Finite Group Databasebeta

← #219600

Subgroups — #219600

Order 512.

513 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22236C211yesmaximal, characteristic
2#22229C211yesmaximal, characteristic, complemented
3#19657C211yesmaximal, characteristic
4#9775C211yesmaximal, characteristic, complemented
5#9758C211yesmaximal, characteristic
6#9301C211yesmaximal, characteristic
7#9300C211yesmaximal, characteristic
8#9298C211yesmaximal, characteristic, complemented
9#9296C211yesmaximal, characteristic, complemented
10#9292C211yesmaximal, characteristic, complemented
11#9291C211yesmaximal, characteristic
12#9289C211yesmaximal, characteristic
13#9288C211yesmaximal, characteristic, complemented
14#8793C211yesmaximal, characteristic, complemented
15#8141C211yesmaximal, characteristic, complemented
16#3230C2211yescharacteristic
17#3229C2211yescharacteristic
18#3226C2211yescharacteristic
19#3206C2211yescharacteristic
20#3131C2211yescharacteristic
21C2×#487C2211yescharacteristic
22#2961C2211yescharacteristic
23#2932C2211yescharacteristic
24#2331C2211yescharacteristic
25#2327C2211yescharacteristic
26#1995C2211yescharacteristic
27#1993C2211yescharacteristic
28#1963C2211yescharacteristic
29#1960C2211yescharacteristic
30#1929—21no—
31#1929—21no—
32#1928—21no—
33#1927—21no—
34#1866—21no—
35#1866—21no—
36#1865—21no—
37#1863—21no—
38#1848C2211yescharacteristic
39#1846C2211yescharacteristic
40#1746C2211yescharacteristic
41#1719C412yescomplemented
42#1718C412yescomplemented
43#1710C412yescomplemented
44#1708C412yescomplemented
45#1549C2211yescharacteristic
46#1548C2211yescharacteristic
47#1546C2211yescharacteristic
48#1546C2211yescharacteristic
49#1518C2211yescharacteristic
50#1518C2211yescharacteristic
51#1517C2211yescharacteristic
52#1517C2211yescharacteristic
53#1452C2211yescharacteristic
54#1450C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1286C2211yescharacteristic
58#1286C2211yescharacteristic
59#1275C2211yescharacteristic
60#1275C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C2×#131—21no—
64#533C2311yescharacteristic
65#532C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4×Q8—21no—
69#500C2×C412yes—
70#500C2×C412yes—
71#498C2×C412yes—
72#492C2×C412yes—
73#488—21no—
74#488—21no—
75#487—21no—
76#487—21no—
77#487C2×C412yes—
78#487C2×C412yes—
79#487C2×C411yescharacteristic
80#487C2×C411yescharacteristic
81#487C2×C412yes—
82#487C2×C412yes—
83#487C2×C412yes—
84#486C2×C411yescharacteristic
85#486C2×C411yescharacteristic
86#486C2×C412yes—
87#484—41no—
88#483—41no—
89#480—41no—
90#477—41no—
91#475—41no—
92#474—41no—
93#461—41no—
94#448—41no—
95#446—41no—
96#446—41no—
97#435—41no—
98#435—41no—
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103C2×#108—21no—
104C2×#108—21no—
105C2×#107—21no—
106C2×#106—21no—
107#424—21no—
108#423C2311yescharacteristic
109#423C2311yescharacteristic
110C2×#104—21no—
111C2×#104—21no—
112#416—21no—
113#416—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#103—21no—
118#408—21no—
119#405—21no—
120C2×#97C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—21no—
123#391—41no—
124#389—21no—
125#387—41no—
126#387—41no—
127#387—41no—
128#382C2311yescharacteristic
129#382C2311yescharacteristic
130C4×C4⋊C4—21no—
131#376C2311yescharacteristic
132#340—21no—
133#340—21no—
134#339—21no—
135#339—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138#336D411yescharacteristic
139#336D411yescharacteristic
140#333—41no—
141#332—41no—
142#332—41no—
143#332—41no—
144#328—41no—
145#328—41no—
146#327—41no—
147#327—41no—
148#322—21no—
149#322—21no—
150#322C2×C412yes—
151C82C2×C412yes—
152C2×#41—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C2×SD16—42no—
156#131—81no—
157#131—81no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—41no—
161C22×C8—41no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C22×C411yescharacteristic
175C4⋊Q8C22×C411yescharacteristic
