Finite Group Databasebeta

← #190323

Subgroups — #190323

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2356C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#8775C214yesmaximal, complemented
5#8774C214yesmaximal, complemented
6#8072C212yesmaximal
7#7975C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2356—21no—
12#2356—21no—
13#2356—21no—
14#2356—21no—
15#2356C412yes—
16#2356C412yes—
17#2356C2212yes—
18#2356C2212yes—
19C2×#391—21no—
20C2×#387—21no—
21C2×#387—21no—
22C2×#383C411yescharacteristic
23C2×#383C2211yescharacteristic
24C2×#382C411yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C411yescharacteristic
27C2×#376C411yescharacteristic
28C2×#376C2211yescharacteristic
29#1821—21no—
30#1801—21no—
31#1801—21no—
32#1784—21no—
33C4×#101C2212yes—
34C4×#101C2212yes—
35#1467C2212yes—
36#1467C2212yes—
37#1463—21no—
38#1463—21no—
39#1461—21no—
40#1461—21no—
41#1300C2218yes—
42#1288C2218yes—
43C23×Q8D411yescharacteristic
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C42—21no—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67#432—22no—
68#432—22no—
69#432—22no—
70#432C2314yes—
71#431—22no—
72#431—22no—
73#431—22no—
74#431—22no—
75#423—22no—
76#423—22no—
77C2×#101—22no—
78C2×#101—22no—
79#407—22no—
80#407—22no—
81C4×M16—22no—
82C4×M16C2314yes—
83#391—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#383—21no—
96#383—21no—
97#383—21no—
98#383—21no—
99#383C2×C412yes—
100#383C2×C412yes—
101#383C2×C412yes—
102#383C2312yes—
103#382—21no—
104#382—21no—
105#382C2×C412yes—
106#382C2×C412yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115#376—21no—
116#376—21no—
117#376—21no—
118#376—21no—
119#376C2×C412yes—
120#376C2×C412yes—
121#376C2×C412yes—
122#376C2312yes—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95C2×C411yescharacteristic
131C2×#95C2×C411yescharacteristic
132C2×#95C2×C411yescharacteristic
133C2×#95C2×C411yescharacteristic
134#343—22no—
135#343—22no—
136#324—22no—
137#324—22no—
138#324—22no—
139#324—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8C22⋊C411yescharacteristic
145C22×Q8C22⋊C411yescharacteristic
146C22×Q8C2×D411yescharacteristic
147C22×Q8C2×D411yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×M16—24no—
160C2×M16—42no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
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×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—44no—
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—24no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238#101—44no—
239#101—44no—
240C22⋊C8—24no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244#97—24no—
245#97—24no—
246C4×C8—24no—
247C4×C8—42no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95C22×C412yes—
267#95C22×C412yes—
268#95C22×C412yes—
269#95C22×C412yes—
270C24C2×C22⋊C411yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—48no—
284C2×Q8—48no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4#11711yescharacteristic
331C22×C4#11711yescharacteristic
332C22×C4#11711yescharacteristic
333C22×C4#11711yescharacteristic
334C22×C4#12312yes—
335C22×C4#12312yes—
336C22×C4#12312yes—
337C22×C4#12312yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341M16—44no—
342M16—44no—
343M16—84no—
344C2×C8—44no—
345C2×C8—44no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C42—21no—
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—21no—
376C42—21no—
377C42—21no—
378C42—44no—
379C42—44no—
380C42—44no—
381C42—44no—
382C42—44no—
383C42—44no—
384C42#11712yes—
385C42#11712yes—
386C42#12312yes—
387C42#12312yes—
388C42#12312yes—
389C42#12312yes—
390C42C2×#4112yes—
391C42C2×#4112yes—
392C23—22no—
393C23—22no—
394C23—21no—
395C23—21no—
396C23#38111yescharacteristic
397C23#38111yescharacteristic
398C23C22×C22⋊C411yescharacteristic
399Q8—44no—
400Q8—44no—
401Q8—44no—
402Q8—44no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—24no—
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—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4#38111yescharacteristic
462C2×C4#38111yescharacteristic
463C2×C4#38112yes—
464C2×C4#38112yes—
465C2×C4#42112yes—
466C2×C4#42112yes—
467C2×C4C2×#11711yescharacteristic
468C2×C4C2×#11711yescharacteristic
469C2×C4C2×#12312yes—
470C2×C4C2×#12312yes—
471C8—84no—
472C8—84no—
473C22—24no—
474C22—24no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22#182712yes—
482C22#183012yes—
483C22C2×#38111yescharacteristic
484C22C2×#42111yescharacteristic
485C22C2×#42111yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—44no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—82no—
504C4—82no—
505C4—82no—
506C2—21no—
507C2—21no—
508C2—22no—
509C2—22no—
510C2#2070511yescharacteristic
511C2C2×#182711yescharacteristic
512C2C2×#183011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
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. #8774C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23562
#207272
#173652
#87752
#87742
#80722
#79752