Finite Group Databasebeta

← #118418

Subgroups — #118418

Order 512.

829 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21938C211yesmaximal, characteristic
2C2×#1996C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic
4C2×#1520C211yesmaximal, characteristic
5#10846C211yesmaximal, characteristic
6#8023C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C2×#380C2211yescharacteristic
12C2×#380C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1996—22no—
16#1996—22no—
17#1853—22no—
18#1853—22no—
19#1807—22no—
20C2×#342—22no—
21C2×#342—22no—
22#1520—22no—
23#1520—22no—
24#1520—22no—
25#1520C414yescomplemented
26C2×#328—22no—
27C2×#328—22no—
28#1294—22no—
29#1282—22no—
30C23×Q8Q811yescharacteristic
31C24×C4C2311yescharacteristic
32C23×C8—21no—
33C23×C8—21no—
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8—21no—
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8C2×C412yes—
45C2×C22⋊Q8D412yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4Q811yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4D412yes—
56C22×C42—21no—
57C2×#106—24no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8C2×C412yes—
67C2×C22⋊C8D412yes—
68C2×C22⋊C8D412yes—
69C2×C22⋊C8D412yes—
70#380—21no—
71#380—21no—
72#380—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380—21no—
79#380C2×C412yes—
80#380D412yes—
81#380D412yes—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90#342—28no—
91#342—44no—
92#340—24no—
93#328—28no—
94#328—28no—
95C25C2×D411yescharacteristic
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—82no—
102C22×Q8C4⋊C412yes—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4—82no—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×Q811yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4#4111yescharacteristic
123C23×C4#4111yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—82no—
131C22×C8—82no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C4⋊C412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4SD1614yes—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42—82no—
196#106—84no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8C22⋊C414yes—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213C24—81no—
214C24—81no—
215C24C4×D411yescharacteristic
216C24#12011yescharacteristic
217C24#12111yescharacteristic
218C24#12111yescharacteristic
219C24C22⋊Q811yescharacteristic
220C24#12311yescharacteristic
221C24#12711yescharacteristic
222C24C2×SD1612yes—
223C24#13512yes—
224C24#13712yes—
225C24#13712yes—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—82no—
235C2×Q8—82no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C4⋊C411yescharacteristic
278C22×C4#11711yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4C22⋊Q811yescharacteristic
291C22×C4C22⋊Q811yescharacteristic
292C22×C4C22⋊Q811yescharacteristic
293C22×C4C22⋊Q811yescharacteristic
294C22×C4C22⋊Q811yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4#12311yescharacteristic
299C22×C4#12411yescharacteristic
300C22×C4#12411yescharacteristic
301C22×C4#12411yescharacteristic
302C22×C4#12611yescharacteristic
303C22×C4#12611yescharacteristic
304C2×C8—24no—
305C2×C8—44no—
306C2×C8—42no—
307C2×C8—82no—
308C2×C8—82no—
309C2×C8—82no—
310C2×C8—82no—
311C2×C8—82no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—82no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C22⋊C4—81no—
342C42—82no—
343C42—82no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23#38511yescharacteristic
363C23#38811yescharacteristic
364C23#39011yescharacteristic
365C23#39211yescharacteristic
366C23#39411yescharacteristic
367C23#39611yescharacteristic
368C23#39911yescharacteristic
369C23C4×SD1612yes—
370C23#44112yes—
371C23#44812yes—
372C23#45212yes—
373C23#46112yes—
374C23#46412yes—
375C23#46512yes—
376C23#47012yes—
377C23#47512yes—
378C23#47912yes—
379C23#48312yes—
380C23#48512yes—
381C23#49211yescharacteristic
382C23#49511yescharacteristic
383C23#49711yescharacteristic
384C23#49711yescharacteristic
385Q8—82no—
386Q8—82no—
387Q8—82no—
388Q8—82no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—84no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—164no—
440C2×C4—164no—
441C2×C4—162no—
442C2×C4—162no—
443C2×C4#38011yescharacteristic
444C2×C4#38511yescharacteristic
445C2×C4#38611yescharacteristic
446C2×C4#38811yescharacteristic
447C2×C4#39311yescharacteristic
448C2×C4#39411yescharacteristic
449C2×C4#39611yescharacteristic
450C2×C4#39711yescharacteristic
451C8—44no—
452C8—82no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—41no—
472C22—41no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—41no—
482C22—41no—
483C22#143911yescharacteristic
484C22#186512yes—
485C22#192712yes—
486C22#196811yescharacteristic
487C22#197111yescharacteristic
488C22#201311yescharacteristic
489C22#201511yescharacteristic
490C22#203512yes—
491C22#206612yes—
492C22#209611yescharacteristic
493C22#209711yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—84no—
502C4—82no—
503C4—82no—
504C4—162no—
505C4—162no—
506C4—162no—
507C4#203212yes—
508C2—22no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—22no—
513C2—21no—
514C2—21no—
515C2—22no—
516C2—22no—
517C2—21no—
518C2#1108911yescharacteristic
519C2#1111911yescharacteristic
520C2#1112211yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1520C4semidirect

Maximal subgroups

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

subgroupindex
#219382
C2×#19962
C2×#15202
#108462
#80232
C2×#12822