Finite Group Databasebeta

← #119169

Subgroups — #119169

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21306C211yesmaximal, characteristic
2#17708C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4#10858C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7#7860C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1986—22no—
16#1986—22no—
17C2×#343—21no—
18C2×#343—21no—
19#1733—21no—
20#1733—21no—
21#1550—22no—
22#1549C412yescomplemented
23#1548C412yescomplemented
24#1542—22no—
25#1518—22no—
26#1518—22no—
27C2×#328—22no—
28#1303—22no—
29#1294—22no—
30#1275—22no—
31C22×M16—21no—
32C22×M16D411yescharacteristic
33C22×M16D411yescharacteristic
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533D412yes—
41#533D412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C2×#117—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#431C2×C412yes—
60#431D412yes—
61#423—21no—
62#423—21no—
63#423—21no—
64#423C2×C412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#384—21no—
80#382—21no—
81#382—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4D412yes—
89C2×#95—21no—
90C2×#95D412yes—
91#343—42no—
92#343—42no—
93#339—24no—
94#339—24no—
95#333—24no—
96#332—24no—
97#328—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C2×M16—21no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—21no—
108C2×M16—21no—
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—22no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16SD1612yes—
121C2×M16Q1612yes—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8C22⋊C412yes—
134C4⋊Q8C22⋊C412yes—
135#125—22no—
136#125—22no—
137#125—41no—
138#125—41no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155#117—22no—
156#117—22no—
157#117—41no—
158#117—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162#117—41no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
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×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C42—22no—
181C2×C42—41no—
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—41no—
207#97—41no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—41no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8—41no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—41no—
218#95—41no—
219#95—41no—
220C24#12111yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—82no—
226C2×Q8—82no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4#11711yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4#12411yescharacteristic
254C22×C4#12611yescharacteristic
255C22×C4C2×SD1611yescharacteristic
256C22×C4C2×Q1611yescharacteristic
257C22×C4#13511yescharacteristic
258C22×C4#13511yescharacteristic
259C22×C4AGL(1,ℤ/8)11yescharacteristic
260C22×C4#13711yescharacteristic
261C22×C4#13711yescharacteristic
262C22×C4#13711yescharacteristic
263M16—22no—
264M16—22no—
265M16—22no—
266M16—22no—
267M16—41no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—42no—
272M16—42no—
273M16—81no—
274M16—81no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—82no—
285C2×C8—82no—
286C2×C8—81no—
287C2×C8—81no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C42—42no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42C2×C22⋊C411yescharacteristic
317C42#11711yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42#12111yescharacteristic
321C42#12111yescharacteristic
322C42#12111yescharacteristic
323C42#12311yescharacteristic
324C42#12311yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12411yescharacteristic
329C42#12611yescharacteristic
330C42#12611yescharacteristic
331C42#12611yescharacteristic
332C23—41no—
333C23—41no—
334C23#38811yescharacteristic
335C23#39411yescharacteristic
336C23#39411yescharacteristic
337C23#46111yescharacteristic
338C23#46211yescharacteristic
339C23#46311yescharacteristic
340C23#46411yescharacteristic
341Q8—82no—
342Q8—82no—
343Q8—82no—
344Q8—82no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—162no—
368C2×C4—162no—
369C2×C4#38111yescharacteristic
370C2×C4#38611yescharacteristic
371C2×C4#38611yescharacteristic
372C2×C4#38811yescharacteristic
373C2×C4#38811yescharacteristic
374C2×C4#38811yescharacteristic
375C2×C4#39411yescharacteristic
376C2×C4#39411yescharacteristic
377C2×C4#39711yescharacteristic
378C2×C4#39711yescharacteristic
379C2×C4#39711yescharacteristic
380C2×C4#39711yescharacteristic
381C2×C4C4×SD1611yescharacteristic
382C2×C4C4×Q1611yescharacteristic
383C2×C4#44011yescharacteristic
384C2×C4#44111yescharacteristic
385C2×C4#44811yescharacteristic
386C2×C4#44911yescharacteristic
387C2×C4#45111yescharacteristic
388C2×C4#45211yescharacteristic
389C2×C4#46011yescharacteristic
390C2×C4#46211yescharacteristic
391C2×C4#46411yescharacteristic
392C2×C4#46411yescharacteristic
393C2×C4#46511yescharacteristic
394C2×C4#46711yescharacteristic
395C2×C4#46811yescharacteristic
396C2×C4#47011yescharacteristic
397C2×C4#48311yescharacteristic
398C2×C4#48411yescharacteristic
399C2×C4#48511yescharacteristic
400C2×C4#48511yescharacteristic
401C2×C4#48711yescharacteristic
402C2×C4#48811yescharacteristic
403C2×C4#48911yescharacteristic
404C2×C4#49111yescharacteristic
405C8—42no—
406C8—42no—
407C8—82no—
408C8—82no—
409C8—82no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22#144011yescharacteristic
421C22#192711yescharacteristic
422C22#192811yescharacteristic
423C22#204211yescharacteristic
424C22#204311yescharacteristic
425C22#205511yescharacteristic
426C22#205611yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—82no—
436C4—82no—
437C4—162no—
438C4—162no—
439C4#187811yescharacteristic
440C4#188011yescharacteristic
441C4#193311yescharacteristic
442C4#193511yescharacteristic
443C4#203511yescharacteristic
444C4#203611yescharacteristic
445C4#207211yescharacteristic
446C4#207411yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2#1115911yescharacteristic
454C2#1118011yescharacteristic
455C2#1119011yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
22. #1549C4semidirect
23. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#213062
#177082
C2×#15182
#108582
#80952
#80262
#78602