Finite Group Databasebeta

← #117178

Subgroups — #117178

Order 512.

790 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21959C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5#10851C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C22×#103—22no—
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1993—22no—
17#1990—22no—
18#1990—22no—
19#1987—22no—
20#1981—22no—
21#1981C414yescomplemented
22#1948—22no—
23#1948—22no—
24#1824—22no—
25#1807—22no—
26#1736—22no—
27C2×#339—22no—
28#1294—22no—
29#1283—22no—
30C23×Q8—21no—
31C22×M16D412yes—
32C23×C8D412yes—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8D412yes—
44C2×C4⋊Q8Q812yes—
45C2×#117—21no—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56C2×#106—24no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C412yes—
63C2×C4⋊C8D412yes—
64C2×#103—28no—
65C2×#103—48no—
66C2×C22⋊C8D412yes—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74#384D412yes—
75#384D412yes—
76#382—21no—
77#382—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2×C412yes—
90C2×#95—21no—
91#340—24no—
92#340—28no—
93#340—28no—
94#339—44no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—44no—
99C22×Q8—44no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16SD1614yes—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—44no—
119C22×C8C22⋊C414yes—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8C4⋊C414yes—
136#117—42no—
137#117—42no—
138#117—42no—
139#117—42no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C42—22no—
165C2×C42—42no—
166C2×C42—42no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×Q811yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182#106—84no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189#103—416no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192#97—41no—
193#97—41no—
194C4×C8—41no—
195C4×C8—41no—
196C4×C8—41no—
197C4×C8—41no—
198#95—22no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204C24#12711yescharacteristic
205C2×Q8—24no—
206C2×Q8—44no—
207C2×Q8—48no—
208C2×Q8—44no—
209C2×Q8—48no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—82no—
214C2×Q8—82no—
215C2×Q8—84no—
216C2×Q8—82no—
217C2×Q8—82no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—82no—
229C22×C4—84no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q811yescharacteristic
245C22×C4C22⋊Q811yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4C2×SD1612yes—
251C22×C4#13512yes—
252C22×C4#13712yes—
253C22×C4#13712yes—
254M16—24no—
255M16—24no—
256M16—42no—
257M16—42no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—44no—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—42no—
266C2×C8—82no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—82no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4—81no—
290C42—44no—
291C42—82no—
292C42—82no—
293C42—82no—
294C42C2×C4⋊C411yescharacteristic
295C42#11711yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42C22⋊Q811yescharacteristic
303C42C22⋊Q811yescharacteristic
304C42#12311yescharacteristic
305C42#12311yescharacteristic
306C42#12411yescharacteristic
307C42#12411yescharacteristic
308C42#12611yescharacteristic
309C42#12611yescharacteristic
310C23—22no—
311C23—22no—
312C23#39011yescharacteristic
313C23#39211yescharacteristic
314C23#39911yescharacteristic
315C23#49211yescharacteristic
316C23#49511yescharacteristic
317C23#49711yescharacteristic
318C23#49711yescharacteristic
319Q8—48no—
320Q8—84no—
321Q8—82no—
322Q8—82no—
323Q8—84no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—82no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#38011yescharacteristic
353C2×C4#38511yescharacteristic
354C2×C4#38511yescharacteristic
355C2×C4#38611yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#39311yescharacteristic
359C2×C4#39411yescharacteristic
360C2×C4#39411yescharacteristic
361C2×C4#39611yescharacteristic
362C2×C4#39611yescharacteristic
363C2×C4#39711yescharacteristic
364C2×C4C4×SD1612yes—
365C2×C4#44112yes—
366C2×C4#44812yes—
367C2×C4#45212yes—
368C2×C4#45312yes—
369C2×C4#45412yes—
370C2×C4#46112yes—
371C2×C4#46412yes—
372C2×C4#46512yes—
373C2×C4#47012yes—
374C2×C4#47512yes—
375C2×C4#47912yes—
376C2×C4#48312yes—
377C2×C4#48512yes—
378C8—44no—
379C8—44no—
380C8—82no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22#143911yescharacteristic
390C22#196811yescharacteristic
391C22#197111yescharacteristic
392C22#200612yes—
393C22#200912yes—
394C22#201311yescharacteristic
395C22#201511yescharacteristic
396C22#209611yescharacteristic
397C22#209711yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—84no—
405C4—84no—
406C4—162no—
407C4—162no—
408C4#186512yes—
409C4#188711yescharacteristic
410C4#188811yescharacteristic
411C4#192712yes—
412C4#202712yes—
413C4#203512yes—
414C4#206612yes—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2#1020112yes—
420C2#1020212yes—
421C2#1105411yescharacteristic
422C2#1112211yescharacteristic
423C2#1113011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1981C4semidirect

Maximal subgroups

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

subgroupindex
#219592
#218342
#213242
C2×#19812
#108512
#80142
C2×#12832