Finite Group Databasebeta

← #189855

Subgroups — #189855

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34196C211yesmaximal, characteristic
2#21820C211yesmaximal, characteristic
3#14346C211yesmaximal, characteristic
4#8717C212yesmaximal
5#8716C212yesmaximal
6#8714C212yesmaximal
7#8714C212yesmaximal
8#8026C211yesmaximal, characteristic
9#8025C211yesmaximal, characteristic
10#8018C211yesmaximal, characteristic
11#8017C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2355C412yescomplemented
16#2355C412yescomplemented
17#2355C412yescomplemented
18Q8×C4⋊C4C412yescomplemented
19#2344—22no—
20#2344—21no—
21Q8×C22⋊C4—21no—
22#2320—22no—
23#2320—21no—
24#2285C2212yes—
25#2285C2211yescharacteristic
26#2283C2212yes—
27#2283C2211yescharacteristic
28C2×#383C2211yescharacteristic
29C2×#382C2211yescharacteristic
30#2276—21no—
31#1985—21no—
32#1985—21no—
33#1984—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1446—21no—
37#1445—21no—
38#1441—22no—
39#1434C2212yes—
40#1434C2212yes—
41#1294C2212yes—
42#1294C2211yescharacteristic
43#1294C2211yescharacteristic
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2211yescharacteristic
49#1294C2212yes—
50#1294C2211yescharacteristic
51C23×Q8D411yescharacteristic
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8D412yes—
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×#117—22no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#432—22no—
71#432—22no—
72#431—22no—
73#431—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C22⋊C8C2311yescharacteristic
77C2×C22⋊C8C2311yescharacteristic
78C2×C22⋊C8C2312yes—
79C2×#97—21no—
80C2×C4×C8—21no—
81#391—22no—
82#391—22no—
83#391—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#384C2×C412yes—
94#384C2×C412yes—
95#384C2×C412yes—
96#383—22no—
97#383—21no—
98#383—22no—
99#383—22no—
100#382—22no—
101#382—21no—
102#382—22no—
103#382—22no—
104#382C2×C412yes—
105#382C2×C412yes—
106#382C2×C412yes—
107#381—22no—
108#381—21no—
109#380—22no—
110#380—21no—
111#380C2311yescharacteristic
112#380C2312yes—
113#380C2311yescharacteristic
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128#376—22no—
129#376—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95C2312yes—
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—44no—
147C22×Q8C22⋊C411yescharacteristic
148C22×Q8C22⋊C412yes—
149C22×Q8C22⋊C411yescharacteristic
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—24no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—24no—
172C4×Q8—22no—
173C4×Q8—48no—
174#117—81no—
175#117—82no—
176#117—81no—
177#117—82no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—81no—
190C2×C4⋊C4—81no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4112yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236#97—42no—
237C4×C8—42no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—82no—
266#95—82no—
267C24C22×D411yescharacteristic
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—48no—
276C2×Q8—48no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4#12312yes—
323C22×C4#12312yes—
324C22×C4#12312yes—
325C22×C4#12312yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—44no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—84no—
356C22⋊C4—42no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—42no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—44no—
371C42—44no—
372C42—44no—
373C42—81no—
374C42—81no—
375C42—82no—
376C42—82no—
377C42#11712yes—
378C42#11712yes—
379C42#11712yes—
380C42#11712yes—
381C42#12312yes—
382C42#12312yes—
383C42#12312yes—
384C42#12312yes—
385C23—22no—
386C23—22no—
387C23C22×C22⋊C411yescharacteristic
388C23C2×#12311yescharacteristic
389C23C2×#12311yescharacteristic
390C23#57611yescharacteristic
391C23#57711yescharacteristic
392C23#57711yescharacteristic
393C23#57811yescharacteristic
394Q8—28no—
395Q8—48no—
396Q8—44no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
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—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4#38112yes—
462C2×C4#38112yes—
463C2×C4#38112yes—
464C2×C4#38112yes—
465C2×C4#42114yes—
466C2×C4C2×#11711yescharacteristic
467C2×C4C2×#11711yescharacteristic
468C2×C4C2×#12311yescharacteristic
469C2×C4C2×#12311yescharacteristic
470C8—81no—
471C8—81no—
472C8—82no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22C2×#38111yescharacteristic
482C22#288412yes—
483C22#289711yescharacteristic
484C22#289811yescharacteristic
485C22C2×#42112yes—
486C22#309611yescharacteristic
487C22#309711yescharacteristic
488C22#309811yescharacteristic
489C22#309911yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—24no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—162no—
507C4—161no—
508C4—161no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2#1760112yes—
514C2#1760212yes—
515C2#2069111yescharacteristic
516C2#2070311yescharacteristic
517C2#2070511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2355C4semidirect
16. #2355C4semidirect
17. #2355C4semidirect
18. Q8×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#341962
#218202
#143462
#87172
#87162
#87142
#80262
#80252
#80182
#80172