Finite Group Databasebeta

← #10691644

Subgroups — #10691644

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5033C211yesmaximal, characteristic
2#73456C211yesmaximal, characteristic, complemented
3C3×#1964C211yesmaximal, characteristic
4#71675C211yesmaximal, characteristic
5#69501C211yesmaximal, characteristic, complemented
6#67625C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9260—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#877—21no—
12C2×C3×#99—21no—
13C2×C3×#99—21no—
14C3×#380C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16C2×#832C412yescomplemented
17#5212C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#830C2211yescharacteristic
20#5033—21no—
21#5033—21no—
22#5033—22no—
23C2×#807C412yescomplemented
24#4943—21no—
25C4×C4⋊Dic3—21no—
26#4928—21no—
27#4653—22no—
28#4537—22no—
29#4529—22no—
30C2×#480—31no—
31#2344—31no—
32#1964D311yescharacteristic
33#1823—31no—
34#1519—31no—
35#1289—31no—
36#1282—31no—
37C22×C3×D4—21no—
38C22×C3×D4C2×C411yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C3×#121—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×#121—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C22⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×#217—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3C2×C412yes—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D412yes—
61C2×Dic3⋊C4—22no—
62C2×C4×Dic3—22no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8C2×C412yes—
65#877—42no—
66#877—42no—
67C3×#99—42no—
68C3×#99—42no—
69C3×#95—21no—
70C3×#95—21no—
71#832—22no—
72#832—22no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95D412yes—
76#830—21no—
77#830—21no—
78#830—21no—
79#830—21no—
80#807—22no—
81#807—22no—
82C2×#121C2×D311yescharacteristic
83C2×#117—61no—
84C22×C4⋊C4—31no—
85#480—62no—
86#480—61no—
87#480—61no—
88C2×#107—32no—
89C2×#102—32no—
90C2×#99—21no—
91C2×#99—21no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8—31no—
94#389—61no—
95#387—62no—
96#381—61no—
97#380—31no—
98#380C2×D311yescharacteristic
99C4×C4⋊C4—61no—
100C4×C22⋊C4C2×D311yescharacteristic
101#340—62no—
102#327—62no—
103C24×C3—21no—
104C24×C3C2×D411yescharacteristic
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—42no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—42no—
118C22×C3×C4—41no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×Dic3—22no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3—41no—
131C22×Dic3—41no—
132C22×Dic3—41no—
133C22×Dic3—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—42no—
137C3×C22⋊C4—21no—
138C3×C22⋊C4—21no—
139C3×C22⋊C4—21no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4D812yes—
145C3×C42—42no—
146#217—41no—
147#217—41no—
148C4⋊Dic3—22no—
149C4⋊Dic3—22no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—42no—
153C4⋊Dic3C22⋊C412yes—
154C4⋊Dic3C22⋊C412yes—
155Dic3⋊C4—44no—
156C4×Dic3—44no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—42no—
159C2×C3⋊C8—42no—
160C2×C3⋊C8—41no—
161C22×D4—21no—
162C22×D4C4×D311yescharacteristic
163C23×C4—61no—
164C23×C4C22×D311yescharacteristic
165C22×C8—32no—
166C22×C8—61no—
167#121—21no—
168#121—21no—
169#121—21no—
170#121—21no—
171#117—122no—
172#117—122no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4C4×D311yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4#6711yescharacteristic
188C2×C22⋊C4#6711yescharacteristic
189C2×C42—21no—
190C2×C42—62no—
191#107—62no—
192#107—62no—
193#102—62no—
194#102—62no—
195#99—42no—
196#99—42no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201#95—21no—
202#95—21no—
203#95—32no—
204#95—62no—
205#95—62no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—81no—
212C23×C3—81no—
213C23×C3C4×D411yescharacteristic
214C23×C3#12011yescharacteristic
215C23×C3#12111yescharacteristic
216C23×C3C2×D811yescharacteristic
217C23×C3#13511yescharacteristic
218C23×C3AGL(1,ℤ/8)11yescharacteristic
219C23×C3AGL(1,ℤ/8)11yescharacteristic
220C3×D4—42no—
221C3×D4—81no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C3×D4—81no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12311yescharacteristic
248C2×Dic3—22no—
249C2×Dic3—22no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C3⋊C8—42no—
262C3⋊C8—81no—
263C24—21no—
264C24C2×#6711yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—62no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—121no—
285C22×C4—121no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4#23511yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4#23711yescharacteristic
294C2×C8—62no—
295C2×C8—122no—
296C2×C8—122no—
297C2×C8—121no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—32no—
301C4⋊C4—32no—
302C4⋊C4—42no—
303C4⋊C4—62no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—62no—
307C4⋊C4—124no—
308C4⋊C4—122no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4#21312yes—
319C42—42no—
320C42—124no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—161no—
335C22×C3—161no—
336C22×C3#38611yescharacteristic
337C22×C3C4×D811yescharacteristic
338C22×C3#44211yescharacteristic
339C22×C3#44711yescharacteristic
340C22×C3#44911yescharacteristic
341C22×C3#45911yescharacteristic
342C22×C3#46311yescharacteristic
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—42no—
347C3×C4—81no—
348C3×C4—81no—
349C3×C4—161no—
350C3×C4#41012yes—
351Dic3—42no—
352Dic3—42no—
353Dic3—82no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—81no—
360C23—81no—
361C23C4×#6711yescharacteristic
362C23C2×#21311yescharacteristic
363C23#93211yescharacteristic
364C23#93711yescharacteristic
365C23#93811yescharacteristic
366C23#94911yescharacteristic
367C23#95011yescharacteristic
368D4—42no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—81no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
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—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—122no—
404C2×C4—122no—
405C2×C4D3×C22⋊C411yescharacteristic
406C2×C4#67⋊C411yescharacteristic
407C2×C4#88311yescharacteristic
408C2×C4#88411yescharacteristic
409C8—122no—
410C8—241no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—161no—
418C2×C3#187911yescharacteristic
419C2×C3#189911yescharacteristic
420C2×C3#190511yescharacteristic
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—161no—
435C22—161no—
436C22#495611yescharacteristic
437C22C4×#21311yescharacteristic
438C22#501411yescharacteristic
439C22#503611yescharacteristic
440C22#503711yescharacteristic
441C22#505711yescharacteristic
442C22#507211yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—42no—
447C4—81no—
448C4—81no—
449C4—122no—
450C4—122no—
451C4—161no—
452C4—242no—
453C4#474012yes—
454C3#926011yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—161no—
462C2#6965711yescharacteristic
463C2#7002911yescharacteristic
464C2#7005211yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1964C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1964D3
hypocenterC3#9260
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #73456C2semidirect
5. #69501C2semidirect
6. #67625C2semidirect
7. #67617C2semidirect
16. C2×#832C4semidirect
23. C2×#807C4semidirect
454. C3#9260semidirect

Maximal subgroups

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

subgroupindex
C2×#50332
#734562
C3×#19642
#716752
#695012
#676252
#676172
#92603