Finite Group Databasebeta

← #117520

Subgroups — #117520

Order 512.

649 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1986C211yesmaximal, characteristic
2#21177C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6#7918C211yesmaximal, characteristic
7#7901C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#430—21no—
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1986—22no—
17#1986—22no—
18#1950—22no—
19#1950—22no—
20#1738—21no—
21C2×#339—21no—
22C2×#339—21no—
23#1550—22no—
24#1549C412yescomplemented
25#1548C412yescomplemented
26#1542—22no—
27#1493—22no—
28#1487—22no—
29#1481—22no—
30#1303—22no—
31#1281—22no—
32#1279—22no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16D411yescharacteristic
36C22×M16D411yescharacteristic
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533D412yes—
44#533Q812yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64#430—42no—
65#430—42no—
66C2×#108—24no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423C2×C412yes—
73#423D412yes—
74#423D412yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8D412yes—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D412yes—
95C2×#95—21no—
96#341—24no—
97#339—24no—
98#339—24no—
99#333—24no—
100#332—24no—
101Q8⋊C8—28no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4—41no—
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C23×C4#4111yescharacteristic
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—21no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16SD1612yes—
131C2×M16Q1612yes—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8C4⋊C412yes—
145C4⋊Q8C4⋊C412yes—
146#125—22no—
147#125—22no—
148#125—41no—
149#125—41no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156C22⋊Q8—41no—
157C22⋊Q8—41no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166#117—22no—
167#117—22no—
168#117—41no—
169#117—41no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C42—22no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×Q811yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199#108—48no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8C22⋊C414yes—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216#97—41no—
217#97—41no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—41no—
221C4×C8—41no—
222C4×C8—41no—
223C4×C8—41no—
224#95—22no—
225#95—41no—
226C24#12411yescharacteristic
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—82no—
232C2×Q8—82no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—81no—
239C22×C4—82no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4#11711yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12011yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12611yescharacteristic
260C22×C4C2×SD1611yescharacteristic
261C22×C4C2×Q1611yescharacteristic
262C22×C4#13511yescharacteristic
263C22×C4#13511yescharacteristic
264C22×C4AGL(1,ℤ/8)11yescharacteristic
265C22×C4#13711yescharacteristic
266C22×C4#13711yescharacteristic
267C22×C4#13711yescharacteristic
268M16—22no—
269M16—22no—
270M16—22no—
271M16—22no—
272M16—42no—
273M16—42no—
274M16—41no—
275M16—41no—
276M16—41no—
277M16—41no—
278M16—42no—
279M16—42no—
280M16—42no—
281M16—42no—
282M16—81no—
283M16—81no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—82no—
297C2×C8—82no—
298C2×C8—81no—
299C2×C8—81no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—81no—
322C42—81no—
323C42C2×C22⋊C411yescharacteristic
324C42C2×C4⋊C411yescharacteristic
325C42C4×D411yescharacteristic
326C42C4×D411yescharacteristic
327C42#12011yescharacteristic
328C42#12111yescharacteristic
329C42#12111yescharacteristic
330C42#12111yescharacteristic
331C42#12111yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42C22⋊Q811yescharacteristic
334C42C22⋊Q811yescharacteristic
335C42#12311yescharacteristic
336C42#12311yescharacteristic
337C42#12611yescharacteristic
338C42#12711yescharacteristic
339C23—41no—
340C23—41no—
341C23#38811yescharacteristic
342C23#39311yescharacteristic
343C23#39411yescharacteristic
344C23#48711yescharacteristic
345C23#48811yescharacteristic
346C23#48911yescharacteristic
347C23#49111yescharacteristic
348Q8—44no—
349Q8—82no—
350Q8—82no—
351Q8—82no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4#38011yescharacteristic
374C2×C4#38511yescharacteristic
375C2×C4#38511yescharacteristic
376C2×C4#38611yescharacteristic
377C2×C4#38811yescharacteristic
378C2×C4#39011yescharacteristic
379C2×C4#39211yescharacteristic
380C2×C4#39411yescharacteristic
381C2×C4#39611yescharacteristic
382C2×C4#39611yescharacteristic
383C2×C4#39711yescharacteristic
384C2×C4#39911yescharacteristic
385C2×C4C4×SD1611yescharacteristic
386C2×C4C4×Q1611yescharacteristic
387C2×C4#44011yescharacteristic
388C2×C4#44111yescharacteristic
389C2×C4#44811yescharacteristic
390C2×C4#44911yescharacteristic
391C2×C4#45011yescharacteristic
392C2×C4#45111yescharacteristic
393C2×C4#45111yescharacteristic
394C2×C4#45211yescharacteristic
395C2×C4#45211yescharacteristic
396C2×C4#45211yescharacteristic
397C2×C4#46111yescharacteristic
398C2×C4#46211yescharacteristic
399C2×C4#46311yescharacteristic
400C2×C4#46411yescharacteristic
401C2×C4#46511yescharacteristic
402C2×C4#46711yescharacteristic
403C2×C4#46811yescharacteristic
404C2×C4#47011yescharacteristic
405C2×C4#47511yescharacteristic
406C2×C4#47711yescharacteristic
407C2×C4#47911yescharacteristic
408C2×C4#47911yescharacteristic
409C8—44no—
410C8—42no—
411C8—42no—
412C8—82no—
413C8—82no—
414C8—82no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22#143911yescharacteristic
426C22#193311yescharacteristic
427C22#193511yescharacteristic
428C22#202411yescharacteristic
429C22#202611yescharacteristic
430C22#204211yescharacteristic
431C22#204311yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—84no—
440C4—82no—
441C4—82no—
442C4#186311yescharacteristic
443C4#186511yescharacteristic
444C4#187711yescharacteristic
445C4#187811yescharacteristic
446C4#193212yes—
447C4#200011yescharacteristic
448C4#200111yescharacteristic
449C4#206611yescharacteristic
450C4#206811yescharacteristic
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2#1106711yescharacteristic
458C2#1113111yescharacteristic
459C2#1114611yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1549C4semidirect
25. #1548C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19862
#211772
#177082
#174862
C2×#13032
#79182
#79012