Finite Group Databasebeta

← #10690198

Subgroups — #10690198

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5222C211yesmaximal, characteristic
2#73867C211yesmaximal, characteristic, complemented
3C3×#1786C211yesmaximal, characteristic
4#72128C211yesmaximal, characteristic
5C2×#4469C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9018—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×#217—21no—
11C22×#217—21no—
12C22×Dic3⋊C4C2211yescharacteristic
13C2×C3×#102—22no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16#5222—22no—
17#5222C414yescomplemented
18C2×#832—22no—
19#5212C2211yescharacteristic
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22#4978—22no—
23#4943—21no—
24Dic3×C22⋊C4—21no—
25#4655—22no—
26#4469—22no—
27C2×#394—31no—
28#2313—31no—
29#1786—31no—
30#1786D311yescharacteristic
31C2×#324—31no—
32#1294—31no—
33#1294—31no—
34C25×C3—21no—
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—44no—
41C22×C3×D4C2×C412yes—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C23×Dic3D411yescharacteristic
46C22×C3×C8—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×#217—24no—
51C2×#217—24no—
52C2×#217—42no—
53C2×#217—42no—
54C2×#217—44no—
55C2×C4⋊Dic3D412yes—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4C2×C412yes—
61C2×Dic3⋊C4D412yes—
62C2×C4×Dic3—22no—
63C22×C3⋊C8—21no—
64C3×#102—44no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#95—21no—
68#832—44no—
69C3⋊#95—22no—
70C3⋊#95—24no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95C2×C412yes—
74#830—21no—
75#830—21no—
76C23×D4C2×D311yescharacteristic
77C22×C4⋊C4—31no—
78C22×C22⋊C4—61no—
79C22×C22⋊C4—61no—
80C2×#102—22no—
81C2×#102—62no—
82C2×C22⋊C8—31no—
83C2×C22⋊C8C2×D311yescharacteristic
84#394—34no—
85#394—62no—
86#387—62no—
87#387—62no—
88#381—61no—
89#380—31no—
90#380C2×D311yescharacteristic
91C4×C22⋊C4—61no—
92C2×#95—31no—
93#324—62no—
94C24×C3—24no—
95C24×C3—22no—
96C24×C3—42no—
97C24×C3—42no—
98C24×C3—42no—
99C24×C3—41no—
100C24×C3—41no—
101C24×C3—41no—
102C24×C3—41no—
103C24×C3—41no—
104C24×C3—41no—
105C24×C3C2×D411yescharacteristic
106C2×C3×D4—24no—
107C2×C3×D4—24no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—42no—
111C2×C3×D4—42no—
112C2×C3×D4—44no—
113C2×C3×D4—44no—
114C2×C3×D4—44no—
115C2×C3×D4—44no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C22×C3×C4—22no—
119C22×C3×C4—42no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—42no—
137C22×Dic3—42no—
138C22×Dic3—42no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3C22⋊C412yes—
142C22×Dic3C22⋊C412yes—
143C2×C3×C8—42no—
144C2×C3×C8—41no—
145C3×C4⋊C4—42no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148#217—48no—
149#217—48no—
150#217—41no—
151#217—41no—
152C4⋊Dic3—42no—
153Dic3⋊C4—22no—
154Dic3⋊C4—22no—
155Dic3⋊C4—22no—
156Dic3⋊C4—22no—
157C4×Dic3—82no—
158C2×C3⋊C8—42no—
159C2×C3⋊C8—41no—
160C25—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—44no—
167C22×D4C4×D312yes—
168C23×C4—31no—
169C23×C4—31no—
170C23×C4—61no—
171C23×C4C22×D311yescharacteristic
172C22×C8—21no—
173C22×C8—61no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4#6712yes—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—64no—
184C2×C22⋊C4—64no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—122no—
187C2×C22⋊C4—122no—
188C2×C22⋊C4—124no—
189C2×C42—62no—
190#102—44no—
191#102—124no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196#95—21no—
197#95—32no—
198#95—62no—
199#95—64no—
200#95—61no—
201#95—61no—
202C23×C3—22no—
203C23×C3—22no—
204C23×C3—24no—
205C23×C3—42no—
206C23×C3—44no—
207C23×C3—44no—
208C23×C3—44no—
209C23×C3—44no—
210C23×C3—44no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—44no—
217C23×C3—82no—
218C23×C3—82no—
219C23×C3—82no—
220C23×C3—82no—
221C23×C3—82no—
222C23×C3—81no—
223C23×C3—81no—
224C23×C3—82no—
225C23×C3—81no—
226C23×C3—81no—
227C23×C3C2×C22⋊C411yescharacteristic
228C23×C3#12011yescharacteristic
229C23×C3#12311yescharacteristic
230C23×C3#13511yescharacteristic
231C23×C3#13511yescharacteristic
