Finite Group Databasebeta

← #115852

Subgroups — #115852

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1960C211yesmaximal, characteristic
4#14677C211yesmaximal, characteristic
5C2×#1295C211yesmaximal, characteristic
6C2×#1295C211yesmaximal, characteristic
7C2×#1295C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C22×#103—21no—
10C22×#103—21no—
11C22×#97C2211yescharacteristic
12C22×#97C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#1993—22no—
18#1993—22no—
19#1993—22no—
20#1993C414yescomplemented
21#1960—22no—
22#1960—22no—
23C2×#340—21no—
24C2×#340—21no—
25C2×#340—22no—
26#1444—21no—
27#1444—21no—
28#1442—21no—
29#1442—21no—
30#1441—22no—
31#1295—22no—
32#1295—22no—
33#1295—22no—
34C23×Q8—21no—
35C23×C8—21no—
36C23×C8—21no—
37C23×C8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—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⋊Q8C2×C412yes—
48C2×C4⋊Q8D412yes—
49C2×C4⋊Q8Q812yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C42C2311yescharacteristic
59C2×#103—24no—
60C2×#103—24no—
61C2×#103—44no—
62C2×#103—44no—
63C2×#97—21no—
64C2×#97—21no—
65C2×#97—21no—
66C2×#97—21no—
67C2×#97—21no—
68C2×#97—21no—
69C2×#97—21no—
70C2×#97—21no—
71C2×#97—21no—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97C2×C412yes—
79C2×#97D412yes—
80C2×#97D412yes—
81#382—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#382—21no—
86#382—21no—
87#382—21no—
88#382—21no—
89#382—21no—
90#382—21no—
91#382C2×C412yes—
92#382D412yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102#340—24no—
103#340—24no—
104#340—28no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8—41no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—41no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—24no—
127C22×C8—24no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—42no—
132C22×C8—42no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
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⋊Q8C4⋊C414yescomplemented
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
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×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C22⋊C414yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×Q811yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200#103—48no—
201#103—48no—
202#97—22no—
203#97—22no—
204#97—22no—
205#97—22no—
206#97—22no—
207#97—22no—
208#97—41no—
209#97—41no—
210#97—41no—
211#97—41no—
212#97—41no—
213#97—41no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—82no—
227#95—82no—
228#95—82no—
229C24#12011yescharacteristic
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8—82no—
247C2×Q8—82no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4C22⋊Q811yescharacteristic
299C22×C4C22⋊Q811yescharacteristic
300C22×C4C22⋊Q811yescharacteristic
301C22×C4#12311yescharacteristic
302C22×C4#12411yescharacteristic
303C22×C4#12411yescharacteristic
304C22×C4#12411yescharacteristic
305C22×C4#12711yescharacteristic
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—44no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—44no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—82no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—88no—
342C42C2×C4⋊C411yescharacteristic
343C42#11711yescharacteristic
344C42C4×D411yescharacteristic
345C42C4×D411yescharacteristic
346C42#12111yescharacteristic
347C42#12111yescharacteristic
348C42#12111yescharacteristic
349C42C22⋊Q811yescharacteristic
350C42C22⋊Q811yescharacteristic
351C42C22⋊Q811yescharacteristic
352C42#12311yescharacteristic
353C42#12311yescharacteristic
354C42#12311yescharacteristic
355C42#12311yescharacteristic
356C42#12611yescharacteristic
357C42#12611yescharacteristic
358C23#38611yescharacteristic
359C23#39211yescharacteristic
360C23#39311yescharacteristic
361C23#45312yes—
362C23#45412yes—
363C23#45512yes—
364C23#45511yescharacteristic
365C23#45511yescharacteristic
366C23#45612yes—
367C23#45611yescharacteristic
368C23#45611yescharacteristic
369Q8—44no—
370Q8—44no—
371Q8—82no—
372Q8—82no—
373Q8—84no—
374Q8—84no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—84no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—84no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4#38011yescharacteristic
436C2×C4#38511yescharacteristic
437C2×C4#38511yescharacteristic
438C2×C4#38811yescharacteristic
439C2×C4#38811yescharacteristic
440C2×C4#39011yescharacteristic
441C2×C4#39411yescharacteristic
442C2×C4#39411yescharacteristic
443C2×C4#39611yescharacteristic
444C2×C4#39611yescharacteristic
445C2×C4#39711yescharacteristic
446C2×C4#39911yescharacteristic
447C8—44no—
448C8—44no—
449C8—44no—
450C22#143911yescharacteristic
451C22#165912yes—
452C22#166012yes—
453C22#166112yes—
454C22#166112yes—
455C22#166212yes—
456C22#166312yes—
457C22#166412yes—
458C22#166412yes—
459C22#188212yes—
460C22#188312yes—
461C22#188511yescharacteristic
462C22#188611yescharacteristic
463C22#200211yescharacteristic
464C22#200311yescharacteristic
465C22#200411yescharacteristic
466C22#200511yescharacteristic
467C22#200711yescharacteristic
468C22#200711yescharacteristic
469C22#202712yes—
470C22#202811yescharacteristic
471C22#202811yescharacteristic
472C22#202812yes—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—84no—
486C4—84no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4—161no—
491C2#899912yes—
492C2#900012yes—
493C2#1008111yescharacteristic
494C2#1008211yescharacteristic
495C2#1008311yescharacteristic
496C2#1008411yescharacteristic
497C2#1035912yes—
498C2#1036112yes—
499C2#1105511yescharacteristic
500C2#1106311yescharacteristic
501C2#1106411yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1993C4semidirect
156. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19932
C2×#19602
#146772
C2×#12952