1303C2214yes—
42#1281C2214yes—
43C22×M16—21no—
44C23×C8D411yescharacteristic
45C23×C8D411yescharacteristic
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533C2×C412yes—
51#533D412yes—
52#533Q812yes—
53#533C2312yes—
54C2×C4⋊Q8Q811yescharacteristic
55C2×C4⋊Q8Q811yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8C2×C412yes—
59C2×C4×Q8—21no—
60C2×C4×Q8D412yes—
61C2×#117—21no—
62C22×C42C2311yescharacteristic
63#446—28no—
64#446—28no—
65#446—28no—
66#446—28no—
67#432—21no—
68#432—21no—
69#432—21no—
70#432C2×C412yes—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431C2312yes—
78#431C2312yes—
79#424—24no—
80#424—24no—
81#423—21no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423D412yes—
86#423C2312yes—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8D412yes—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94#407—21no—
95#407C2×C412yes—
96C2×C22⋊C8—21no—
97#405—24no—
98#405—24no—
99C4×M16—21no—
100C4×M16—21no—
101C2×#97—22no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97D412yes—
107C2×#97C2312yes—
108C2×C4×C8—22no—
109C2×C4×C8—21no—
110#383C2×C412yes—
111#382—21no—
112#381C2×C412yes—
113#380—21no—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—21no—
116#376—21no—
117#336—22no—
118#336—22no—
119#335—24no—
120#334—24no—
121#328C2×C4116yes—
122Q8⋊C8—24no—
123Q8⋊C8—24no—
124C22×Q8—21no—
125C22×Q8C4⋊C412yes—
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4111yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—44no—
133C2×M16—41no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—24no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—42no—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8C4⋊C412yes—
150C4⋊Q8C4⋊C412yes—
151C4⋊Q8C2×Q812yes—
152C4⋊Q8C2×Q812yes—
153#125—22no—
154#125—22no—
155#125—22no—
156#125—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8C4⋊C414yes—
164C22⋊Q8C22×C414yes—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8C4⋊C414yes—
172C4×Q8C2×D414yes—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C4⋊C412yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×Q811yescharacteristic
195C2×C42C2×Q811yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42C2411yescharacteristic
200C4⋊C8—216no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—44no—
212C4⋊C8—44no—
213C4⋊C8—41no—
214C4⋊C8C2×D414yes—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220C22⋊C8C22×C414yes—
221#97—28no—
222#97—22no—
223#97—22no—
224#97—22no—
225#97—22no—
226#97—22no—
227#97—41no—
228C4×C8—28no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—41no—
232C4×C8—41no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95C22×C414yes—
237C24C4×D411yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—44no—
242C2×Q8—82no—
243C2×Q8C2×C4⋊C414yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C4×D411yescharacteristic
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×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C2×#4111yescharacteristic
269M16—44no—
270M16—44no—
271M16—44no—
272M16—82no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—44no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8#13714yes—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—82no—
303C4⋊C4C2×C4⋊C414yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—44no—
307C42—24no—
308C42—24no—
309C42—42no—
310C42—42no—
311C42C2×C4⋊C411yescharacteristic
312C42C2×C4⋊C411yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42#12011yescharacteristic
316C42#12011yescharacteristic
317C42C22⋊Q811yescharacteristic
318C42C22⋊Q811yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42#12311yescharacteristic
322C42#12311yescharacteristic
323C42C22×D411yescharacteristic
324C42C22×D411yescharacteristic
325C42C22×Q811yescharacteristic
326C42C2×#4111yescharacteristic
327C23—22no—
328C23—22no—
329C23—41no—
330C23#38511yescharacteristic
331C23#38511yescharacteristic
332C23C2×C4×D411yescharacteristic
333Q8—44no—
334Q8—82no—
335Q8—82no—
336Q8—82no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4#38511yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38511yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4#38511yescharacteristic
369C2×C4C22×C4⋊C411yescharacteristic
370C2×C4C2×C4×D411yescharacteristic
371C2×C4C2×#12011yescharacteristic
372C2×C4C2×C22⋊Q811yescharacteristic
373C2×C4C2×C22⋊Q811yescharacteristic
374C2×C4C2×#12311yescharacteristic
375C2×C4C2×#13712yes—
376C2×C4#57512yes—
377C8—24no—
378C8—44no—
379C8—42no—
380C8—44no—
381C8—82no—
382C8—81no—
383C8—81no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22#186812yes—
393C22C2×#38511yescharacteristic
394C22C4×#13712yes—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—44no—
406C4—44no—
407C4—84no—
408C4#300411yescharacteristic
409C4#300511yescharacteristic
410C4#307812yes—
411C2—22no—
412C2—22no—
413C2—41no—
414C2#2086711yescharacteristic
415C2#2087411yescharacteristic
416C2C2×#186811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ 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. #8881C2semidirect
7. #8879C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29612
#213532
#178092
#175622
#174642
#88812
#88792
#80932
#79192