Finite Group Databasebeta

← #101401

Subgroups — #101401

Order 512.

521 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27644C211yesmaximal, characteristic
2#22176C211yesmaximal, characteristic
3#11373C211yesmaximal, characteristic, complemented
4#8723C211yesmaximal, characteristic, complemented
5#8571C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic, complemented
7#7020C211yesmaximal, characteristic, complemented
8C2×#480—21no—
9C2×#435—21no—
10C4×#121C2211yescharacteristic
11C2×#384C2211yescharacteristic
12#2083C412yescomplemented
13#2071C412yescomplemented
14#1947—21no—
15#1924—21no—
16#1917—21no—
17#1870—21no—
18#1869—21no—
19#1851—21no—
20#1823—21no—
21#1802—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25C2×#340—21no—
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#327—21no—
29C4⋊#95C2211yescharacteristic
30#1296—21no—
31#1296—21no—
32#1294—21no—
33C22×M16—21no—
34C23×C8D411yescharacteristic
35C23×C8D411yescharacteristic
36C2×#121D411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4D411yescharacteristic
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C42C2311yescharacteristic
44#480—42no—
45#480—42no—
46#435—42no—
47#435—42no—
48C2×#107—22no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D411yescharacteristic
53C2×C4⋊C8D411yescharacteristic
54C2×#102—22no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×#97—21no—
62#390C2×C411yescharacteristic
63#389C2×C411yescharacteristic
64#388—21no—
65#386—21no—
66#384—21no—
67#384—21no—
68#384—21no—
69#384C2×C412yes—
70#380D411yescharacteristic
71C4×C4⋊C4D411yescharacteristic
72C4×C22⋊C4—21no—
73C2×#95—21no—
74C2×#95—21no—
75#340—22no—
76#340—22no—
77#340—42no—
78#340—42no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336C2×C412yes—
86#327—42no—
87C22×D4—21no—
88C22×D4C22⋊C411yescharacteristic
89C23×C4—21no—
90C23×C4—41no—
91C23×C4—41no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4#4111yescharacteristic
95C2×M16—42no—
96C2×M16—81no—
97C2×M16—81no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8D812yes—
112#121—21no—
113#121—21no—
114#121—21no—
115#121—21no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121C4×D4—41no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—81no—
144C2×C4⋊C4—81no—
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C42—21no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42C22×C411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4111yescharacteristic
160#107—44no—
161C4⋊C8—41no—
162C4⋊C8—41no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—42no—
168#102—44no—
169C22⋊C8—42no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—81no—
177C22⋊C8—81no—
178#97—42no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—41no—
183#95—41no—
184#95—81no—
185C24—21no—
186C24#12111yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4#13511yescharacteristic
238C22×C4AGL(1,ℤ/8)11yescharacteristic
239C22×C4AGL(1,ℤ/8)11yescharacteristic
240C22×C4AGL(1,ℤ/8)11yescharacteristic
241C22×C4AGL(1,ℤ/8)11yescharacteristic
242C22×C4AGL(1,ℤ/8)11yescharacteristic
243C22×C4#13711yescharacteristic
244M16—84no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—81no—
263C2×C8—81no—
264C2×C8—81no—
265C2×C8—81no—
266C2×C8—81no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—82no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C4⋊C4—82no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23#38611yescharacteristic
315C23#45911yescharacteristic
316C23#46311yescharacteristic
317C23#46811yescharacteristic
318C23#46911yescharacteristic
319C23#47111yescharacteristic
320C23#47211yescharacteristic
321D4—42no—
322D4—81no—
323D4—81no—
324D4—81no—
325D4—81no—
326D4—81no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—161no—
373C2×C4—161no—
374C2×C4—161no—
375C2×C4—161no—
376C2×C4—161no—
377C2×C4—161no—
378C2×C4—161no—
379C2×C4—161no—
380C2×C4#41711yescharacteristic
381C2×C4#41811yescharacteristic
382C2×C4C4×D811yescharacteristic
383C2×C4#44211yescharacteristic
384C2×C4#44711yescharacteristic
385C2×C4#44911yescharacteristic
386C2×C4#45511yescharacteristic
387C2×C4#45611yescharacteristic
388C2×C4#45811yescharacteristic
389C2×C4#45811yescharacteristic
390C2×C4#46811yescharacteristic
391C2×C4#46911yescharacteristic
392C8—42no—
393C8—82no—
394C8—81no—
395C8—81no—
396C8—81no—
397C8—161no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#168511yescharacteristic
418C22#168711yescharacteristic
419C22#169711yescharacteristic
420C22#170111yescharacteristic
421C22#187911yescharacteristic
422C22#188211yescharacteristic
423C22#189011yescharacteristic
424C22#189511yescharacteristic
425C22#190211yescharacteristic
426C22#190711yescharacteristic
427C22#190911yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—162no—
440C4—162no—
441C4#160711yescharacteristic
442C4#160911yescharacteristic
443C4#161011yescharacteristic
444C4#161211yescharacteristic
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—161no—
450C2#904311yescharacteristic
451C2#922911yescharacteristic
452C2#929111yescharacteristic
453C2#932011yescharacteristic
454C2#937511yescharacteristic
455C2#947411yescharacteristic
456C2#952211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11373C2semidirect
4. #8723C2semidirect
6. #7039C2semidirect
7. #7020C2semidirect
12. #2083C4semidirect
13. #2071C4semidirect

Maximal subgroups

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

subgroupindex
#276442
#221762
#113732
#87232
#85712
#70392
#70202