Finite Group Databasebeta

← #134430

Subgroups — #134430

Order 512.

630 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 106 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#14312C211yesmaximal, characteristic
2C2×#1345C212yesmaximal
3#10847C211yesmaximal, characteristic, complemented
4#8558C211yesmaximal, characteristic, complemented
5#7251C212yesmaximal, complemented
6C2×#423—21no—
7C2×#394C2211yescharacteristic
8C2×#390C411yescharacteristic
9C2×#388—21no—
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797—21no—
12C2×#343—21no—
13C2×#343C2212yes—
14C2×#324C2212yes—
15#1444C411yescharacteristic
16#1441C2211yescharacteristic
17#1439—21no—
18#1439—21no—
19#1438C411yescharacteristic
20#1437—21no—
21#1434C411yescharacteristic
22#1345—24no—
23#1280—21no—
24C23×D4C2×C411yescharacteristic
25C22×M16—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4C2311yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C42C2×C411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42D411yescharacteristic
39#423—24no—
40#423—42no—
41C2×C4⋊C8—41no—
42C2×C4⋊C8—41no—
43C2×C4⋊C8—41no—
44C2×C22⋊C8—22no—
45#394—21no—
46#394—21no—
47#394D412yes—
48#394Q812yes—
49#390—41no—
50#390—41no—
51#388—42no—
52#388—42no—
53C4×C4⋊C4—41no—
54C4×C4⋊C4—41no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C411yescharacteristic
63C2×#95D411yescharacteristic
64#343—22no—
65#343—22no—
66#343—42no—
67#336—41no—
68#324—22no—
69#324—22no—
70C25—21no—
71C22×D4—21no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—81no—
75C22×D4—81no—
76C22×D4C4⋊C412yes—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C2×M16—24no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C2×M16—42no—
93C22×C8—42no—
94C22×C8—42no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4—81no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4C2×D411yescharacteristic
107C2×C4⋊C4C2×Q811yescharacteristic
108C2×C4⋊C4#4111yescharacteristic
109C2×C4⋊C4#4111yescharacteristic
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—81no—
126C2×C22⋊C4—81no—
127C2×C22⋊C4—81no—
128C2×C22⋊C4—81no—
129C2×C22⋊C4—81no—
130C2×C22⋊C4—81no—
131C2×C22⋊C4—81no—
132C2×C22⋊C4—81no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—22no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—81no—
141C2×C42—81no—
142C2×C42D812yes—
143C2×C42SD1612yes—
144C4⋊C8—44no—
145C4⋊C8—44no—
146C4⋊C8—82no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149#95—21no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—42no—
154#95—41no—
155#95—41no—
156#95—81no—
157#95—81no—
158#95—81no—
159#95C4⋊C412yes—
160C24—22no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24—41no—
174C24—41no—
175C24C2×C22⋊C411yescharacteristic
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—82no—
179C2×D4—82no—
180C2×D4—82no—
181C2×D4—84no—
182C2×D4—82no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
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—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4#9911yescharacteristic
233C22×C4#9911yescharacteristic
234C22×C4#10212yes—
235C22×C4#10212yes—
236C22×C4#10412yes—
237C22×C4C2×C4⋊C411yescharacteristic
238C22×C4#11711yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C2×D811yescharacteristic
244C22×C4C2×SD1611yescharacteristic
245C22×C4AGL(1,ℤ/8)11yescharacteristic
246C22×C4#13711yescharacteristic
247M16—44no—
248M16—44no—
249M16—82no—
250M16—82no—
251C2×C8—44no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C2×C8—82no—
256C2×C8—82no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C4⋊C4—84no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C22⋊C4—84no—
272C22⋊C4—84no—
273C22⋊C4—84no—
274C22⋊C4—82no—
275C22⋊C4—82no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—42no—
281C42—81no—
282C42—81no—
283C42—81no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23#38011yescharacteristic
324C23C2×#9911yescharacteristic
325C23#41011yescharacteristic
326C23C2×#10211yescharacteristic
327C23#41711yescharacteristic
328C23C2×#10411yescharacteristic
329C23#42111yescharacteristic
330D4—84no—
331D4—84no—
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—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
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#32712yes—
401C2×C4#32712yes—
402C2×C4#45911yescharacteristic
403C2×C4#46111yescharacteristic
404C2×C4#47411yescharacteristic
405C2×C4#47611yescharacteristic
406C8—84no—
407C8—162no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22C2×#32711yescharacteristic
436C22#150911yescharacteristic
437C22#151311yescharacteristic
438C22#151711yescharacteristic
439C22#179911yescharacteristic
440C22#180211yescharacteristic
441C22#180611yescharacteristic
442C22#212111yescharacteristic
443C22#212211yescharacteristic
444C22#213311yescharacteristic
445C22#213411yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—82no—
458C4—81no—
459C4—81no—
460C4—161no—
461C4—161no—
462C2—21no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2—162no—
467C2#857111yescharacteristic
468C2#1169811yescharacteristic
469C2#1169911yescharacteristic
470C2#1183211yescharacteristic
471C2#1183311yescharacteristic
472C2#1279511yescharacteristic
473C2#1280311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10847C2semidirect
4. #8558C2semidirect
5. #7251C2semidirect

Maximal subgroups

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

subgroupindex
#143122
C2×#13452
#108472
#85582
#72512