Finite Group Databasebeta

← #192372

Subgroups — #192372

Order 512.

1076 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21385C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#14150C211yesmaximal, characteristic
6#8872C214yesmaximal
7#8871C214yesmaximal
8#8026C211yesmaximal, characteristic
9#7908C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2370—22no—
14#2369—21no—
15#2365—21no—
16#2355—21no—
17#2355—22no—
18Q8×C4⋊C4—21no—
19#2329C2214yes—
20#2317—21no—
21C4×C22⋊Q8—22no—
22C2×#389—21no—
23#2285C2211yescharacteristic
24C2×#382C2211yescharacteristic
25#2274—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1987—21no—
29#1981—21no—
30#1948—21no—
31#1948—21no—
32#1872C2212yes—
33#1872C2212yes—
34#1853C2212yes—
35#1853C2212yes—
36C4×#106—22no—
37C4×#103—22no—
38#1518C418yescomplemented
39C2×#328C2214yes—
40#1444C2212yes—
41#1442C2212yes—
42#1436—22no—
43C4⋊#95—21no—
44#1294C2214yes—
45#1280C2214yes—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533Q814yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8D412yes—
58C2×#117—21no—
59C2×#117—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69#431—22no—
70#431—22no—
71#431—22no—
72#431C2×C414yes—
73C2×#106—22no—
74C2×#106—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×#103—22no—
78C2×#103—22no—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83#391—22no—
84#391—24no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—24no—
89#389—44no—
90#387—24no—
91#387—24no—
92#387—24no—
93#385—24no—
94#384—22no—
95#384D411yescharacteristic
96#384D411yescharacteristic
97#382—22no—
98#382—21no—
99#382—22no—
100#382C2×C414yes—
101#380—21no—
102#380C2312yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4D412yes—
119C4×C22⋊C4—24no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95C2312yes—
126C2×#95C2312yes—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#328C2×C4116yes—
132C22×Q8—21no—
133C22×Q8C2×D412yes—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8C4⋊C414yescomplemented
147C4⋊Q8C4⋊C414yescomplemented
148#125—24no—
149#125—24no—
150C22⋊Q8—24no—
151C22⋊Q8—24no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8C22×C414yes—
155C22⋊Q8C2×Q814yes—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—84no—
160C4×Q8C4⋊C418yescomplemented
161#117—24no—
162#117—24no—
163#117—42no—
164#117—42no—
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—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4SD1614yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—82no—
212C2×C42C22×C412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220#106—44no—
221#106—44no—
222C4⋊C8—24no—
223C4⋊C8—42no—
224#103—44no—
225#103—44no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8C22×C414yes—
230C4×C8—24no—
231C4×C8—42no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—44no—
243#95—41no—
244#95—41no—
245#95—44no—
246#95—82no—
247#95C22×C414yes—
248C24C22×D411yescharacteristic
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8C2×C4⋊C414yes—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4C22⋊Q812yes—
301C22×C4C22⋊Q812yes—
302C22×C4C2×SD1614yes—
303C22×C4C2×SD1612yes—
304C22×C4#13512yes—
305C22×C4#13712yes—
306C22×C4#13712yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×Q811yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C2×C8—24no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—48no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C4⋊C4—88no—
348C4⋊C4C2×C4⋊C414yes—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C42—28no—
354C42—44no—
355C42—44no—
356C42—82no—
357C42—84no—
358C42—84no—
359C42—82no—
360C42—82no—
361C42C2×C4⋊C412yes—
362C42C2×C4⋊C412yes—
363C42C4×D412yes—
364C42C4×D412yes—
365C42C22⋊Q812yes—
366C42C22⋊Q812yes—
367C42#12312yes—
368C42#12312yes—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23C2×C4×D411yescharacteristic
374C23C2×#12011yescharacteristic
375C23C2×C22⋊Q811yescharacteristic
376C23C22×SD1611yescharacteristic
377C23C2×#13511yescharacteristic
378C23C2×#13711yescharacteristic
379C23C2×#13711yescharacteristic
380Q8—44no—
381Q8—48no—
382Q8—82no—
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—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4#38512yes—
441C2×C4#38512yes—
442C2×C4#38512yes—
443C2×C4#38512yes—
444C2×C4C4×SD1614yes—
445C2×C4#44114yes—
446C2×C4#44812yes—
447C2×C4#44812yes—
448C2×C4#45212yes—
449C2×C4#45212yes—
450C2×C4#47514yes—
451C2×C4#47914yes—
452C2×C4C22×C4⋊C411yescharacteristic
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C8—44no—
457C8—82no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22C2×#38511yescharacteristic
471C22C2×C4×SD1611yescharacteristic
472C22C2×#44111yescharacteristic
473C22C2×#44811yescharacteristic
474C22C2×#45211yescharacteristic
475C22#301911yescharacteristic
476C22#302011yescharacteristic
477C22#302511yescharacteristic
478C22#302511yescharacteristic
479C22C2×#47511yescharacteristic
480C22C2×#47911yescharacteristic
481C4—22no—
482C4—22no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—44no—
488C4—44no—
489C4—44no—
490C4—84no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4—82no—
496C4#186514yes—
497C4#186514yes—
498C2—21no—
499C2—21no—
500C2—21no—
501C2—21no—
502C2#1821211yescharacteristic
503C2#1821511yescharacteristic
504C2#1822011yescharacteristic
505C2#1822311yescharacteristic
506C2#2086411yescharacteristic
507C2C2×#186511yescharacteristic
508C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
38. #1518C4semidirect
146. C4⋊Q8C4⋊C4semidirect
147. C4⋊Q8C4⋊C4semidirect
160. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#213852
#213592
C2×#15182
#141502
#88722
#88712
#80262
#79082