Finite Group Databasebeta

← #190227

Subgroups — #190227

Order 512.

892 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34189C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#8763C214yesmaximal, complemented
5#8762C214yesmaximal, complemented
6#8017C212yesmaximal, complemented
7#7984C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C2×#432C2212yes—
10#2357—21no—
11#2357C412yescomplemented
12#2356—21no—
13#2356C412yescomplemented
14#2353—21no—
15#2353—21no—
16#2353C2212yes—
17#2353C2212yes—
18#2314—21no—
19#2314—21no—
20#2311—21no—
21#2285C2211yescharacteristic
22#2283C411yescharacteristic, complemented
23#2283C411yescharacteristic, complemented
24#2283C411yescharacteristic, complemented
25#2283C411yescharacteristic, complemented
26#2283C2212yes—
27C2×#382C2211yescharacteristic
28#2274—21no—
29#1820—21no—
30#1799C2212yes—
31#1799C2212yes—
32#1782—21no—
33C4×#99—21no—
34C4×#99—21no—
35#1467—21no—
36#1466—21no—
37#1464—21no—
38#1464—21no—
39#1459C2212yes—
40#1459C2212yes—
41#1294C2218yes—
42#1289C2218yes—
43C22×#41D411yescharacteristic
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×#117—21no—
57C2×#117D411yescharacteristic
58C2×#117D411yescharacteristic
59C2×#117D411yescharacteristic
60C22×C4⋊C4—21no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2311yescharacteristic
64#432—22no—
65#432—22no—
66#432—22no—
67#432—22no—
68C2×C4⋊C8—22no—
69C2×#99—22no—
70C2×#99—22no—
71C2×C22⋊C8C2314yes—
72C2×#97—22no—
73#391—22no—
74#390—22no—
75#388—22no—
76#388—22no—
77#388—22no—
78#388—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#386—22no—
84#386—22no—
85#384—21no—
86#383—21no—
87#383—21no—
88#383—22no—
89#383C2×C412yes—
90#383C2×C412yes—
91#382—21no—
92#382—21no—
93#382—22no—
94#382C2×C412yes—
95#382C2×C411yescharacteristic
96#382C2×C411yescharacteristic
97#382C2×C411yescharacteristic
98#382C2×C411yescharacteristic
99#381—21no—
100#381—21no—
101#381—22no—
102#381C2×C412yes—
103#381C2×C412yes—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380C2314yes—
110#380C2312yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C412yes—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4C2×C412yes—
127#376—21no—
128#376—21no—
129#376—22no—
130#376C2×C412yes—
131#376C2×C412yes—
132C2×#95—21no—
133C2×#95C2312yes—
134#342—22no—
135#342—22no—
136#324—22no—
137#324—22no—
138#323—22no—
139#323—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—42no—
143C2×#41—42no—
144C22×Q8C22⋊C411yescharacteristic
145C22×D4C22⋊C411yescharacteristic
146C22×D4C2×D411yescharacteristic
147C22×D4C2×D411yescharacteristic
148C23×C4—22no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—24no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C22⋊C411yescharacteristic
204C2×C22⋊C4C22⋊C411yescharacteristic
205C2×C22⋊C4C2×D411yescharacteristic
206C2×C22⋊C4C2×D411yescharacteristic
207C2×C22⋊C4C2×D411yescharacteristic
208C2×C22⋊C4C2×D411yescharacteristic
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—42no—
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C4⋊C8—42no—
238C4⋊C8—42no—
239#99—44no—
240#99—44no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243#97—42no—
244#97—42no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—24no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95C22×C412yes—
264#95C22×C412yes—
265#95C22×C412yes—
266#95C22×C412yes—
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C411yescharacteristic
269C24C22×D411yescharacteristic
270#41—48no—
271#41—48no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—41no—
283C2×D4—41no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4#12312yes—
327C22×C4#12312yes—
328C22×C4#12312yes—
329C22×C4#12312yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4112yes—
333C22×C4C2×#4112yes—
334C2×C8—44no—
335C2×C8—84no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—84no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—44no—
376C42—82no—
377C42—82no—
378C42—82no—
379C42—82no—
380C42—84no—
381C42#11711yescharacteristic
382C42#11711yescharacteristic
383C42#11711yescharacteristic
384C42#11711yescharacteristic
385C42#11712yes—
386C42#11712yes—
387C42#12312yes—
388C42#12312yes—
389C42#12312yes—
390C42#12312yes—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C22×C22⋊C411yescharacteristic
404C23C2×#12312yes—
405C23#57512yes—
406C23#57512yes—
407Q8—44no—
408D4—44no—
409D4—44no—
410D4—44no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—164no—
467C2×C4#38111yescharacteristic
468C2×C4#38111yescharacteristic
469C2×C4#38111yescharacteristic
470C2×C4#38111yescharacteristic
471C2×C4#38112yes—
472C2×C4#38112yes—
473C2×C4C2×#11711yescharacteristic
474C2×C4C2×#11711yescharacteristic
475C2×C4C2×#12312yes—
476C8—84no—
477C22—21no—
478C22—22no—
479C22—21no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22C2×#38111yescharacteristic
494C22#289611yescharacteristic
495C22#289611yescharacteristic
496C22#309314yes—
497C4—21no—
498C4—21no—
499C4—22no—
500C4—21no—
501C4—21no—
502C4—22no—
503C4—22no—
504C4—21no—
505C4—21no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—41no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—164no—
517C2—21no—
518C2—21no—
519C2—24no—
520C2—81no—
521C2—81no—
522C2#2070111yescharacteristic
523C2#2070811yescharacteristic
524C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8763C2semidirect
5. #8762C2semidirect
6. #8017C2semidirect
7. #7984C2semidirect
11. #2357C4semidirect
13. #2356C4semidirect
22. #2283C4semidirect
23. #2283C4semidirect
24. #2283C4semidirect
25. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#341892
#207232
#173632
#87632
#87622
#80172
#79842