Finite Group Databasebeta

← #192539

Subgroups — #192539

Order 512.

800 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
1#60251C211yesmaximal, characteristic
2#21977C211yesmaximal, characteristic
3#17807C211yesmaximal, characteristic
4#17562C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#8881C214yesmaximal, complemented
7#8879C214yesmaximal, complemented
8#8096C211yesmaximal, characteristic
9#7922C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983—22no—
15#2962—21no—
16#2962—21no—
17#2961C2212yes—
18#2961C2212yes—
19#2924—22no—
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25#2283C2211yescharacteristic
26#1940C2214yes—
27#1850—22no—
28#1843—21no—
29#1832—21no—
30#1573—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1565—22no—
34#1515C414yes—
35#1511C414yes—
36#1503C2214yes—
37#1481—22no—
38#1481C2214yes—
39C2×Q8⋊C8—22no—
40#1303C2212yes—
41#1303C2212yes—
42#1281C2214yes—
43C23×C8—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
50#533Q812yes—
51#533C2312yes—
52C2×C4⋊Q8Q811yescharacteristic
53C2×C4⋊Q8Q811yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×Q8D412yes—
59C2×#117—21no—
60C22×C42C2311yescharacteristic
61#446—24no—
62#446—24no—
63#446—22no—
64#446—22no—
65#446—24no—
66#446—24no—
67#446—24no—
68#446—24no—
69C8×Q8—22no—
70C8×Q8—22no—
71#433—24no—
72#433—24no—
73#432—24no—
74#432—21no—
75#432—21no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432—21no—
80#432C2×C412yes—
81#432D412yes—
82#431—24no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431D412yes—
89#431C2312yes—
90#431C2312yes—
91#424—22no—
92#424—22no—
93#423—21no—
94#423C2311yescharacteristic
95#423C2311yescharacteristic
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—24no—
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8C2312yes—
106#407—21no—
107#407C2×C412yes—
108C2×C22⋊C8—21no—
109C2×C22⋊C8D412yes—
110#405—22no—
111#405—22no—
112C4×M16—21no—
113C4×M16—21no—
114C2×#97C2312yes—
115C2×C4×C8—21no—
116#383C2×C412yes—
117#382—21no—
118#381C2×C412yes—
119#380—21no—
120C4×C4⋊C4C2312yes—
121C4×C22⋊C4—21no—
122#376—21no—
123#336—24no—
124#335—24no—
125#334—24no—
126#328C2×C4116yes—
127Q8⋊C8—28no—
128C22×Q8—21no—
129C22×Q8C4⋊C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C2×M16—22no—
134C2×M16—21no—
135C2×M16—21no—
136C22×C8—22no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—82no—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8C4⋊C412yes—
148C4⋊Q8C4⋊C412yes—
149C4⋊Q8C2×Q812yes—
150C4⋊Q8C2×Q812yes—
151#125—22no—
152#125—22no—
153#125—22no—
154#125—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8C4⋊C414yes—
162C22⋊Q8C22×C414yes—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8C4⋊C414yes—
170C4×Q8C2×D414yes—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C4⋊C412yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42C2×Q811yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42C2411yescharacteristic
198C4⋊C8—24no—
199C4⋊C8—24no—
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—22no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—84no—
217C4⋊C8C2×D412yes—
218C4⋊C8C2×D412yes—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—41no—
226C22⋊C8C22×C414yes—
227#97—24no—
228#97—24no—
229#97—22no—
230#97—22no—
231#97—22no—
232#97—22no—
233C4×C8—24no—
234C4×C8—24no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—41no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95C22×C414yes—
243C24C4×D411yescharacteristic
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—44no—
248C2×Q8—82no—
249C2×Q8C2×C4⋊C414yes—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—42no—
256C22×C4—81no—
257C22×C4C2×C4⋊C411yescharacteristic
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4C22⋊Q811yescharacteristic
269C22×C4C22⋊Q811yescharacteristic
270C22×C4#12311yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4111yescharacteristic
275M16—44no—
276M16—42no—
277M16—42no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—44no—
288C2×C8—82no—
289C2×C8—82no—
290C2×C8—82no—
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—44no—
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—82no—
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×C4#57712yes—
376C2×C4#57812yes—
377C8—44no—
378C8—42no—
379C8—42no—
380C8—42no—
381C8—42no—
382C8—82no—
383C8—82no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22#186812yes—
393C22C2×#38511yescharacteristic
394C22#294712yes—
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#301011yescharacteristic
409C4#301111yescharacteristic
410C4#308212yes—
411C2—22no—
412C2—22no—
413C2—41no—
414C2#2086711yescharacteristic
415C2#2087711yescharacteristic
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
#602512
#219772
#178072
#175622
C2×#14812
#88812
#88792
#80962
#79222