Finite Group Databasebeta

← #508356

Subgroups — #508356

Order 512.

595 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2220C211yesmaximal, characteristic
2C2×#2150C211yesmaximal, characteristic, complemented
3#33130C211yesmaximal, characteristic, complemented
4#33129C211yesmaximal, characteristic
5#13515C211yesmaximal, characteristic, complemented
6#13514C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13476C211yesmaximal, characteristic, complemented
9#13475C211yesmaximal, characteristic, complemented
10#13474C211yesmaximal, characteristic, complemented
11#13449C211yesmaximal, characteristic, complemented
12#13449C211yesmaximal, characteristic, complemented
13#13062C211yesmaximal, characteristic, complemented
14#13061C211yesmaximal, characteristic
15#12836C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C22×M32C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2855—21no—
23#2854—21no—
24#2220—24no—
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28#2150—24no—
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1969C2211yescharacteristic
42#1969C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1934—21no—
48#1934—21no—
49#1933—21no—
50#1933—21no—
51#1893—21no—
52#1892—21no—
53#1808C2211yescharacteristic
54#1808C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8D411yescharacteristic
64C23×C8C2311yescharacteristic
65C2×#123—41no—
66C2×C22⋊Q8—21no—
67C2×#121—21no—
68C22×C42D411yescharacteristic
69C2×SD32—22no—
70C2×SD32—41no—
71C2×SD32—41no—
72C2×M32—22no—
73C2×M32—41no—
74C2×M32—41no—
75C2×M32—41no—
76C2×M32—41no—
77C2×M32D414yes—
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#467—22no—
83#467—22no—
84#466—22no—
85#466—22no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8D411yescharacteristic
101C2×C4×C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#391—21no—
105#391—21no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#388—41no—
113#388—41no—
114#387—41no—
115#387—41no—
116#386—41no—
117#385—41no—
118#381—41no—
119#380—21no—
120#380—21no—
121#367—41no—
122#366—22no—
123#366—41no—
124C4⋊C16—22no—
125C4⋊C16—22no—
126C4⋊C16—22no—
127C4⋊C16—22no—
128#358—22no—
129#358—41no—
130#358—41no—
131#357—22no—
132#357—41no—
133#357—41no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4—41no—
143C23×C4D811yescharacteristic
144C23×C4D811yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D412yes—
169C22×C8#4112yes—
170C22×C8C2411yescharacteristic
171#123—84no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174#121—42no—
175#121—42no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42D811yescharacteristic
198C2×C42D811yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201SD32—44no—
202M32—44no—
203M32—44no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16C2×D412yes—
215C2×C16C2×D412yes—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#103—42no—
229#103—42no—
230#103—42no—
231#102—42no—
232#102—42no—
233#102—42no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250C24—41no—
251C24C2×D811yescharacteristic
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4#12112yes—
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4#13512yes—
291C22×C4C22×D411yescharacteristic
292Q16—41no—
293Q16—41no—
294Q16—41no—
295Q16—41no—
296D8—41no—
297D8—41no—
298D8—41no—
299D8—41no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—44no—
313C2×C8#12112yes—
314C2×C8C22×D411yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8H(5,𝔽2)11yescharacteristic
317C2×C8H(5,𝔽2)11yescharacteristic
318C2×C82−1+411yescharacteristic
319C2×C82−1+411yescharacteristic
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—22no—
350C42—22no—
351C16—22no—
352C16—22no—
353C16—41no—
354C16—41no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23#46612yes—
369C23C22×D811yescharacteristic
370Q8—81no—
371Q8—81no—
372Q8—81no—
373Q8—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—164no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4#46612yes—
419C2×C4#50912yes—
420C2×C4#51012yes—
421C2×C4C2×#12111yescharacteristic
422C2×C4#53711yescharacteristic
423C2×C4#53711yescharacteristic
424C2×C4C2×#13511yescharacteristic
425C2×C4#57511yescharacteristic
426C2×C4#57511yescharacteristic
427C8—22no—
428C8—41no—
429C8—41no—
430C8#53411yescharacteristic
431C8#53611yescharacteristic
432C8#53911yescharacteristic
433C8#53911yescharacteristic
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22C2×#46611yescharacteristic
449C22#324511yescharacteristic
450C22#324511yescharacteristic
451C22C2×#50911yescharacteristic
452C22C2×#51011yescharacteristic
453C22#341411yescharacteristic
454C22#341411yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4#261811yescharacteristic
468C4#305411yescharacteristic
469C4#325011yescharacteristic
470C4#325011yescharacteristic
471C2—42no—
472C2—321no—
473C2#2977011yescharacteristic
474C2#3360511yescharacteristic
475C2#3360611yescharacteristic
476C2#3381711yescharacteristic
477C2#3381811yescharacteristic
478C2#3382011yescharacteristic
479C2#3382211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2150C2semidirect
3. #33130C2semidirect
5. #13515C2semidirect
6. #13514C2semidirect
7. #13477C2semidirect
8. #13476C2semidirect
9. #13475C2semidirect
10. #13474C2semidirect
11. #13449C2semidirect
12. #13449C2semidirect
13. #13062C2semidirect
15. #12836C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22202
C2×#21502
#331302
#331292
#135152
#135142
#134772
#134762
#134752
#134742
#134492
#130622
#130612
#128362