Finite Group Databasebeta

← #224169

Subgroups — #224169

Order 512.

919 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#124C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic, complemented
3#20520C211yesmaximal, characteristic, complemented
4#9895C212yesmaximal, complemented
5#9890C212yesmaximal, complemented
6#9890C211yesmaximal, characteristic, complemented
7#9890C211yesmaximal, characteristic, complemented
8#8581C212yesmaximal
9#8579C211yesmaximal, characteristic
10#8226C212yesmaximal
11C4×#329C211yesmaximal, characteristic
12C2×Q82C411yescharacteristic
13C2×D4×Q8C411yescharacteristic
14C2×#446—21no—
15#2862C2211yescharacteristic
16#2862C2212yes—
17#2568C412yes—
18#2568C411yescharacteristic
19#2563C411yescharacteristic
20#2563C412yes—
21#2492—22no—
22#2492—21no—
23#2492—21no—
24#2492—22no—
25Q8×C22⋊C4—21no—
26Q8×C22⋊C4—21no—
27C4×#124C2211yescharacteristic
28C4×#124C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1771—22no—
34C8×C4⋊C4—21no—
35#1767C2211yescharacteristic
36#1767C2212yes—
37#1767C2211yescharacteristic
38#1767C2212yes—
39#1745C2212yes—
40C4×#97C2211yescharacteristic
41C4×#97C2211yescharacteristic
42C4×#97C2212yes—
43C2×#329C2211yescharacteristic
44C2×#329C2212yes—
45C2×#329C2211yescharacteristic
46#1295C2211yescharacteristic
47#1295C2211yescharacteristic
48#1295C2212yes—
49#1295C2212yes—
50#1295C2212yes—
51#1295C2212yes—
52#1295C2212yes—
53C23×Q8—21no—
54Q82—44no—
55D4×Q8—44no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×#124C2311yescharacteristic
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C412yes—
73#446—28no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×#97—22no—
79C2×#97—21no—
80C2×#97—21no—
81C2×#97C2311yescharacteristic
82C2×#97C2311yescharacteristic
83C2×#97C2312yes—
84C2×#97C2312yes—
85C2×#97C2311yescharacteristic
86C2×#97C2311yescharacteristic
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90#400—22no—
91#400—22no—
92#400—22no—
93#393—22no—
94#393—22no—
95#393—22no—
96#391—22no—
97#391—21no—
98#391—21no—
99#391—22no—
100#391—21no—
101#391—21no—
102#391—22no—
103#391—22no—
104#391—22no—
105#391—21no—
106#391—21no—
107#391—22no—
108#391C2×C411yescharacteristic
109#391C2×C412yes—
110#390C2×C411yescharacteristic
111#390C2×C412yes—
112#388—22no—
113#388—21no—
114#388—21no—
115#388—22no—
116#382C2×C412yes—
117#382C2×C412yes—
118#382C2×C412yes—
119#382C2311yescharacteristic
120#382C2312yes—
121#380—22no—
122#380—21no—
123#380—21no—
124#380—22no—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D412yes—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C43C2311yescharacteristic
136C43C2312yes—
137#336—21no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—21no—
143#329—22no—
144#329—24no—
145#329Q814yescomplemented
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—44no—
151C22×Q8—44no—
152C22×Q8—44no—
153C22×Q8C22⋊C411yescharacteristic
154C22×Q8C22×C411yescharacteristic
155C22×D4C22×C411yescharacteristic
156C23×C4—22no—
157C23×C4—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C4⋊Q8—42no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—42no—
168C4⋊Q8—48no—
169C4⋊Q8—44no—
170#124—21no—
171#124—21no—
172#124C2×Q812yes—
173#124#4112yes—
174C22⋊Q8—48no—
175C22⋊Q8—44no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—42no—
179C4×Q8—44no—
180C4×Q8—42no—
181C4×Q8—44no—
182C4×D4—42no—
183C4×D4—44no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2411yescharacteristic
223C4⋊C8—24no—
224C4⋊C8—28no—
225C4⋊C8—42no—
226C4⋊C8—44no—
227#97—28no—
228#97—21no—
229#97—21no—
230#97—22no—
231#97—22no—
232#97—21no—
233#97—21no—
234#97—21no—
235#97—21no—
236#97—22no—
237#97—22no—
238#97—42no—
239#97—42no—
240#97C2×Q812yescomplemented
241#97C2×Q812yescomplemented
242C4×C8—24no—
243C4×C8—42no—
244C4×C8—44no—
245#95—21no—
246#95—22no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95—21no—
258#95—22no—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262C24—21no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—82no—
277C2×Q8—82no—
278C2×Q8—81no—
279C2×Q8—81no—
280C2×Q8—816no—
281C2×Q8#10414yes—
282C2×Q8C4×Q812yes—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4C4×Q812yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—24no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4#10014yes—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—82no—
350C22⋊C4—81no—
351C22⋊C4—81no—
352C22⋊C4—82no—
353C42—24no—
354C42—22no—
355C42—22no—
356C42—24no—
357C42—22no—
358C42—42no—
359C42—41no—
360C42—41no—
361C42—42no—
362C42—41no—
363C42—42no—
364C42—42no—
365C42—44no—
366C42—42no—
367C42C22×Q811yescharacteristic
368C42C2×#4111yescharacteristic
369C422−1+411yescharacteristic
370C422−1+411yescharacteristic
371C23—22no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23C22×C22⋊C411yescharacteristic
379Q8—28no—
380Q8—42no—
381Q8—42no—
382Q8—88no—
383Q8—84no—
384Q8—164no—
385Q8—162no—
386D4—42no—
387D4—42no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—88no—
429C2×C4—84no—
430C2×C4C2×#10014yes—
431C2×C4C2×#10012yes—
432C2×C4C2×#10414yes—
433C2×C4C2×#10412yes—
434C2×C4#42112yes—
435C2×C4C2×C4×Q811yescharacteristic
436C2×C4#51911yescharacteristic
437C2×C4#54811yescharacteristic
438C2×C4#54811yescharacteristic
439C2×C4D4×Q811yescharacteristic
440C2×C4D4×Q811yescharacteristic
441C8—24no—
442C8—42no—
443C8—44no—
444C8—42no—
445C8—42no—
446C8—82no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22Q8×C22⋊C411yescharacteristic
455C22C22×#10011yescharacteristic
456C22#288911yescharacteristic
457C22C22×#10411yescharacteristic
458C22C2×#42111yescharacteristic
459C22#290111yescharacteristic
460C22#290111yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—24no—
466C4—24no—
467C4—21no—
468C4—21no—
469C4—41no—
470C4—42no—
471C4—41no—
472C4—42no—
473C4—41no—
474C4—42no—
475C4—81no—
476C4—82no—
477C4—82no—
478C4—81no—
479C4#152714yes—
480C2—82no—
481C2C2×#152711yescharacteristic
482C2#1765411yescharacteristic
483C2#1768311yescharacteristic
484C2#1768511yescharacteristic
485C2Q8×#10011yescharacteristic
486C2Q8×#10411yescharacteristic
487C2#2252911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21345C2semidirect
3. #20520C2semidirect
4. #9895C2semidirect
5. #9890C2semidirect
6. #9890C2semidirect
7. #9890C2semidirect
145. #329Q8semidirect
240. #97C2×Q8semidirect
241. #97C2×Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×#1242
#213452
#205202
#98952
#98902
#85812
#85792
#82262
C4×#3292