Finite Group Databasebeta

← #224770

Subgroups — #224770

Order 512.

920 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#557C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic
3#21339C211yesmaximal, characteristic
4#9677C212yesmaximal
5#9673C211yesmaximal, characteristic
6#9643C212yesmaximal
7#9637C211yesmaximal, characteristic
8#8564C211yesmaximal, characteristic
9#8564C212yesmaximal
10#8175C212yesmaximal
11C4×Q8⋊C8C211yesmaximal, characteristic
12C2×#557C411yescharacteristic
13C2×#446—21no—
14#2370—22no—
15#2370—21no—
16#2365C412yes—
17#2365C411yescharacteristic
18#2355C412yes—
19#2355C411yescharacteristic
20#2351—21no—
21Q8×C4⋊C4C411yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2212yes—
24C4×#125—21no—
25C4×#125—22no—
26C42×Q8—21no—
27C42×Q8C2211yescharacteristic
28C42×Q8C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1939—21no—
34#1916—22no—
35#1842C2211yescharacteristic
36#1842C2212yes—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53#557—24no—
54#557—24no—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—22no—
58C2×#125—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2312yes—
69C2×C4×Q8C2311yescharacteristic
70#446—44no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×#97—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2312yes—
86C2×C4×C8C2311yescharacteristic
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2×C412yes—
90#389—21no—
91#389—22no—
92#389—21no—
93#389—22no—
94#389—21no—
95#389—22no—
96#389—21no—
97#389—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#384C2×C412yes—
105#384C2×C411yescharacteristic
106#382—21no—
107#382—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C43—21no—
134C43—22no—
135C43C2311yescharacteristic
136C43C2312yes—
137#336—21no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—21no—
143Q8⋊C8—24no—
144Q8⋊C8—22no—
145Q8⋊C8—24no—
146Q8⋊C8—22no—
147Q8⋊C8—42no—
148Q8⋊C8—42no—
149Q8⋊C8—42no—
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22×C412yes—
152C22×Q8C22×C411yescharacteristic
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—22no—
162C4⋊Q8M1614yes—
163#125—216no—
164#125—28no—
165C4×Q8—28no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—24no—
173C4×Q8—21no—
174C4×Q8—22no—
175C4×Q8—21no—
176C4×Q8—22no—
177C4×Q8—21no—
178C4×Q8—22no—
179C4×Q8—21no—
180C4×Q8—22no—
181C4×Q8—42no—
182C4×Q8—41no—
183C4×Q8—42no—
184C4×Q8—44no—
185C4×Q8M1614yes—
186C4×Q8SD1614yes—
187C4×Q8Q1614yes—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2411yescharacteristic
236C4⋊C8—21no—
237C4⋊C8—22no—
238C4⋊C8—21no—
239C4⋊C8—22no—
240C4⋊C8—21no—
241C4⋊C8—42no—
242C4⋊C8—44no—
243C4⋊C8—42no—
244C4⋊C8—44no—
245C4⋊C8—82no—
246C4⋊C8—81no—
247C4⋊C8#4112yes—
248#97—44no—
249C4×C8—21no—
250C4×C8—22no—
251C4×C8—21no—
252C4×C8—22no—
253C4×C8—21no—
254C4×C8—42no—
255C4×C8—42no—
256C4×C8—42no—
257C4×C8#4112yes—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95—22no—
263#95—22no—
264#95—21no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—21no—
284C2×Q8#10314yes—
285C2×Q8C2×M1614yes—
286C2×Q8C2×M1612yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—42no—
322C4⋊C4—28no—
323C4⋊C4—28no—
324C4⋊C4—28no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—21no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—81no—
347C4⋊C4—82no—
348C4⋊C4#10318yes—
349C4⋊C4#10314yes—
350C4⋊C4C2×M1612yes—
351C4⋊C4#13112yes—
352C42—28no—
353C42—22no—
354C42—22no—
355C42—21no—
356C42—22no—
357C42—21no—
358C42—22no—
359C42—21no—
360C42—22no—
361C42—21no—
362C42—22no—
363C42—24no—
364C42—22no—
365C42—41no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—44no—
370C42—44no—
371C42—41no—
372C42—42no—
373C42—41no—
374C42—42no—
375C42—41no—
376C42—41no—
377C42—42no—
378C42—81no—
379C42—82no—
380C42C2×M1612yes—
381C42C2×M1612yes—
382C42C2×SD1612yes—
383C42C2×SD1614yes—
384C42C2×Q1614yes—
385C42C2×Q1612yes—
386C42C2×#4111yescharacteristic
387C42C2×#4111yescharacteristic
388C42H(5,𝔽2)11yescharacteristic
389C422−1+411yescharacteristic
390C23C22×C22⋊C411yescharacteristic
391Q8—28no—
392Q8—22no—
393Q8—24no—
394Q8—22no—
395Q8—24no—
396Q8—42no—
397Q8—41no—
398Q8—42no—
399Q8—41no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4#40712yes—
438C2×C4#40712yes—
439C2×C4C2×#10312yes—
440C2×C4C2×#10314yes—
441C2×C4C2×#10314yes—
442C2×C4C2×#10312yes—
443C2×C4#42112yes—
444C2×C4C4×#4111yescharacteristic
445C2×C4#52011yescharacteristic
446C2×C4#54811yescharacteristic
447C2×C4#54811yescharacteristic
448C2×C4#55011yescharacteristic
449C2×C4#55011yescharacteristic
450C2×C4C22×M1611yescharacteristic
451C2×C4C22×M1611yescharacteristic
452C2×C4C2×#13111yescharacteristic
453C2×C4#56811yescharacteristic
454C2×C4C22×SD1611yescharacteristic
455C2×C4C22×Q1611yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57811yescharacteristic
458C8—42no—
459C8—42no—
460C8—42no—
461C8—82no—
462C8—81no—
463C8—82no—
464C22#234111yescharacteristic
465C22C2×#40711yescharacteristic
466C22#288011yescharacteristic
467C22C22×#10311yescharacteristic
468C22#289811yescharacteristic
469C22C2×#42111yescharacteristic
470C22#290111yescharacteristic
471C4—21no—
472C4—21no—
473C4—24no—
474C4—22no—
475C4—22no—
476C4—24no—
477C4—42no—
478C4—44no—
479C4—42no—
480C4—42no—
481C4—41no—
482C4—42no—
483C4—42no—
484C4—82no—
485C4—81no—
486C4#148714yes—
487C4#296511yescharacteristic
488C4#297011yescharacteristic
489C4#298911yescharacteristic
490C4#299011yescharacteristic
491C4#335911yescharacteristic
492C4#336311yescharacteristic
493C4#339211yescharacteristic
494C4#339511yescharacteristic
495C2C2×#148711yescharacteristic
496C2#1745411yescharacteristic
497C2#1750911yescharacteristic
498C2#1752311yescharacteristic
499C2#2253211yescharacteristic
500C2#2254211yescharacteristic
501C2#2255111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socle, prime radicalC23C22×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C4×#5572
#227222
#213392
#96772
#96732
#96432
#96372
#85642
#81752
C4×Q8⋊C82