Finite Group Databasebeta

← #117576

Subgroups — #117576

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21324C211yesmaximal, characteristic
2#21177C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6#7957C211yesmaximal, characteristic
7#7932C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#430—21no—
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2285C2211yescharacteristic
16#1993—22no—
17#1990—22no—
18#1950—22no—
19#1950—22no—
20#1738—21no—
21#1736—21no—
22#1736—21no—
23#1549—22no—
24#1549C414yescomplemented
25#1548—22no—
26#1548—22no—
27C2×#333—22no—
28C2×#332—22no—
29#1303—22no—
30#1286—22no—
31#1283—22no—
32C23×Q8—21no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16D411yescharacteristic
36C22×M16D411yescharacteristic
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⋊Q8D412yes—
48C2×C4⋊Q8Q812yes—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C42C2311yescharacteristic
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431C2×C412yes—
61#430—42no—
62#430—42no—
63C2×#108—24no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423C2×C412yes—
70#423D412yes—
71#423D412yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8D412yes—
77C2×C4⋊C8D412yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C2×#97—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88#384—21no—
89#382D412yes—
90#380—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94#341—24no—
95#340—24no—
96#340—24no—
97#333—28no—
98#332—28no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—41no—
105C22×Q8—84no—
106C23×C4—41no—
107C23×C4C2×D411yescharacteristic
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—22no—
111C2×M16—22no—
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—21no—
121C2×M16—21no—
122C2×M16—22no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16SD1612yes—
133C2×M16Q1612yes—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—41no—
139C22×C8—41no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4⋊Q8C4⋊C414yes—
156#117—41no—
157#117—41no—
158#117—41no—
159#117—41no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42—41no—
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
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
189#108—48no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8C22⋊C414yes—
202C22⋊C8—22no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—41no—
207#97—41no—
208C4×C8—22no—
209C4×C8—41no—
210C4×C8—41no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214#95—22no—
215#95—22no—
216C24#12411yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—84no—
222C2×Q8—84no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×Q8—84no—
226C2×Q8—84no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—84no—
234C22×C4#11711yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
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×C4#12411yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4C2×SD1611yescharacteristic
253C22×C4C2×Q1611yescharacteristic
254C22×C4#13511yescharacteristic
255C22×C4#13511yescharacteristic
256C22×C4AGL(1,ℤ/8)11yescharacteristic
257C22×C4#13711yescharacteristic
258C22×C4#13711yescharacteristic
259C22×C4#13711yescharacteristic
260M16—22no—
261M16—22no—
262M16—22no—
263M16—22no—
264M16—42no—
265M16—42no—
266M16—41no—
267M16—41no—
268M16—42no—
269M16—42no—
270M16—41no—
271M16—41no—
272M16—42no—
273M16—42no—
274M16—81no—
275M16—81no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—82no—
289C2×C8—81no—
290C2×C8—81no—
291C2×C8—82no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C42—81no—
307C42—81no—
308C42C2×C22⋊C411yescharacteristic
309C42C2×C4⋊C411yescharacteristic
310C42C4×D411yescharacteristic
311C42C4×D411yescharacteristic
312C42#12011yescharacteristic
313C42#12111yescharacteristic
314C42#12111yescharacteristic
315C42#12111yescharacteristic
316C42#12111yescharacteristic
317C42C22⋊Q811yescharacteristic
318C42C22⋊Q811yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42#12311yescharacteristic
321C42#12311yescharacteristic
322C42#12611yescharacteristic
323C42#12711yescharacteristic
324C23—41no—
325C23—41no—
326C23#38811yescharacteristic
327C23#39311yescharacteristic
328C23#39411yescharacteristic
329C23#48711yescharacteristic
330C23#48811yescharacteristic
331C23#48911yescharacteristic
332C23#49111yescharacteristic
333Q8—84no—
334Q8—84no—
335Q8—84no—
336Q8—84no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4#38011yescharacteristic
357C2×C4#38511yescharacteristic
358C2×C4#38511yescharacteristic
359C2×C4#38611yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#39011yescharacteristic
362C2×C4#39211yescharacteristic
363C2×C4#39411yescharacteristic
364C2×C4#39611yescharacteristic
365C2×C4#39611yescharacteristic
366C2×C4#39711yescharacteristic
367C2×C4#39911yescharacteristic
368C2×C4C4×SD1611yescharacteristic
369C2×C4C4×Q1611yescharacteristic
370C2×C4#44011yescharacteristic
371C2×C4#44111yescharacteristic
372C2×C4#44811yescharacteristic
373C2×C4#44911yescharacteristic
374C2×C4#45011yescharacteristic
375C2×C4#45111yescharacteristic
376C2×C4#45111yescharacteristic
377C2×C4#45211yescharacteristic
378C2×C4#45211yescharacteristic
379C2×C4#45211yescharacteristic
380C2×C4#46111yescharacteristic
381C2×C4#46211yescharacteristic
382C2×C4#46311yescharacteristic
383C2×C4#46411yescharacteristic
384C2×C4#46511yescharacteristic
385C2×C4#46711yescharacteristic
386C2×C4#46811yescharacteristic
387C2×C4#47011yescharacteristic
388C2×C4#47511yescharacteristic
389C2×C4#47711yescharacteristic
390C2×C4#47911yescharacteristic
391C2×C4#47911yescharacteristic
392C8—44no—
393C8—42no—
394C8—42no—
395C8—82no—
396C8—82no—
397C8—82no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22#143911yescharacteristic
409C22#193311yescharacteristic
410C22#193511yescharacteristic
411C22#202411yescharacteristic
412C22#202611yescharacteristic
413C22#204211yescharacteristic
414C22#204311yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—84no—
422C4—162no—
423C4#186311yescharacteristic
424C4#186511yescharacteristic
425C4#187711yescharacteristic
426C4#187811yescharacteristic
427C4#200011yescharacteristic
428C4#200111yescharacteristic
429C4#204912yes—
430C4#206611yescharacteristic
431C4#206811yescharacteristic
432C4#208112yes—
433C2—21no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—21no—
438C2—21no—
439C2#1106711yescharacteristic
440C2#1113311yescharacteristic
441C2#1114311yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1549C4semidirect

Maximal subgroups

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

subgroupindex
#213242
#211772
C2×#15492
C2×#15482
#80952
#79572
#79322