Finite Group Databasebeta

← #111424

Subgroups — #111424

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3C2×#1966C211yesmaximal, characteristic
4C2×#1549C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6#7874C211yesmaximal, characteristic
7#7848C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10C2×#498—22no—
11C2×#498—22no—
12C2×#498—22no—
13C2×#498—22no—
14C2×#498—22no—
15C2×#498—22no—
16C2×#431C2211yescharacteristic
17C22×#106C414yes—
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×#101—22no—
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#1997—22no—
24#1997—22no—
25#1966—22no—
26#1966—22no—
27C2×#343—22no—
28#1549—22no—
29#1549—22no—
30#1546—22no—
31#1546—22no—
32C2×#332—22no—
33C2×#332—22no—
34#1277—22no—
35#1273—22no—
36C23×Q8—21no—
37C23×Q8—21no—
38C22×M16—21no—
39C22×M16—21no—
40C22×M16D412yes—
41C23×C8C2×C412yes—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8D412yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C412yes—
70C22×C42C2311yescharacteristic
71#498—28no—
72#498—28no—
73#498—28no—
74#498—28no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79C2×#106—28no—
80C2×#106—28no—
81C2×#106—28no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86#423—21no—
87#423—21no—
88#423D412yes—
89#423D412yes—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8D412yes—
95C2×#101—28no—
96C2×#101—44no—
97C2×#101—44no—
98C2×C22⋊C8—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16—21no—
102C4×M16—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8D412yes—
111C2×C4×C8D412yes—
112#343—28no—
113#332—28no—
114#332—28no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—41no—
120C22×Q8—41no—
121C22×Q8—41no—
122C22×Q8—41no—
123C22×Q8—41no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—84no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4#4111yescharacteristic
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—41no—
145C2×M16—41no—
146C2×M16—42no—
147C2×M16—42no—
148C2×M16—42no—
149C2×M16—42no—
150C2×M16SD1614yes—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—28no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—41no—
157C22×C8—41no—
158C22×C8C22⋊C414yes—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C4⋊Q8—41no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C22⋊C414yes—
214C2×C4⋊C4C22⋊C414yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227#106—232no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—41no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237C4⋊C8—41no—
238#101—416no—
239C22⋊C8—41no—
240C22⋊C8—41no—
241#97—41no—
242#97—41no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8—22no—
247C4×C8—41no—
248C4×C8—41no—
249C4×C8—41no—
250C4×C8—41no—
251C24C4×D411yescharacteristic
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—84no—
265C2×Q8—84no—
266C2×Q8—84no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—82no—
275C2×Q8—84no—
276C2×Q8—81no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×Q8—81no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—84no—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12411yescharacteristic
308C22×C4#12711yescharacteristic
309C22×C4C2×SD1612yes—
310C22×C4#13512yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4#13712yes—
313M16—24no—
314M16—24no—
315M16—42no—
316M16—42no—
317M16—44no—
318M16—44no—
319M16—82no—
320M16—82no—
321C2×C8—28no—
322C2×C8—28no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—44no—
332C2×C8—81no—
333C2×C8—81no—
334C2×C8—81no—
335C2×C8—81no—
336C2×C8#10118yescomplemented
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C42#12011yescharacteristic
354C42#12011yescharacteristic
355C42#12011yescharacteristic
356C42#12011yescharacteristic
357C42#12011yescharacteristic
358C42#12011yescharacteristic
359C42#12111yescharacteristic
360C42#12111yescharacteristic
361C42#12111yescharacteristic
362C42#12111yescharacteristic
363C42#12111yescharacteristic
364C42#12111yescharacteristic
365C42#12311yescharacteristic
366C42#12311yescharacteristic
367C42#12411yescharacteristic
368C42#12711yescharacteristic
369C23—22no—
370C23—22no—
371C23#38611yescharacteristic
372C23#38611yescharacteristic
373C23#39011yescharacteristic
374C23C4×SD1612yes—
375C23#44012yes—
376Q8—84no—
377Q8—84no—
378Q8—84no—
379Q8—84no—
380Q8—82no—
381Q8—82no—
382Q8—82no—
383Q8—82no—
384Q8—84no—
385Q8—84no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—48no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4#37911yescharacteristic
406C2×C4#38611yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#39011yescharacteristic
409C2×C4#39211yescharacteristic
410C2×C4#39211yescharacteristic
411C2×C4#39211yescharacteristic
412C2×C4#39211yescharacteristic
413C2×C4#39411yescharacteristic
414C2×C4#39411yescharacteristic
415C2×C4#39411yescharacteristic
416C2×C4#39411yescharacteristic
417C2×C4C2×#10114yes—
418C2×C4#44812yes—
419C2×C4#44912yes—
420C2×C4#45012yes—
421C2×C4#45212yes—
422C2×C4#46012yes—
423C2×C4#46112yes—
424C2×C4#46312yes—
425C2×C4#46412yes—
426C2×C4#46512yes—
427C2×C4#46812yes—
428C2×C4#47114yes—
429C2×C4#49211yescharacteristic
430C2×C4#49511yescharacteristic
431C2×C4#49611yescharacteristic
432C2×C4#49711yescharacteristic
433C8—28no—
434C8—44no—
435C8—82no—
436C8—82no—
437C8—82no—
438C22—21no—
439C22—21no—
440C22—24no—
441C22—24no—
442C22—21no—
443C22—21no—
444C22#143811yescharacteristic
445C22#187712yes—
446C22#188012yes—
447C22#196612yes—
448C22#196811yescharacteristic
449C22#197311yescharacteristic
450C22#197712yes—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—84no—
456C4—84no—
457C4#178412yes—
458C4#196612yes—
459C4#200012yes—
460C4#201211yescharacteristic
461C4#201311yescharacteristic
462C4#201411yescharacteristic
463C4#201511yescharacteristic
464C4#201912yes—
465C4#204612yes—
466C4#205512yes—
467C2—22no—
468C2—22no—
469C2#979412yes—
470C2#980612yes—
471C2#1098311yescharacteristic
472C2#1102311yescharacteristic
473C2#1102811yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ 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
336. C2×C8#101semidirect

Maximal subgroups

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

subgroupindex
C22×#4982
C2×#19972
C2×#19662
C2×#15492
C2×#15462
#78742
#78482