Finite Group Databasebeta

← #165028

Subgroups — #165028

Order 512.

821 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36474C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic
3#22736C211yesmaximal, characteristic
4#7722C212yesmaximal
5#7682C212yesmaximal
6#7662C211yesmaximal, characteristic
7#7662C211yesmaximal, characteristic
8#7653C211yesmaximal, characteristic
9#7624C212yesmaximal
10#7590C212yesmaximal
11#7571C211yesmaximal, characteristic
12C2×Q82C411yescharacteristic
13C2×#557C411yescharacteristic
14C2×#446—21no—
15#2862C2212yes—
16#2862C2211yescharacteristic
17#2567C412yes—
18#2567C411yescharacteristic
19#2550—22no—
20#2550—21no—
21#2550—22no—
22#2550—21no—
23#2546C412yes—
24#2546C411yescharacteristic
25#2351—21no—
26#2351—21no—
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1916—22no—
34#1839C2212yes—
35#1831C2212yes—
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38C8×C4⋊C4—21no—
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42#1744C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2211yescharacteristic
48#1272C2212yes—
49#1272C2212yes—
50#1272C2211yescharacteristic
51#1272C2211yescharacteristic
52#1272C2212yes—
53Q82—44no—
54#557—44no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×#125—22no—
64C2×#125—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74#446—44no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—21no—
81C2×#97—22no—
82C2×C4×C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#398—22no—
86#398—22no—
87#398—22no—
88#395—22no—
89#395—22no—
90#395—22no—
91#391—22no—
92#391—22no—
93#391—21no—
94#391—21no—
95#391C2×C412yes—
96#391C2×C412yes—
97#391C2×C412yes—
98#391C2311yescharacteristic
99#391C2311yescharacteristic
100#391C2312yes—
101#389—21no—
102#389—22no—
103#389—22no—
104#389—21no—
105#389—21no—
106#389—22no—
107#389—22no—
108#389—21no—
109#389—22no—
110#389—21no—
111#389—21no—
112#389—22no—
113#384C2×C411yescharacteristic
114#384C2×C412yes—
115#382—22no—
116#382—21no—
117#382—22no—
118#382—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C43C2311yescharacteristic
130C43C2312yes—
131#336—21no—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8C22⋊C411yescharacteristic
146C22×Q8C22×C412yes—
147C22×Q8C22×C411yescharacteristic
148C22×Q8C22×C411yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—24no—
157C4⋊Q8—41no—
158C4⋊Q8—42no—
159C4⋊Q8—44no—
160C4⋊Q8—44no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—48no—
164C4⋊Q8—44no—
165#125—48no—
166#125—44no—
167C4×Q8—24no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—44no—
175C4×Q8—44no—
176C4×Q8SD1614yes—
177C4×Q8Q1614yes—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C412yes—
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2411yescharacteristic
222C4⋊C8—42no—
223C4⋊C8—44no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C4⋊C8—44no—
227C4⋊C8—42no—
228#97—44no—
229C4×C8—44no—
230C4×C8—42no—
231C4×C8—42no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C2×Q8—21no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—21no—
253C2×Q8—22no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—42no—
260C2×Q8—41no—
261C2×Q8—44no—
262C2×Q8—82no—
263C2×Q8—82no—
264C2×Q8—81no—
265C2×Q8—81no—
266C2×Q8#10314yes—
267C2×Q8#13112yes—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—42no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—82no—
324C4⋊C4—81no—
325C4⋊C4—81no—
326C4⋊C4—82no—
327C4⋊C4#10014yes—
328C4⋊C4#10318yes—
329C4⋊C4#10314yes—
330C42—21no—
331C42—22no—
332C42—21no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—24no—
337C42—22no—
338C42—42no—
339C42—41no—
340C42—41no—
341C42—42no—
342C42—41no—
343C42—44no—
344C42—44no—
345C42—42no—
346C42—42no—
347C42C2×SD1612yes—
348C42C2×SD1614yes—
349C42C2×Q1614yes—
350C42C2×Q1612yes—
351C42#13712yes—
352C42#13712yes—
353C23C22×C22⋊C411yescharacteristic
354Q8—28no—
355Q8—42no—
356Q8—42no—
357Q8—82no—
358Q8—81no—
359Q8—82no—
360Q8—81no—
361Q8—164no—
362Q8—162no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4C2×#10014yes—
404C2×C4C2×#10012yes—
405C2×C4C2×#10312yes—
406C2×C4C2×#10314yes—
407C2×C4C2×#10314yes—
408C2×C4C2×#10312yes—
409C2×C4#41912yes—
410C2×C4#41912yes—
411C2×C4C2×#13111yescharacteristic
412C2×C4#56811yescharacteristic
413C2×C4C22×SD1611yescharacteristic
414C2×C4C22×Q1611yescharacteristic
415C2×C4C2×#13711yescharacteristic
416C2×C4C2×#13711yescharacteristic
417C8—42no—
418C8—42no—
419C8—81no—
420C8—82no—
421C8—81no—
422C8—82no—
423C22#288011yescharacteristic
424C22#288411yescharacteristic
425C22C22×#10011yescharacteristic
426C22C22×#10311yescharacteristic
427C22C2×#41911yescharacteristic
428C22#290111yescharacteristic
429C22#290111yescharacteristic
430C4—21no—
431C4—21no—
432C4—24no—
433C4—24no—
434C4—24no—
435C4—22no—
436C4—22no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—41no—
443C4—42no—
444C4—82no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—82no—
449C4#154814yes—
450C2#1742511yescharacteristic
451C2#1750911yescharacteristic
452C2#1751411yescharacteristic
453C2#1761911yescharacteristic
454C2#1763211yescharacteristic
455C2#1768511yescharacteristic
456C2C2×#154811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#364742
#228282
#227362
#77222
#76822
#76622
#76532
#76242
#75902
#75712