Finite Group Databasebeta

← #189922

Subgroups — #189922

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34193C211yesmaximal, characteristic
2#21379C211yesmaximal, characteristic
3#20719C211yesmaximal, characteristic
4#8722C212yesmaximal, complemented
5#8721C212yesmaximal, complemented
6#8720C212yesmaximal, complemented
7#8720C212yesmaximal, complemented
8#8036C211yesmaximal, characteristic, complemented
9#8034C211yesmaximal, characteristic, complemented
10#8032C211yesmaximal, characteristic, complemented
11#8031C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2354C412yes—
18#2354C412yes—
19#2354C412yes—
20Q8×C4⋊C4C412yes—
21#2347—22no—
22#2347—21no—
23#2341—21no—
24#2319—22no—
25#2319—21no—
26#2285C2212yes—
27#2285C2211yescharacteristic
28#2283C2212yes—
29#2283C2211yescharacteristic
30#2276—21no—
31#1985—21no—
32#1955—21no—
33#1869—22no—
34#1818—21no—
35#1798—22no—
36#1780—21no—
37#1749C2212yes—
38#1749C2212yes—
39#1749C2212yes—
40#1749C2212yes—
41#1295C2211yescharacteristic
42#1295C2211yescharacteristic
43#1295C2212yes—
44#1295C2212yes—
45#1295C2212yes—
46#1295C2212yes—
47#1295C2212yes—
48#1295C2211yescharacteristic
49#1295C2212yes—
50#1295C2211yescharacteristic
51C22×#41D411yescharacteristic
52C23×C8—21no—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C4×#41—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D411yescharacteristic
64C2×#117C2×C412yes—
65C2×#117C2×C412yes—
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117D412yes—
69C2×#117D411yescharacteristic
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72#432—22no—
73#432—22no—
74#432—22no—
75#432—22no—
76#432—22no—
77#432—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×#97—22no—
89C2×#97—22no—
90C2×#97—21no—
91C2×#97C2312yes—
92C2×#97C2311yescharacteristic
93C2×#97C2311yescharacteristic
94C2×#97C2311yescharacteristic
95C2×#97C2311yescharacteristic
96C2×#97C2312yes—
97#390—22no—
98#390—22no—
99#390—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#384C2×C412yes—
110#384C2×C412yes—
111#384C2×C412yes—
112#383—22no—
113#383—21no—
114#382C2311yescharacteristic
115#382C2312yes—
116#382C2311yescharacteristic
117#381—22no—
118#381—21no—
119#380—22no—
120#380—21no—
121#380—22no—
122#380—21no—
123#380C2×C412yes—
124#380C2×C412yes—
125#380C2×C412yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2×C412yes—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—21no—
138#376—22no—
139#376—21no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—44no—
147C22×Q8C22⋊C411yescharacteristic
148C22×D4C22⋊C412yes—
149C22×D4C22⋊C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—44no—
163C22×C8—42no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—48no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177#117—22no—
178#117—22no—
179#117—24no—
180#117—24no—
181#117—81no—
182#117—81no—
183#117—82no—
184#117—82no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C22×C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42C2411yescharacteristic
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C22⋊C8—44no—
235C22⋊C8—44no—
236C22⋊C8—44no—
237C22⋊C8—44no—
238C22⋊C8—42no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241#97—22no—
242#97—22no—
243#97—21no—
244#97—21no—
245#97—22no—
246#97—21no—
247#97—21no—
248#97—21no—
249#97—21no—
250#97—21no—
251#97—21no—
252#97—22no—
253#97—42no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—21no—
266C24C2×C22⋊C412yes—
267C24C2×C22⋊C411yescharacteristic
268#41—48no—
269#41—48no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—42no—
276C2×D4—42no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4#11712yes—
304C22×C4#11712yes—
305C22×C4#12312yes—
306C22×C4#12312yes—
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—48no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—44no—
358C42—44no—
359C42—44no—
360C42—81no—
361C42—81no—
362C42—82no—
363C42—82no—
364C42#11712yes—
365C42#11712yes—
366C42#12312yes—
367C42#12312yes—
368C42#12312yes—
369C42#12312yes—
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C42C2×#4111yescharacteristic
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23#38112yes—
386C23C22×C22⋊C411yescharacteristic
387Q8—28no—
388D4—48no—
389D4—44no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4#38112yes—
431C2×C4#38112yes—
432C2×C4#38112yes—
433C2×C4#42114yes—
434C2×C4C2×#11711yescharacteristic
435C2×C4C2×#11711yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4C2×#12311yescharacteristic
438C2×C4C2×#12311yescharacteristic
439C2×C4C2×#12311yescharacteristic
440C8—44no—
441C8—81no—
442C8—82no—
443C8—81no—
444C8—81no—
445C8—81no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22C2×#38111yescharacteristic
459C22#288811yescharacteristic
460C22#288911yescharacteristic
461C22C2×#42112yes—
462C22#290112yes—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—21no—
467C4—21no—
468C4—22no—
469C4—24no—
470C4—21no—
471C4—21no—
472C4—22no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—82no—
479C2—42no—
480C2—82no—
481C2#1768312yes—
482C2#1768412yes—
483C2#2069511yescharacteristic
484C2#2070511yescharacteristic
485C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8722C2semidirect
5. #8721C2semidirect
6. #8720C2semidirect
7. #8720C2semidirect
8. #8036C2semidirect
9. #8034C2semidirect
10. #8032C2semidirect
11. #8031C2semidirect

Maximal subgroups

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

subgroupindex
#341932
#213792
#207192
#87222
#87212
#87202
#80362
#80342
#80322
#80312