Finite Group Databasebeta

← #118685

Subgroups — #118685

Order 512.

689 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#419C211yesmaximal, characteristic
2C2×#1479C211yesmaximal, characteristic
3C2×#1461C211yesmaximal, characteristic
4#7932C211yesmaximal, characteristic
5#7924C211yesmaximal, characteristic
6#7882C211yesmaximal, characteristic
7#7854C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#419C411yescharacteristic
14C2×#419C411yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1763—22no—
18C4×#103—22no—
19#1736C411yescharacteristic
20#1736C411yescharacteristic
21#1479—22no—
22#1479—22no—
23C2×Q8⋊C8—22no—
24#1461—22no—
25#1461—22no—
26C2×#324—22no—
27#1283—22no—
28#1282—22no—
29#1278—22no—
30#1274—22no—
31C23×Q8C2×C411yescharacteristic
32C22×M16C2×C411yescharacteristic
33C22×M16C2×C411yescharacteristic
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—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×Q8D412yes—
45C2×C4×Q8D412yes—
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431D412yes—
67#431D412yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72#419—22no—
73#419—22no—
74#419—22no—
75#419C814yescomplemented
76C2×#103—22no—
77C2×#103—22no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8D412yes—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8D412yes—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96#376—21no—
97C43—21no—
98#340—22no—
99#340—22no—
100Q8⋊C8—28no—
101#324—28no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8C22⋊C412yes—
114C22×Q8C2×C812yes—
115C22×Q8M1612yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C2×M16—21no—
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—21no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—21no—
131C2×M16C2×C812yes—
132C2×M16M1612yes—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—41no—
138C22×C8—41no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155#117—21no—
156#117—21no—
157#117—21no—
158#117—21no—
159#117—21no—
160#117—21no—
161#117C2×C812yes—
162#117M1612yes—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209#103—28no—
210#103—28no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219#97—41no—
220#97—41no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—41no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8—41no—
229C4×C8—41no—
230C4×C8—41no—
231#95—41no—
232#95—41no—
233C24C4×D411yescharacteristic
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8C22⋊C814yes—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—81no—
276C22×C4—82no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4#11711yescharacteristic
279C22×C4#11711yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12311yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#12411yescharacteristic
292C22×C4#12411yescharacteristic
293C22×C4#12611yescharacteristic
294C22×C4#12611yescharacteristic
295C22×C4C22×C811yescharacteristic
296C22×C4C2×M1611yescharacteristic
297C22×C4C2×M1611yescharacteristic
298C22×C4C2×M1611yescharacteristic
299C22×C4#13111yescharacteristic
300C22×C4#13111yescharacteristic
301C22×C4#13111yescharacteristic
302C22×C4#13111yescharacteristic
303M16—22no—
304M16—22no—
305M16—22no—
306M16—22no—
307M16—41no—
308M16—41no—
309M16—41no—
310M16—41no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—82no—
322C2×C8—82no—
323C2×C8—82no—
324C2×C8—82no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C42—22no—
342C42—22no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—41no—
348C42—41no—
349C42—82no—
350C42—82no—
351C42#12011yescharacteristic
352C42#12111yescharacteristic
353C42#12111yescharacteristic
354C42#12111yescharacteristic
355C42#12111yescharacteristic
356C42#12311yescharacteristic
357C42#12311yescharacteristic
358C42#12311yescharacteristic
359C42#12311yescharacteristic
360C42#12311yescharacteristic
361C42#12311yescharacteristic
362C42#12311yescharacteristic
363C42#12411yescharacteristic
364C42#12411yescharacteristic
365C42#12611yescharacteristic
366C42#12611yescharacteristic
367C23—41no—
368C23—41no—
369C23#38611yescharacteristic
370C23#38811yescharacteristic
371C23#38811yescharacteristic
372C23C8×D411yescharacteristic
373C23#43511yescharacteristic
374C23#43511yescharacteristic
375C23#43611yescharacteristic
376Q8—24no—
377Q8—24no—
378Q8—42no—
379Q8—42no—
380Q8—42no—
381Q8—42no—
382Q8—42no—
383Q8—42no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4#38111yescharacteristic
414C2×C4#38611yescharacteristic
415C2×C4#38811yescharacteristic
416C2×C4#38811yescharacteristic
417C2×C4#39411yescharacteristic
418C2×C4#39411yescharacteristic
419C2×C4#39411yescharacteristic
420C2×C4#39411yescharacteristic
421C2×C4#39711yescharacteristic
422C2×C4#39711yescharacteristic
423C2×C4#39711yescharacteristic
424C2×C4#39711yescharacteristic
425C2×C4C2×C22⋊C811yescharacteristic
426C2×C4#40711yescharacteristic
427C2×C4#40811yescharacteristic
428C2×C4#40811yescharacteristic
429C2×C4#43111yescharacteristic
430C2×C4#43111yescharacteristic
431C2×C4#43211yescharacteristic
432C2×C4#43211yescharacteristic
433C2×C4#43311yescharacteristic
434C2×C4#43311yescharacteristic
435C2×C4#43311yescharacteristic
436C2×C4#43311yescharacteristic
437C2×C4C8×D411yescharacteristic
438C2×C4C8×D411yescharacteristic
439C2×C4C8×D411yescharacteristic
440C2×C4#43511yescharacteristic
441C2×C4#43511yescharacteristic
442C2×C4#43511yescharacteristic
443C2×C4#43511yescharacteristic
444C2×C4#43511yescharacteristic
445C2×C4#43511yescharacteristic
446C2×C4#43611yescharacteristic
447C2×C4#43611yescharacteristic
448C2×C4#43611yescharacteristic
449C8—42no—
450C8—42no—
451C8—82no—
452C8—82no—
453C8—82no—
454C8—82no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22#144011yescharacteristic
466C22#186911yescharacteristic
467C22#187011yescharacteristic
468C22#192311yescharacteristic
469C22#192311yescharacteristic
470C22#192411yescharacteristic
471C22#192411yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—44no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4#181711yescharacteristic
484C4#181811yescharacteristic
485C4#186911yescharacteristic
486C4#187011yescharacteristic
487C4#192311yescharacteristic
488C4#192311yescharacteristic
489C4#192411yescharacteristic
490C4#192411yescharacteristic
491C4#208212yes—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—21no—
498C2#1114711yescharacteristic
499C2#1118011yescharacteristic
500C2#1118211yescharacteristic
subgroupG/H
centerC2×C4#397
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC22#1440
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
75. #419C8semidirect

Maximal subgroups

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

subgroupindex
C4×#4192
C2×#14792
C2×#14612
#79322
#79242
#78822
#78542