Finite Group Databasebeta

← #112674

Subgroups — #112674

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2#21838C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic
5C2×#1470C211yesmaximal, characteristic
6C2×#1302C211yesmaximal, characteristic
7#8034C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×#125C2211yescharacteristic
10C2×#501—22no—
11C2×#501—22no—
12C2×#501—22no—
13C2×#501—22no—
14C2×#501—22no—
15C2×#501C414yescomplemented
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×#107—21no—
20C22×#106—21no—
21C22×#97C2211yescharacteristic
22#2285C2211yescharacteristic
23#1992—22no—
24#1990—22no—
25#1989—22no—
26#1988—22no—
27#1824—21no—
28#1823—22no—
29#1823—21no—
30#1558—22no—
31#1558—22no—
32C2×#333—22no—
33#1470—22no—
34#1470—22no—
35C2×#324—22no—
36#1302—22no—
37#1295—22no—
38C23×Q8—21no—
39C23×C8—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⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8D412yes—
51C2×C4⋊Q8D412yes—
52C2×#125—21no—
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125—21no—
58C2×#125—21no—
59C2×#125—21no—
60C2×#125—21no—
61C2×#125—21no—
62C2×#125C2×C412yes—
63C2×#125D412yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C42C2311yescharacteristic
72#501—28no—
73#501—28no—
74#501—28no—
75#501—28no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432—21no—
80#432—21no—
81#432—21no—
82#432—21no—
83#432D412yes—
84#431—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88C2×#107—24no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8D412yes—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8D412yes—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97C2×C412yes—
106C2×#97D412yes—
107C2×#97D412yes—
108C2×C4×C8—21no—
109#384D412yes—
110#382—21no—
111#380—21no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115#340—28no—
116#340—24no—
117#340—24no—
118#333—28no—
119#324—28no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—41no—
124C22×Q8—41no—
125C22×Q8—41no—
126C22×Q8—82no—
127C22×Q8—82no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4#4111yescharacteristic
134C22×C8—28no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—41no—
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—22no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4⋊Q8—41no—
156C4⋊Q8C22⋊C414yes—
157#125—22no—
158#125—22no—
159#125—22no—
160#125—22no—
161#125—22no—
162#125—22no—
163#125—22no—
164#125—22no—
165#125—41no—
166#125—41no—
167#125—41no—
168#125—41no—
169#125—41no—
170#125—41no—
171#125—41no—
172#125—41no—
173#117—82no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
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⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C22⋊C4—41no—
214C2×C42—41no—
215C2×C42C22×C411yescharacteristic
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#107—216no—
228#106—216no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—41no—
233C4⋊C8—41no—
234C4⋊C8—41no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—41no—
239C22⋊C8—41no—
240C22⋊C8—41no—
241#97—22no—
242#97—22no—
243#97—22no—
244#97—22no—
245#97—41no—
246#97—41no—
247#97—41no—
248#97C22⋊C414yes—
249C4×C8—41no—
250C4×C8—41no—
251#95—41no—
252#95—41no—
253C24#12111yescharacteristic
254C2×Q8—24no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×Q8—82no—
261C2×Q8—82no—
262C2×Q8—84no—
263C2×Q8—84no—
264C2×Q8—82no—
265C2×Q8—82no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12311yescharacteristic
299C22×C4#12311yescharacteristic
300C22×C4#12311yescharacteristic
301C22×C4#12411yescharacteristic
302C22×C4#12711yescharacteristic
303C2×C8—28no—
304C2×C8—28no—
305C2×C8—28no—
306C2×C8—24no—
307C2×C8—44no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—44no—
312C2×C8—82no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—82no—
340C22⋊C4—82no—
341C42—82no—
342C42C2×C22⋊C411yescharacteristic
343C42C2×C22⋊C411yescharacteristic
344C42C4×D411yescharacteristic
345C42C4×D411yescharacteristic
346C42#12011yescharacteristic
347C42#12011yescharacteristic
348C42#12011yescharacteristic
349C42#12111yescharacteristic
350C42#12111yescharacteristic
351C42#12111yescharacteristic
352C42#12111yescharacteristic
353C42#12111yescharacteristic
354C42#12111yescharacteristic
355C42#12311yescharacteristic
356C42#12411yescharacteristic
357C42#12711yescharacteristic
358C23—22no—
359C23—22no—
360C23#38611yescharacteristic
361C23#39211yescharacteristic
362C23#39411yescharacteristic
363C23#47112yes—
364C23#47212yes—
365Q8—84no—
366Q8—84no—
367Q8—84no—
368Q8—84no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#37911yescharacteristic
406C2×C4#38611yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#38611yescharacteristic
409C2×C4#39011yescharacteristic
410C2×C4#39011yescharacteristic
411C2×C4#39211yescharacteristic
412C2×C4#39211yescharacteristic
413C2×C4#39211yescharacteristic
414C2×C4#39411yescharacteristic
415C2×C4#39411yescharacteristic
416C2×C4#39411yescharacteristic
417C2×C4#45614yes—
418C8—28no—
419C8—82no—
420C8—82no—
421C8—82no—
422C22—21no—
423C22—21no—
424C22—24no—
425C22—24no—
426C22—21no—
427C22—21no—
428C22#143811yescharacteristic
429C22#190712yes—
430C22#200512yes—
431C22#200912yes—
432C22#201811yescharacteristic
433C22#201911yescharacteristic
434C22#205112yes—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—82no—
446C4—84no—
447C4—82no—
448C4—162no—
449C4#188612yes—
450C4#200512yes—
451C2—22no—
452C2—22no—
453C2#1014212yes—
454C2#1014312yes—
455C2#1097811yescharacteristic
456C2#1102011yescharacteristic
457C2#1103111yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ 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
15. C2×#501C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
#218382
#218332
C2×#15582
C2×#14702
C2×#13022
#80342