232C23×C3AGL(1,ℤ/8)11yescharacteristic
233C23×C3AGL(1,ℤ/8)11yescharacteristic
234C3×D4—24no—
235C3×D4—24no—
236C3×D4—44no—
237C3×D4—44no—
238C3×D4—42no—
239C3×D4—42no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—81no—
270C2×Dic3—82no—
271C2×Dic3—82no—
272C3×C8—81no—
273C3⋊C8—81no—
274C24—24no—
275C24—22no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24D3×D411yescharacteristic
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—41no—
297C2×D4—41no—
298C22×C4—22no—
299C22×C4—32no—
300C22×C4—32no—
301C22×C4—42no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—122no—
313C22×C4—122no—
314C22×C4—122no—
315C22×C4—121no—
316C22×C4—121no—
317C22×C4C2×C4×D311yescharacteristic
318C22×C4#23511yescharacteristic
319C22×C4D3×D411yescharacteristic
320C22×C4D3×D411yescharacteristic
321C22×C4#23711yescharacteristic
322C22×C4C2×#6711yescharacteristic
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—122no—
326C2×C8—121no—
327C4⋊C4—42no—
328C4⋊C4—62no—
329C4⋊C4—62no—
330C4⋊C4—62no—
331C4⋊C4—62no—
332C4⋊C4—122no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—128no—
336C22⋊C4—128no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C42—242no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—44no—
351C22×C3—82no—
352C22×C3—82no—
353C22×C3—84no—
354C22×C3—84no—
355C22×C3—84no—
356C22×C3—84no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3#38611yescharacteristic
360C22×C3#41611yescharacteristic
361C22×C3#41811yescharacteristic
362C22×C3#44911yescharacteristic
363C22×C3#44911yescharacteristic
364C22×C3#45412yes—
365C22×C3B(4,𝔽2)12yes—
366C22×C3#48211yescharacteristic
367C22×C3#48211yescharacteristic
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—82no—
373Dic3—42no—
374Dic3—42no—
375Dic3—82no—
376Dic3—82no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—42no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—81no—
398C23—81no—
399C23—82no—
400C23—81no—
401C23—81no—
402C23D3×C22⋊C411yescharacteristic
403C23#88311yescharacteristic
404C23#91111yescharacteristic
405C23#91211yescharacteristic
406C23#91411yescharacteristic
407C23#91611yescharacteristic
408C23#93711yescharacteristic
409D4—24no—
410D4—24no—
411D4—44no—
412D4—44no—
413D4—42no—
414D4—42no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—122no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—241no—
441C2×C4—242no—
442C2×C4—242no—
443C2×C4#67⋊C411yescharacteristic
444C2×C4#88411yescharacteristic
445C2×C4C4×#6711yescharacteristic
446C2×C4#93811yescharacteristic
447C8—81no—
448C8—241no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—84no—
454C2×C3#159712yes—
455C2×C3#159712yes—
456C2×C3#187511yescharacteristic
457C2×C3#188811yescharacteristic
458C2×C3#188911yescharacteristic
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—82no—
471C22—82no—
472C22—84no—
473C22—84no—
474C22—82no—
475C22—84no—
476C22—84no—
477C22—82no—
478C22#477011yescharacteristic
479C22#477211yescharacteristic
480C22#477911yescharacteristic
481C22#478411yescharacteristic
482C22#495611yescharacteristic
483C22#503212yes—
484C22#515811yescharacteristic
485C22#517211yescharacteristic
486C22#520112yes—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—82no—
492C4—122no—
493C4—122no—
494C4—242no—
495C4—242no—
496C3#901811yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—21no—
501C2—84no—
502C2#6949911yescharacteristic
503C2#7020111yescharacteristic
504C2#7032511yescharacteristic
505C2#7097012yes—
506C2#7097812yes—
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1786C2
socleC23×C3#120
prime radicalC23×C3×D4C22
cosocleC23×C3×C4C23
hypercenter#1786D3
hypocenterC3#9018
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. #73867C2semidirect
5. C2×#4469C2semidirect
6. #67675C2semidirect
7. #67624C2semidirect
17. #5222C4semidirect
496. C3#9018semidirect

Maximal subgroups

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

subgroupindex
C2×#52222
#738672
C3×#17862
#721282
C2×#44692
#676752
#676242
#90183