Finite Group Databasebeta

← #106979

Subgroups — #106979

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34128C211yesmaximal, characteristic
2C2×#1491C211yesmaximal, characteristic
3C2×#1491C211yesmaximal, characteristic
4C4×#344C211yesmaximal, characteristic
5#8073C211yesmaximal, characteristic
6#7972C211yesmaximal, characteristic
7#7930C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2356—22no—
14#2356—21no—
15#2351—21no—
16#2344—22no—
17#2344C412yes—
18Q8×C22⋊C4C412yes—
19#2295—22no—
20#2295—21no—
21#2283—21no—
22#2283C2211yescharacteristic
23C42×Q8—21no—
24C4×#117—21no—
25C4×#117C2211yescharacteristic
26C2×#344—22no—
27#1491—21no—
28#1491—21no—
29#1491—21no—
30#1491—21no—
31#1491—22no—
32#1491C414yescomplemented
33C2×Q8⋊C8—22no—
34C2×Q8⋊C8C414yescomplemented
35#1300—22no—
36#1288—22no—
37#1282—21no—
38#1282—21no—
39C23×Q8C2×C411yescharacteristic
40C22×M16—21no—
41C22×M16—21no—
42C2×C4×Q8—22no—
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×Q8C2×C412yes—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×Q8Q812yes—
57C2×#117—22no—
58C2×#117D412yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C42—21no—
62C22×C42C2311yescharacteristic
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432—21no—
70#432C2×C412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2×C412yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16C2×C412yes—
83C2×#97—21no—
84C2×#97—21no—
85C2×#97—21no—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2×C412yes—
89#391—22no—
90#391—24no—
91#389—24no—
92#389—22no—
93#387—24no—
94#387—24no—
95#387—24no—
96#383—22no—
97#383D412yes—
98#382—22no—
99#382—21no—
100#381—21no—
101#381—22no—
102#380—22no—
103#380C2×C412yes—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4D411yescharacteristic
116C4×C22⋊C4D411yescharacteristic
117#376—22no—
118#376—22no—
119#376—21no—
120#376D412yes—
121C43—22no—
122C43—21no—
123#344—28no—
124Q8⋊C8—24no—
125Q8⋊C8—24no—
126Q8⋊C8—24no—
127Q8⋊C8—24no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—44no—
132C22×Q8C4212yes—
133C22×Q8C22⋊C411yescharacteristic
134C22×Q8C22⋊C411yescharacteristic
135C22×Q8C4⋊C412yes—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—22no—
144C2×M16—42no—
145C2×M16—42no—
146C2×M16—41no—
147C2×M16—41no—
148C2×M16—41no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—41no—
152C22×C8—41no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—48no—
170C4×Q8C22⋊C412yes—
171C4×Q8C22⋊C412yes—
172C4×Q8C22⋊C412yes—
173C4×Q8C22⋊C412yes—
174C4×Q8C4⋊C412yes—
175C4×Q8C4⋊C412yes—
176C4×Q8C4⋊C412yes—
177C4×Q8C4⋊C412yes—
178#117—24no—
179#117—42no—
180#117—44no—
181#117—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C4212yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C4⋊C412yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×Q811yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—41no—
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239#97—22no—
240#97—22no—
241#97—41no—
242#97—41no—
243#97—41no—
244#97—41no—
245#97—41no—
246#97—41no—
247C4×C8—22no—
248C4×C8—22no—
249C4×C8—22no—
250C4×C8—41no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—21no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95C22⋊C414yes—
262C24C2×C22⋊C411yescharacteristic
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8#9512yes—
277C2×Q8#9512yes—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4C2×C4211yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300C22×C4#11711yescharacteristic
301C22×C4#11711yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4C22⋊Q811yescharacteristic
313C22×C4C22⋊Q811yescharacteristic
314M16—24no—
315M16—44no—
316M16—44no—
317M16—42no—
318M16—44no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—44no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—82no—
327C2×C8—82no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—44no—
344C4⋊C4#9512yes—
345C4⋊C4#9512yes—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4#10414yes—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—44no—
361C42—42no—
362C42—42no—
363C42—44no—
364C42—48no—
365C42—48no—
366C42—44no—
367C42C2×C22⋊C411yescharacteristic
368C42C2×C22⋊C411yescharacteristic
369C42C2×C4⋊C411yescharacteristic
370C42C2×C4⋊C411yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42C22⋊Q811yescharacteristic
378C42C22⋊Q811yescharacteristic
379C42#12311yescharacteristic
380C42#12311yescharacteristic
381C42#12311yescharacteristic
382C42#12311yescharacteristic
383C23—41no—
384C23—41no—
385C23C4×C22⋊C411yescharacteristic
386C23#37911yescharacteristic
387C23#38011yescharacteristic
388C23C2×#10412yes—
389C23#42112yes—
390Q8—24no—
391Q8—22no—
392Q8—22no—
393Q8—22no—
394Q8—44no—
395Q8—44no—
396Q8—42no—
397Q8#34414yescomplemented
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4C2×#9511yescharacteristic
441C2×C4C4×C22⋊C411yescharacteristic
442C2×C4#38011yescharacteristic
443C2×C4#38111yescharacteristic
444C2×C4#38511yescharacteristic
445C2×C4#38511yescharacteristic
446C2×C4#38511yescharacteristic
447C2×C4#38511yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C8—44no—
453C8—44no—
454C8—82no—
455C8—82no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—42no—
463C22—42no—
464C22C22⋊#9511yescharacteristic
465C22C4×#10411yescharacteristic
466C22#175911yescharacteristic
467C22#178912yes—
468C22#180612yes—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—24no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—82no—
485C4—84no—
486C4C2×#34412yes—
487C4#190512yes—
488C2—22no—
489C2—22no—
490C2—21no—
491C2—21no—
492C2#1073811yescharacteristic
493C2#1078611yescharacteristic
494C2#1079011yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
32. #1491C4semidirect
34. C2×Q8⋊C8C4semidirect
397. Q8#344semidirect

Maximal subgroups

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

subgroupindex
#341282
C2×#14912
C4×#3442
#80732
#79722
#79302