Finite Group Databasebeta

← #117148

Subgroups — #117148

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21834C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#8043C211yesmaximal, characteristic
6#7897C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#428—22no—
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×#384C2211yescharacteristic
16#1997—22no—
17#1997C414yescomplemented
18#1990—22no—
19#1990—22no—
20#1990—21no—
21#1990—21no—
22#1987—22no—
23#1847—22no—
24#1845—21no—
25#1844—21no—
26#1824—22no—
27C2×#343—22no—
28C2×#340—22no—
29#1296—22no—
30#1279—22no—
31#1278—22no—
32C23×Q8—21no—
33C22×M16—21no—
34C22×M16D412yes—
35C23×C8—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⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D412yes—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#428—44no—
60#428—44no—
61C2×#107—24no—
62C2×#106—24no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423C2×C412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8D412yes—
70C2×C22⋊C8—21no—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16C2×C412yes—
74C4×M16D412yes—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8D412yes—
84#384—21no—
85#384—21no—
86#384—21no—
87#384D411yescharacteristic
88#384D411yescharacteristic
89#384D412yes—
90#382—21no—
91#380—21no—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95#343—28no—
96#340—24no—
97#340—24no—
98#340—28no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—41no—
105C22×Q8—82no—
106C22×Q8—82no—
107C23×C4—22no—
108C23×C4—41no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—41no—
117C2×M16—42no—
118C2×M16—42no—
119C2×M16C22⋊C414yes—
120C2×M16SD1614yes—
121C2×M16Q1614yes—
122C22×C8—22no—
123C22×C8—24no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—42no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146#117—42no—
147#117—42no—
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—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C4⋊C414yes—
176C2×C22⋊C4—41no—
177C2×C42—41no—
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×Q811yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190#107—48no—
191#106—48no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200#97—22no—
201#97—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—41no—
209C4×C8—41no—
210#95—41no—
211#95—41no—
212C24#12111yescharacteristic
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—84no—
218C2×Q8—84no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—84no—
222C2×Q8—84no—
223C2×Q8—82no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C4⋊C411yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12411yescharacteristic
255C22×C4#12611yescharacteristic
256C22×C4#12611yescharacteristic
257C22×C4#12711yescharacteristic
258C22×C4C2×SD1612yes—
259C22×C4C2×Q1612yes—
260C22×C4AGL(1,ℤ/8)12yes—
261C22×C4#13712yes—
262M16—24no—
263M16—24no—
264M16—24no—
265M16—44no—
266M16—82no—
267M16#10318yes—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—42no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—81no—
281C2×C8—81no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C22⋊C4—82no—
304C42—81no—
305C42—81no—
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42#12111yescharacteristic
311C42#12111yescharacteristic
312C42#12111yescharacteristic
313C42#12111yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42C22⋊Q811yescharacteristic
318C42#12311yescharacteristic
319C42#12311yescharacteristic
320C42#12411yescharacteristic
321C42#12411yescharacteristic
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23#38011yescharacteristic
327C23#39211yescharacteristic
328C23#39711yescharacteristic
329C23#46012yes—
330C23#46212yes—
331Q8—84no—
332Q8—84no—
333Q8—84no—
334Q8—84no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—44no—
348C2×C4—84no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—164no—
362C2×C4#38511yescharacteristic
363C2×C4#38511yescharacteristic
364C2×C4#38611yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#39011yescharacteristic
368C2×C4#39311yescharacteristic
369C2×C4#39411yescharacteristic
370C2×C4#39411yescharacteristic
371C2×C4#39611yescharacteristic
372C2×C4#39611yescharacteristic
373C2×C4#39911yescharacteristic
374C2×C4C2×#10312yes—
375C2×C4#41712yes—
376C2×C4#45012yes—
377C2×C4#45112yes—
378C2×C4#45412yes—
379C2×C4#45612yes—
380C2×C4#47112yes—
381C2×C4#47212yes—
382C2×C4#47512yes—
383C2×C4#47712yes—
384C2×C4#48312yes—
385C2×C4#48412yes—
386C2×C4#48712yes—
387C2×C4#48912yes—
388C2×C4#49211yescharacteristic
389C2×C4#49411yescharacteristic
390C2×C4#49611yescharacteristic
391C2×C4#49711yescharacteristic
392C8—28no—
393C8—44no—
394C8—82no—
395C8—82no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22#143911yescharacteristic
405C22#180312yes—
406C22#200311yescharacteristic
407C22#200511yescharacteristic
408C22#201411yescharacteristic
409C22#201611yescharacteristic
410C22#201811yescharacteristic
411C22#201911yescharacteristic
412C22#207212yes—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—84no—
418C4—161no—
419C4—162no—
420C4—161no—
421C4#185712yes—
422C4#187212yes—
423C4#193112yes—
424C4#195611yescharacteristic
425C4#195911yescharacteristic
426C4#202412yes—
427C4#203812yes—
428C4#206312yes—
429C4#209511yescharacteristic
430C4#209611yescharacteristic
431C2—22no—
432C2—22no—
433C2#1014811yescharacteristic
434C2#1015711yescharacteristic
435C2#1016511yescharacteristic
436C2#1017211yescharacteristic
437C2#1105011yescharacteristic
438C2#1113611yescharacteristic
439C2#1114411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
17. #1997C4semidirect

Maximal subgroups

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

subgroupindex
#218342
C2×#19972
C2×#19902
#208002
#80432
#78972
C2×#12782