Finite Group Databasebeta

← #10692838

Subgroups — #10692838

Order 768.

516 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 89 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#82663C211yesmaximal, characteristic
2#78620C211yesmaximal, characteristic, complemented
3C3×#1510C211yesmaximal, characteristic
4#71674C211yesmaximal, characteristic
5#69534C211yesmaximal, characteristic, complemented
6#67682C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9446—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#381C2211yescharacteristic
13#5222—21no—
14#5222—21no—
15#5221C411yescharacteristic, complemented
16#5221C411yescharacteristic, complemented
17#5211C2211yescharacteristic
18C4×#217C2211yescharacteristic
19#5207C2211yescharacteristic
20#5118—21no—
21Dic3×M16—21no—
22#4977C411yescharacteristic, complemented
23#4975C411yescharacteristic, complemented
24#4944—21no—
25#4944—21no—
26#4720—22no—
27#4720—22no—
28C3×#327—22no—
29#4537—22no—
30#4527—21no—
31#4527—21no—
32#2921—31no—
33#2500—31no—
34#1820—31no—
35#1510—31no—
36#1510D311yescharacteristic
37#1284—31no—
38#1284—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C2×C3×M16—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×#121—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217D411yescharacteristic
57C2×#217D411yescharacteristic
58#926—21no—
59#926—21no—
60#926—21no—
61#926—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3C2×C411yescharacteristic
67C2×C4×Dic3C2×C411yescharacteristic
68C2×C4×Dic3D411yescharacteristic
69C2×C4×Dic3D411yescharacteristic
70C2×#208—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#95—21no—
74C3×#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830D412yes—
82#815—22no—
83Dic3⋊C8—24no—
84C8×Dic3—22no—
85C2×#121C2×D311yescharacteristic
86C2×#117—31no—
87#433—62no—
88#433—62no—
89#423—61no—
90#407—31no—
91#407C2×D311yescharacteristic
92C4×M16—61no—
93#394—61no—
94#394—61no—
95#390—31no—
96#390—31no—
97#388—61no—
98#388—61no—
99#384—31no—
100#382—31no—
101#381—31no—
102#381C2×D311yescharacteristic
103C4×C22⋊C4—31no—
104#342—61no—
105#342—61no—
106#327—22no—
107#327—62no—
108C24×C3—21no—
109C24×C3C2×D411yescharacteristic
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—81no—
117C2×C3×D4—81no—
118C22×C3×C4—21no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C22×C411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—41no—
131C22×Dic3C22⋊C411yescharacteristic
132C22×Dic3C22⋊C411yescharacteristic
133C22×Dic3C22⋊C411yescharacteristic
134C22×Dic3C22⋊C411yescharacteristic
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144#217—21no—
145#217—21no—
146#217—21no—
147#217—21no—
148#217—42no—
149#217—42no—
150#217—42no—
151#217—41no—
152#217—41no—
153#217—81no—
154#217—81no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159Dic3⋊C4—81no—
160Dic3⋊C4—81no—
161C4×Dic3—21no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3—41no—
166C4×Dic3—41no—
167#208—41no—
168#208—41no—
169C2×C3⋊C8—22no—
170C22×D4—21no—
171C22×D4C4×D311yescharacteristic
172C23×C4C22×D311yescharacteristic
173C2×M16—21no—
174C2×M16—61no—
175#121—21no—
176#121—21no—
177#121—21no—
178#121—21no—
179#117—61no—
180#117—61no—
181#117—61no—
182#117—61no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4C4×D311yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—31no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C42—31no—
199C2×C42—31no—
200C2×C42—31no—
201C2×C42—31no—
202C4⋊C8—64no—
203C22⋊C8—21no—
204C22⋊C8—32no—
205C22⋊C8—61no—
206C22⋊C8#6712yes—
207#97—62no—
208C4×C8—62no—
209#95—21no—
210#95—21no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—61no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3C2×C22⋊C411yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12111yescharacteristic
230C3×D4—81no—
231C3×D4—81no—
232C3×D4—82no—
233C3×D4—82no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12311yescharacteristic
252C2×Dic3—22no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—81no—
267C2×Dic3—81no—
268C2×Dic3—81no—
269C3×C8—42no—
270C3⋊C8—42no—
271C24—21no—
272C24D3×D411yescharacteristic
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—81no—
280C2×D4—81no—
281C22×C4—21no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—121no—
292C22×C4C2×C4×D311yescharacteristic
293C22×C4#23511yescharacteristic
294C22×C4D3×D411yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4#23711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298M16—41no—
299M16—41no—
300M16—121no—
301M16—121no—
302C2×C8—22no—
303C2×C8—62no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—241no—
311C4⋊C4—241no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—61no—
317C22⋊C4—61no—
318C22⋊C4—61no—
319C22⋊C4—61no—
320C22⋊C4—122no—
321C22⋊C4—122no—
322C22⋊C4—122no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C22⋊C4—241no—
326C22⋊C4—241no—
327C42—61no—
328C42—61no—
329C42—61no—
330C42—61no—
331C42—121no—
332C42—121no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3#38611yescharacteristic
351C22×C3#41011yescharacteristic
352C22×C3#41011yescharacteristic
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—81no—
358C3×C4—161no—
359Dic3—42no—
360Dic3—41no—
361Dic3—41no—
362Dic3—41no—
363Dic3—41no—
364Dic3—81no—
365Dic3—161no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23D3×C22⋊C411yescharacteristic
378C23#88411yescharacteristic
379C23#93811yescharacteristic
380D4—81no—
381D4—81no—
382D4—82no—
383D4—82no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—62no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—241no—
413C2×C4—241no—
414C2×C4—241no—
415C2×C4#67⋊C411yescharacteristic
416C2×C4#88311yescharacteristic
417C2×C4C4×#6711yescharacteristic
418C2×C4#93711yescharacteristic
419C8—42no—
420C8—122no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—161no—
426C2×C3#188411yescharacteristic
427C2×C3#189911yescharacteristic
428C2×C3#191011yescharacteristic
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22#474011yescharacteristic
447C22#474011yescharacteristic
448C22#495611yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—81no—
454C4—122no—
455C4—121no—
456C4—121no—
457C4—121no—
458C4—121no—
459C4—161no—
460C4—241no—
461C4—481no—
462C3#944611yescharacteristic, complemented, Sylow
463C2—21no—
464C2—21no—
465C2—41no—
466C2—41no—
467C2—161no—
468C2#7002911yescharacteristic
469C2#7033011yescharacteristic
470C2#7041711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1510C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1510D3
hypocenterC3#9446
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
2. #78620C2semidirect
5. #69534C2semidirect
6. #67682C2semidirect
7. #67628C2semidirect
15. #5221C4semidirect
16. #5221C4semidirect
22. #4977C4semidirect
23. #4975C4semidirect
462. C3#9446semidirect

Maximal subgroups

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

subgroupindex
#826632
#786202
C3×#15102
#716742
#695342
#676822
#676282
#94463