Finite Group Databasebeta

← #190324

Subgroups — #190324

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#8775C214yesmaximal, complemented
5#8772C214yesmaximal, complemented
6#8071C212yesmaximal, complemented
7#7977C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10C2×#432C2212yes—
11#2357C412yes—
12#2356C2212yes—
13#2355—21no—
14#2354—21no—
15#2353C412yes—
16#2353C2212yes—
17#2352—21no—
18#2352—21no—
19#2298—21no—
20#2298—21no—
21#2297—21no—
22#2285C411yescharacteristic
23#2283C411yescharacteristic
24#2283C2211yescharacteristic
25C4×#117—21no—
26C4×#117C411yescharacteristic
27C4×#117C2211yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic
29#1822—21no—
30#1801—21no—
31#1800—21no—
32#1785—21no—
33C4×#101C2212yes—
34C4×#100C2212yes—
35#1467C2212yes—
36#1466C2212yes—
37#1464—21no—
38#1464—21no—
39#1459—21no—
40#1459—21no—
41#1300C2218yes—
42#1288C2218yes—
43C22×#41D411yescharacteristic
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8C2311yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×#117—21no—
57C2×#117—21no—
58C2×#117—21no—
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C2×#117D411yescharacteristic
62C22×C4⋊C4—21no—
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67#432—22no—
68#432—22no—
69#432—22no—
70#432—22no—
71#432—22no—
72#432C2314yes—
73#431—22no—
74#431—22no—
75#423—22no—
76C2×C4⋊C8—22no—
77C2×#101—22no—
78C2×#100—22no—
79#407—22no—
80C2×C22⋊C8—22no—
81C4×M16C2314yes—
82C2×#97—22no—
83#391—22no—
84#390—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#386—22no—
94#386—22no—
95#384—21no—
96#383C2×C412yes—
97#383C2312yes—
98#382—21no—
99#382—21no—
100#382C2×C412yes—
101#381C2×C412yes—
102#381C2×C412yes—
103#380—21no—
104#380—21no—
105#380—21no—
106#380C2×C412yes—
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⋊C4C2×C412yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4C2×C412yes—
127#376—21no—
128#376—21no—
129#376C2×C412yes—
130#376C2312yes—
131C2×#95C2×C412yes—
132C43—21no—
133C43—21no—
134#343—22no—
135#343—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×Q8C2×D411yescharacteristic
145C22×D4C22⋊C411yescharacteristic
146C22×D4C22⋊C411yescharacteristic
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×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—24no—
159C22×C8—42no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—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—
175C4×D4—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—44no—
183#117—44no—
184#117—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4C22⋊C411yescharacteristic
205C2×C22⋊C4C22⋊C411yescharacteristic
206C2×C22⋊C4C22⋊C411yescharacteristic
207C2×C22⋊C4C22⋊C411yescharacteristic
208C2×C22⋊C4C2×D411yescharacteristic
209C2×C22⋊C4C2×D411yescharacteristic
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—44no—
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C411yescharacteristic
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242C4⋊C8—24no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246#101—44no—
247#100—44no—
248C22⋊C8—24no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252#97—24no—
253#97—24no—
254#97—42no—
255C4×C8—24no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95C22×C412yes—
267#95C22×C412yes—
268#95C22×C412yes—
269#95C22×C412yes—
270C24C2×C22⋊C411yescharacteristic
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273#41—48no—
274#41—48no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—81no—
315C22×C4—82no—
316C22×C4—81no—
317C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
327C22×C4#11712yes—
328C22×C4#11712yes—
329C22×C4#12312yes—
330C22×C4#12312yes—
331C22×C4#12312yes—
332C22×C4#12312yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336M16—44no—
337M16—44no—
338C2×C8—44no—
339C2×C8—44no—
340C2×C8—84no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—82no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—42no—
378C42—42no—
379C42—44no—
380C42—42no—
381C42—42no—
382C42—44no—
383C42—44no—
384C42—82no—
385C42—82no—
386C42#11711yescharacteristic
387C42#11711yescharacteristic
388C42#11711yescharacteristic
389C42#11711yescharacteristic
390C42#12312yes—
391C42#12312yes—
392C42#12312yes—
393C42#12312yes—
394C42C2×#4112yes—
395C42C2×#4112yes—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23#38112yes—
410C23#42112yes—
411C23C22×C22⋊C411yescharacteristic
412Q8—44no—
413D4—44no—
414D4—44no—
415D4—44no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—88no—
468C2×C4#38111yescharacteristic
469C2×C4#38111yescharacteristic
470C2×C4#38111yescharacteristic
471C2×C4#38112yes—
472C2×C4#38111yescharacteristic
473C2×C4#42112yes—
474C2×C4C2×#11711yescharacteristic
475C2×C4C2×#11711yescharacteristic
476C2×C4C2×#12312yes—
477C2×C4C2×#12312yes—
478C8—84no—
479C8—84no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—42no—
487C22—42no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22C2×#38111yescharacteristic
498C22C2×#42111yescharacteristic
499C22C2×#42111yescharacteristic
500C4—22no—
501C4—21no—
502C4—21no—
503C4—22no—
504C4—22no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—22no—
509C4—44no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4#182712yes—
519C2—21no—
520C2—21no—
521C2—42no—
522C2—42no—
523C2—81no—
524C2#2070511yescharacteristic
525C2C2×#182711yescharacteristic
526C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8775C2semidirect
5. #8772C2semidirect
6. #8071C2semidirect
7. #7977C2semidirect

Maximal subgroups

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

subgroupindex
#341212
#207282
#173632
#87752
#87722
#80712
#79772