Finite Group Databasebeta

← #148412

Subgroups — #148412

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33409C211yesmaximal, characteristic
2C2×#2141C211yesmaximal, characteristic
3C2×#2141C211yesmaximal, characteristic
4#21232C211yesmaximal, characteristic
5#12567C211yesmaximal, characteristic
6C2×#1378C211yesmaximal, characteristic
7#10871C211yesmaximal, characteristic
8C23×Q16C2211yescharacteristic
9C22×M32C2211yescharacteristic
10C22×M32C2211yescharacteristic
11C2×#427C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C2×#419—21no—
15C2×#405C2211yescharacteristic
16#2310—21no—
17#2159—22no—
18#2159—22no—
19#2158—22no—
20#2141—22no—
21#2141—22no—
22#2141—22no—
23#2141C414yescomplemented
24C2×#358—22no—
25C2×#358—22no—
26#1971—22no—
27#1969—22no—
28#1740—21no—
29#1733—21no—
30#1386—22no—
31#1378—22no—
32C23×Q8—21no—
33C22×Q16—21no—
34C22×Q16—21no—
35C22×Q16—21no—
36C22×Q16—21no—
37C22×Q16—21no—
38C22×Q16—21no—
39C22×Q16—21no—
40C22×Q16—21no—
41C22×Q16—21no—
42C22×Q16—21no—
43C22×Q16—21no—
44C22×Q16C2×C412yes—
45C22×Q16D412yes—
46C22×Q16Q812yes—
47C22×M16D411yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117—21no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117D411yescharacteristic
53C2×M32—21no—
54C2×M32—21no—
55C2×M32—21no—
56C2×M32—21no—
57C2×M32—21no—
58C2×M32—21no—
59C2×M32—21no—
60C2×M32—21no—
61C2×M32—21no—
62C2×M32C2×C412yes—
63C2×M32D412yes—
64C2×M32D412yes—
65C22×C16—21no—
66C22×C16D412yes—
67#427—21no—
68#427—21no—
69#427—21no—
70#427—21no—
71#427—21no—
72#427—21no—
73#427—21no—
74#427—21no—
75#427—21no—
76#427—21no—
77#427—21no—
78#427C2×C412yes—
79C2×#107—21no—
80C2×#107D412yes—
81C2×#107D412yes—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85#419—42no—
86#419—42no—
87C2×#103—24no—
88#405—21no—
89#405—21no—
90#405—21no—
91#405—21no—
92C2×#97—21no—
93C2×C4×C8—21no—
94#391—24no—
95#391—42no—
96#381—42no—
97#367—24no—
98#366—24no—
99#358—28no—
100#358—28no—
101#342—42no—
102#339—42no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×Q8—41no—
109C22×Q8—41no—
110C22×Q8—41no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8—41no—
115C22×Q8—41no—
116C22×Q8—41no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C2×Q16—22no—
120C2×Q16—22no—
121C2×Q16—22no—
122C2×Q16—22no—
123C2×Q16—22no—
124C2×Q16—22no—
125C2×Q16—22no—
126C2×Q16—22no—
127C2×Q16—22no—
128C2×Q16—22no—
129C2×Q16—22no—
130C2×Q16—22no—
131C2×Q16—22no—
132C2×Q16—22no—
133C2×Q16—22no—
134C2×Q16—41no—
135C2×Q16—41no—
136C2×Q16—41no—
137C2×Q16—41no—
138C2×Q16—41no—
139C2×Q16—41no—
140C2×Q16—41no—
141C2×Q16—41no—
142C2×Q16—41no—
143C2×Q16—41no—
144C2×Q16—41no—
145C2×Q16—41no—
146C2×Q16C4⋊C414yes—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16D812yes—
153C22×C8—21no—
154C22×C8C22×C411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×Q811yescharacteristic
162C22×C8#4111yescharacteristic
163C22×C8#4111yescharacteristic
164C22×C8#4111yescharacteristic
165C22×C8#4111yescharacteristic
166C22×C8#4111yescharacteristic
167C22×C8#4111yescharacteristic
168#117—21no—
169#117—21no—
170#117—21no—
171#117—21no—
172#117—22no—
173#117—41no—
174#117—41no—
175#117—41no—
176#117—41no—
177#117—41no—
178#117—41no—
179#117—41no—
180#117—41no—
181#117—41no—
182#117—41no—
183#117—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4D812yes—
192C2×C42—21no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42—41no—
196M32—22no—
197M32—22no—
198M32—22no—
199M32—22no—
200M32—22no—
201M32—22no—
202M32—41no—
203M32—41no—
204C2×C16—22no—
205C2×C16—22no—
206C2×C16—22no—
207C2×C16—41no—
208C2×C16—41no—
209C2×C16C22⋊C414yes—
210#107—22no—
211#107—22no—
212#107—22no—
213#107—22no—
214#107—22no—
215#107—41no—
216#106—22no—
217#106—41no—
218#106—41no—
219#106—41no—
220#106—41no—
221#106—41no—
222#103—48no—
223#97—41no—
224#97—41no—
225C4×C8—41no—
226C4×C8—41no—
227#95—48no—
228C24C2×D811yescharacteristic
229C2×Q8—24no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—81no—
246C2×Q8—81no—
247C2×Q8—81no—
248C2×Q8—81no—
249C2×Q8—81no—
250C2×Q8—81no—
251C2×Q8—81no—
252C2×Q8—81no—
253C2×Q8—81no—
254C2×Q8—81no—
255C2×Q8—81no—
256C2×Q8—81no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4C4×D411yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4#12311yescharacteristic
