Finite Group Databasebeta

← #387064

Subgroups — #387064

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2#21303C212yesmaximal
3#11354C212yesmaximal
4#11204C212yesmaximal
5#11045C212yesmaximal
6#10963C212yesmaximal
7#10913C212yesmaximal
8#10763C211yesmaximal, characteristic
9#10763C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#423C2212yes—
13C22×#101C2212yes—
14C22×#101C2211yescharacteristic
15C22×#101C2211yescharacteristic
16C2×C4×M16C2212yes—
17#2582—22no—
18#2582—22no—
19#2582—22no—
20#2582—22no—
21C2×#398C414yes—
22C2×#391C412yes—
23C2×#391C2212yes—
24C2×#389C412yes—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27#1985—22no—
28#1985—22no—
29#1984—22no—
30#1984—22no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2211yescharacteristic
34C2×#343C2212yes—
35C2×#343C2211yescharacteristic
36C2×#343C2212yes—
37C2×#343C2212yes—
38#1728C2212yes—
39#1728C2212yes—
40#1728C2212yes—
41#1728C2212yes—
42C23×Q8C2×C411yescharacteristic
43C23×Q8C2311yescharacteristic
44C22×M16C2312yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47C22×M16C2312yes—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—41no—
51C2×C4×Q8—42no—
52C2×C4×Q8—41no—
53C2×C4×Q8—41no—
54C2×C4×Q8—41no—
55C2×C4×Q8—42no—
56C2×C4×Q8D412yes—
57C2×C4×Q8D412yes—
58C2×C4×Q8D412yes—
59C2×C4×Q8D412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C414yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C42C2×C412yes—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#423—42no—
72#423—42no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×#101—24no—
76C2×#101—22no—
77C2×#101—24no—
78C2×#101—41no—
79C2×#101—41no—
80C2×#101—42no—
81C2×#101—41no—
82C2×#101—41no—
83C2×#101Q814yescomplemented
84C4×M16—42no—
85C4×M16—42no—
86C2×#97—22no—
87C2×C4×C8—22no—
88#398—24no—
89#398—24no—
90#398—24no—
91#398—24no—
92#391—24no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391—22no—
97#391—22no—
98#391—22no—
99#389—24no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#384—22no—
107#384—22no—
108C2×#95C2×C414yes—
109C2×#95C2×C414yes—
110C2×#95C2×C414yes—
111#343—24no—
112#343—24no—
113#343—41no—
114#343—42no—
115#343—41no—
116#343—42no—
117#343—42no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8C22⋊C412yes—
139C22×Q8C22⋊C411yescharacteristic
140C22×Q8C22⋊C411yescharacteristic
141C22×Q8C2×Q812yes—
142C22×Q8#4112yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C412yes—
145C23×C4C22×C412yes—
146C23×C4C22×C414yes—
147C23×C4C22×C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C2411yescharacteristic
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16C2×Q814yes—
174C2×M16#4114yes—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8—44no—
191C2×C4⋊C4—24no—
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—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—84no—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C412yes—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C4⋊C8—44no—
241C4⋊C8—44no—
242#101—44no—
243#101—48no—
244#101—44no—
245#97—44no—
246C4×C8—44no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253C24C23×C411yescharacteristic
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8#10114yescomplemented
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C4×Q814yes—
321C22×C4C22×D412yes—
322C22×C4C22×Q811yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4112yes—
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)12yes—
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C42−1+412yes—
330C22×C42−1+411yescharacteristic
331M16—24no—
332M16—24no—
333M16—24no—
334M16—24no—
335M16—42no—
336M16—42no—
337M16—42no—
338M16—42no—
339M16—42no—
340M16—42no—
341M16—42no—
342M16—42no—
343M16—42no—
344M16—42no—
345M16—42no—
346M16—42no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—44no—
391C42—44no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23C22×C22⋊C411yescharacteristic
397C23C2×C4×Q811yescharacteristic
398C23C4×#4111yescharacteristic
399C23#51811yescharacteristic
400C23#51811yescharacteristic
401C23#51911yescharacteristic
402C23#52011yescharacteristic
403Q8—44no—
404Q8—48no—
405Q8—44no—
406Q8—44no—
407Q8—44no—
408Q8—44no—
409Q8—44no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4C2×#10112yes—
454C2×C4C2×#10114yes—
455C2×C4#41312yes—
456C2×C4#53412yes—
457C2×C4#53412yes—
458C2×C4#53811yescharacteristic
459C2×C4#53911yescharacteristic
460C2×C4#53912yes—
461C2×C4#54212yes—
462C2×C4#54311yescharacteristic
463C2×C4#54411yescharacteristic
464C2×C4#54812yes—
465C2×C4#54812yes—
466C2×C4D4×Q812yes—
467C2×C4#55012yes—
468C2×C4#55112yes—
469C2×C4#55412yes—
470C2×C4#56212yes—
471C2×C4#56312yes—
472C8—44no—
473C8—44no—
474C8—82no—
475C8—82no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22#230811yescharacteristic
489C22#232911yescharacteristic
490C22#233211yescharacteristic
491C22Q8×C22⋊C411yescharacteristic
492C22#234111yescharacteristic
493C22#236811yescharacteristic
494C22#237811yescharacteristic
495C22C22×#10111yescharacteristic
496C22C2×#41311yescharacteristic
497C22#288811yescharacteristic
498C22#288911yescharacteristic
499C4—22no—
500C4—22no—
501C4—24no—
502C4—44no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—84no—
508C4—82no—
509C4#272812yes—
510C4#272912yes—
511C4#273012yes—
512C4#273212yes—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—21no—
517C2#1444911yescharacteristic
518C2#2175311yescharacteristic
519C2#2175711yescharacteristic
520C2Q8×#10111yescharacteristic
521C2#2251711yescharacteristic
522C2#2253811yescharacteristic
523C2#2253911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
83. C2×#101Q8semidirect
280. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
#213032
#113542
#112042
#110452
#109632
#109132
#107632