176C4⋊Q8C22×C411yescharacteristic
177#127C22×C411yescharacteristic
178#125—21no—
179#124—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182#121—21no—
183#121—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—81no—
187C4×Q8—81no—
188C4×D4—21no—
189C4×D4—21no—
190#117—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2411yescharacteristic
204#108—81no—
205#108—81no—
206#107—42no—
207#107—42no—
208#107—81no—
209#106—42no—
210#106—81no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—42no—
220#104—81no—
221#104—81no—
222#103—22no—
223#103—22no—
224#103—22no—
225#103—22no—
226#103—21no—
227#103—21no—
228#103—21no—
229#103—21no—
230#103—42no—
231#103—42no—
232#103—42no—
233#103—81no—
234#103—81no—
235#102—21no—
236#102—21no—
237#102—21no—
238#102—21no—
239#102—42no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—42no—
245#97—21no—
246#97—21no—
247#97—41no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—41no—
251C4×C8C22×C412yes—
252C4×C8C22×C412yes—
253C4×C8C22×C411yescharacteristic
254C4×C8C22×C411yescharacteristic
255C4×C8#4111yescharacteristic
256C4×C8#4111yescharacteristic
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—41no—
264#95—42no—
265#95—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—41no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—161no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—41no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4C22×D411yescharacteristic
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C4H(5,𝔽2)11yescharacteristic
296SD16—162no—
297M16—41no—
298M16—41no—
299M16—41no—
300M16—41no—
301M16—82no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—82no—
319C2×C8C4×D411yescharacteristic
320C2×C8C4×D411yescharacteristic
321C2×C8C4×D411yescharacteristic
322C2×C8C4×D411yescharacteristic
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—82no—
344C4⋊C4—161no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—41no—
353C42—82no—
354C42—161no—
355C42C23×C411yescharacteristic
356C42C2×#4111yescharacteristic
357C42H(5,𝔽2)11yescharacteristic
358C42H(5,𝔽2)11yescharacteristic
359C23—41no—
360C23#53511yescharacteristic
361Q8—42no—
362Q8—41no—
363Q8—42no—
364Q8—41no—
365Q8—82no—
366Q8—81no—
367Q8—81no—
368Q8—81no—
369Q8—81no—
370Q8—161no—
371D4—41no—
372D4—41no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4C2×C4×D411yescharacteristic
403C2×C4#51811yescharacteristic
404C2×C4#51811yescharacteristic
405C2×C4#53511yescharacteristic
406C2×C4#54011yescharacteristic
407C2×C4#54011yescharacteristic
408C2×C4#57611yescharacteristic
409C2×C4#57711yescharacteristic
410C2×C4#57711yescharacteristic
411C2×C4#57811yescharacteristic
412C8—21no—
413C8—21no—
414C8—21no—
415C8—21no—
416C8—42no—
417C8—42no—
418C8—41no—
419C8—41no—
420C8—81no—
421C8—81no—
422C8—81no—
423C8—81no—
424C22—21no—
425C22—21no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#232811yescharacteristic
430C22#322411yescharacteristic
431C22#322511yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—82no—
440C4—82no—
441C4—81no—
442C4—81no—
443C4—161no—
444C4—161no—
445C4#295111yescharacteristic
446C4#295211yescharacteristic
447C4#322411yescharacteristic
448C4#322511yescharacteristic
449C2—21no—
450C2—21no—
451C2—161no—
452C2#2227411yescharacteristic
453C2#2228411yescharacteristic
454C2#2228511yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#532C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22229C2semidirect
4. #9775C2semidirect
8. #9298C2semidirect
9. #9296C2semidirect
10. #9292C2semidirect
13. #9288C2semidirect
14. #8793C2semidirect
15. #8141C2semidirect
41. #1719C4semidirect
42. #1718C4semidirect
43. #1710C4semidirect
44. #1708C4semidirect

Maximal subgroups

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

subgroupindex
#222362
#222292
#196572
#97752
#97582
#93012
#93002
#92982
#92962
#92922
#92912
#92892
#92882
#87932
#81412