274C22×C4#12711yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4#13511yescharacteristic
277C22×C4#13511yescharacteristic
278C22×C4AGL(1,ℤ/8)11yescharacteristic
279C22×C4AGL(1,ℤ/8)11yescharacteristic
280C22×C4#13711yescharacteristic
281C22×C4#13711yescharacteristic
282Q16—24no—
283Q16—24no—
284Q16—42no—
285Q16—42no—
286Q16—42no—
287Q16—42no—
288Q16—42no—
289Q16—42no—
290M16—22no—
291M16—22no—
292M16—41no—
293M16—41no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8C2×C22⋊C411yescharacteristic
298C2×C8C2×C4⋊C411yescharacteristic
299C2×C8#11711yescharacteristic
300C2×C8C4×D411yescharacteristic
301C2×C8C4×D411yescharacteristic
302C2×C8C4×D411yescharacteristic
303C2×C8#12011yescharacteristic
304C2×C8#12011yescharacteristic
305C2×C8#12111yescharacteristic
306C2×C8#12111yescharacteristic
307C2×C8#12111yescharacteristic
308C2×C8#12111yescharacteristic
309C2×C8#12111yescharacteristic
310C2×C8#12111yescharacteristic
311C2×C8C22⋊Q811yescharacteristic
312C2×C8C22⋊Q811yescharacteristic
313C2×C8C22⋊Q811yescharacteristic
314C2×C8C22⋊Q811yescharacteristic
315C2×C8C22⋊Q811yescharacteristic
316C2×C8#12311yescharacteristic
317C2×C8#12311yescharacteristic
318C2×C8#12311yescharacteristic
319C2×C8#12311yescharacteristic
320C2×C8#12411yescharacteristic
321C2×C8#12411yescharacteristic
322C2×C8#12411yescharacteristic
323C2×C8#12611yescharacteristic
324C2×C8#12611yescharacteristic
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—81no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—42no—
336C22⋊C4—81no—
337C22⋊C4—81no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C42—41no—
342C42—41no—
343C42—42no—
344C42—81no—
345C42—81no—
346C42—81no—
347C42—81no—
348C42—81no—
349C16—24no—
350C16—42no—
351C23—42no—
352C23C4×D811yescharacteristic
353C23#44711yescharacteristic
354C23#45911yescharacteristic
355C23#46611yescharacteristic
356C23#47411yescharacteristic
357C23#48011yescharacteristic
358C23#49311yescharacteristic
359Q8—44no—
360Q8—44no—
361Q8—82no—
362Q8—82no—
363Q8—82no—
364Q8—82no—
365Q8—82no—
366Q8—82no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—162no—
381C2×C4—162no—
382C2×C4#38511yescharacteristic
383C2×C4#38811yescharacteristic
384C2×C4#39011yescharacteristic
385C2×C4#39211yescharacteristic
386C2×C4#39411yescharacteristic
387C2×C4#39611yescharacteristic
388C2×C4#39911yescharacteristic
389C2×C4C4×D811yescharacteristic
390C2×C4#44011yescharacteristic
391C2×C4#44011yescharacteristic
392C2×C4#44711yescharacteristic
393C2×C4#45211yescharacteristic
394C2×C4#45211yescharacteristic
395C2×C4#45911yescharacteristic
396C2×C4#46411yescharacteristic
397C2×C4#46411yescharacteristic
398C2×C4#46611yescharacteristic
399C2×C4#46811yescharacteristic
400C2×C4#46811yescharacteristic
401C2×C4#47411yescharacteristic
402C2×C4#47811yescharacteristic
403C2×C4#47811yescharacteristic
404C2×C4#48011yescharacteristic
405C2×C4#48211yescharacteristic
406C2×C4#48211yescharacteristic
407C2×C4#49511yescharacteristic
408C2×C4#49611yescharacteristic
409C2×C4#49711yescharacteristic
410C8—42no—
411C8#38011yescharacteristic
412C8#38511yescharacteristic
413C8#38611yescharacteristic
414C8#38811yescharacteristic
415C8#39311yescharacteristic
416C8#39411yescharacteristic
417C8#39611yescharacteristic
418C8#39711yescharacteristic
419C22—22no—
420C22—22no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22#186211yescharacteristic
427C22#192511yescharacteristic
428C22#197011yescharacteristic
429C22#201111yescharacteristic
430C22#203311yescharacteristic
431C22#206711yescharacteristic
432C22#209311yescharacteristic
433C4—82no—
434C4—84no—
435C4—162no—
436C4—162no—
437C4—162no—
438C4#143911yescharacteristic
439C4#186211yescharacteristic
440C4#192511yescharacteristic
441C4#197311yescharacteristic
442C4#201511yescharacteristic
443C4#203311yescharacteristic
444C4#206711yescharacteristic
445C4#209211yescharacteristic
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2#1112311yescharacteristic
453C2#1333512yes—
subgroupG/H
center, socleC22#2011
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #2141C4semidirect

Maximal subgroups

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

subgroupindex
#334092
C2×#21412
#212322
#125672
C2×#13782
#108712