Finite Group Databasebeta

← #114156

Subgroups — #114156

Order 512.

886 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2C2×#1550C211yesmaximal, characteristic
3C2×#1549C211yesmaximal, characteristic
4C22×#333C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#10853C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3165—22no—
11#3165—22no—
12#3165—22no—
13#3165C414yes—
14C2×#501—22no—
15C2×#500—21no—
16C2×#498—21no—
17C2×#431C2211yescharacteristic
18C2×#428—22no—
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C2×#341—22no—
24#1550—22no—
25#1550—22no—
26#1549—22no—
27#1549—22no—
28C2×#333—22no—
29C2×#333—22no—
30C2×#333—22no—
31C2×#333—21no—
32C2×#332—22no—
33C2×#332—21no—
34#1481—22no—
35#1481—22no—
36C2×Q8⋊C8—22no—
37#1283—22no—
38C23×Q8—21no—
39C22×M16—21no—
40C22×M16D412yes—
41C23×C8—21no—
42C23×C8—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533C2×C412yes—
50#533D412yes—
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⋊Q8C2×C412yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8D412yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#501—28no—
75#501—28no—
76#500—24no—
77#500—24no—
78#498—24no—
79#498—24no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431D412yes—
84#428—28no—
85#428—28no—
86C2×#107—24no—
87C2×#106—24no—
88#423—21no—
89#423—21no—
90#423—21no—
91#423D412yes—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8D412yes—
106C2×C4⋊C8D412yes—
107C2×C22⋊C8—21no—
108C4×M16—21no—
109C4×M16—21no—
110C4×M16C2×C412yes—
111C4×M16D412yes—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115#341—44no—
116#333—28no—
117#333—24no—
118#332—28no—
119#332—24no—
120Q8⋊C8—28no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—84no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C23×C4#4111yescharacteristic
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—41no—
139C2×M16—42no—
140C2×M16—42no—
141C2×M16C22⋊C414yes—
142C2×M16SD1614yes—
143C2×M16Q1614yes—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—41no—
148C22×C8—41no—
149C22×C8—41no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—41no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—82no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—21no—
167C4⋊Q8—22no—
168C4⋊Q8—21no—
169C4⋊Q8—22no—
170C4⋊Q8—21no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177C4⋊Q8C22⋊C414yes—
178C4⋊Q8C22⋊C412yes—
179C4⋊Q8C22⋊C412yes—
180#125—22no—
181#125—22no—
182#125—41no—
183#125—41no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200#117—22no—
201#117—22no—
202#117—41no—
203#117—41no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C42—41no—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42#4111yescharacteristic
236#107—216no—
237#106—216no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C22⋊C8—22no—
253C22⋊C8—41no—
254#97—22no—
255#97—22no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—41no—
260C24#12411yescharacteristic
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—84no—
272C2×Q8—82no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8—84no—
276C2×Q8—84no—
277C2×Q8—84no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—84no—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12311yescharacteristic
301C22×C4#12311yescharacteristic
302C22×C4#12311yescharacteristic
303C22×C4#12311yescharacteristic
304C22×C4#12711yescharacteristic
305C22×C4C2×SD1612yes—
306C22×C4C2×Q1612yes—
307C22×C4#13712yes—
308C22×C4#13712yes—
309M16—24no—
310M16—24no—
311M16—24no—
312M16—44no—
313M16—81no—
314M16—81no—
315M16#10318yes—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—44no—
324C2×C8—82no—
325C2×C8—82no—
326C2×C8—82no—
327C2×C8—82no—
328C2×C8—82no—
329C2×C8—81no—
330C2×C8—81no—
331C2×C8—82no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C42—42no—
355C42—42no—
356C42C2×C22⋊C411yescharacteristic
357C42C2×C22⋊C411yescharacteristic
358C42C4×D411yescharacteristic
359C42C4×D411yescharacteristic
360C42#12011yescharacteristic
361C42#12011yescharacteristic
362C42#12011yescharacteristic
363C42#12011yescharacteristic
364C42#12011yescharacteristic
365C42#12011yescharacteristic
366C42#12111yescharacteristic
367C42#12111yescharacteristic
368C42#12111yescharacteristic
369C42#12111yescharacteristic
370C42#12411yescharacteristic
371C42#12711yescharacteristic
372C23—22no—
373C23—22no—
374C23#39011yescharacteristic
375C23#39411yescharacteristic
376C23#39411yescharacteristic
377C23#48712yes—
378C23#49112yes—
379Q8—44no—
380Q8—82no—
381Q8—84no—
382Q8—82no—
383Q8—84no—
384Q8—84no—
385Q8—84no—
386Q8—81no—
387Q8—81no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#37911yescharacteristic
410C2×C4#38611yescharacteristic
411C2×C4#38611yescharacteristic
412C2×C4#38611yescharacteristic
413C2×C4#38611yescharacteristic
414C2×C4#39011yescharacteristic
415C2×C4#39211yescharacteristic
416C2×C4#39211yescharacteristic
417C2×C4#39211yescharacteristic
418C2×C4#39211yescharacteristic
419C2×C4#39411yescharacteristic
420C2×C4#39411yescharacteristic
421C2×C4C2×#10312yes—
422C2×C4#41912yes—
423C2×C4#44812yes—
424C2×C4#44812yes—
425C2×C4#45112yes—
426C2×C4#45112yes—
427C2×C4#46112yes—
428C2×C4#46112yes—
429C2×C4#46212yes—
430C2×C4#46212yes—
431C2×C4#47112yes—
432C2×C4#47312yes—
433C2×C4#49211yescharacteristic
434C2×C4#49411yescharacteristic
435C2×C4#49711yescharacteristic
436C2×C4#49711yescharacteristic
437C8—28no—
438C8—82no—
439C8—82no—
440C8—82no—
441C8—82no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—24no—
447C22—24no—
448C22#143811yescharacteristic
449C22#195611yescharacteristic
450C22#196011yescharacteristic
451C22#204312yes—
452C22#204312yes—
453C22#204812yes—
454C22#205212yes—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—82no—
462C4—84no—
463C4—82no—
464C4#178712yes—
465C4#190812yes—
466C4#198112yes—
467C4#201311yescharacteristic
468C4#201311yescharacteristic
469C4#201611yescharacteristic
470C4#201611yescharacteristic
471C4#201911yescharacteristic
472C4#202011yescharacteristic
473C2—21no—
474C2—21no—
475C2—22no—
476C2#1055514yes—
477C2#1101011yescharacteristic
478C2#1102111yescharacteristic
479C2#1103411yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#31652
C2×#15502
C2×#15492
C22×#3332
C2×#14812
#108532
C2×#12832