Finite Group Databasebeta

← #10692840

Subgroups — #10692840

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#78620C211yesmaximal, characteristic, complemented
2#73515C211yesmaximal, characteristic
3#69978C211yesmaximal, characteristic
4#69534C211yesmaximal, characteristic, complemented
5#69531C211yesmaximal, characteristic
6C3×#1284C211yesmaximal, characteristic, complemented
7#67616C211yesmaximal, characteristic, complemented
8#9446—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#381C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5211C2211yescharacteristic
15#5207C2211yescharacteristic
16#5118—21no—
17Dic3×M16—21no—
18#4961—21no—
19#4961—21no—
20#4939C411yescharacteristic, complemented
21#4939C411yescharacteristic, complemented
22#4936C411yescharacteristic, complemented
23#4935C411yescharacteristic, complemented
24#4720—22no—
25#4720—22no—
26#4670—21no—
27#4670—21no—
28#4527—21no—
29#4527—21no—
30#4470—22no—
31#4451—22no—
32#2921—31no—
33#2500—31no—
34#1820—31no—
35#1510—31no—
36#1510—31no—
37#1284—31no—
38#1284D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×D12C2×C411yescharacteristic
42C2×C3×M16—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C42C2×C411yescharacteristic
46C2×C3×C42C2×C411yescharacteristic
47C2×#217—21no—
48C2×#217D411yescharacteristic
49C2×#217D411yescharacteristic
50#930—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59#926—21no—
60#926—21no—
61#926—21no—
62#926—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3C2×C411yescharacteristic
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×#121—31no—
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⋊C4C2×D311yescharacteristic
104#342—61no—
105#342—61no—
106#327—62no—
107#327—62no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—41no—
122C2×#67—41no—
123C2×#67—81no—
124C2×#67—81no—
125C22×Dic3—21no—
126C22×Dic3—41no—
127C22×Dic3C22⋊C411yescharacteristic
128C22×Dic3C22⋊C411yescharacteristic
129C22×Dic3C22⋊C411yescharacteristic
130C22×Dic3C22⋊C411yescharacteristic
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—41no—
134C2×D12—41no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C3×C22⋊C4—42no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—41no—
142C3×C42—41no—
143#217—21no—
144#217—21no—
145#217—21no—
146#217—21no—
147#217—41no—
148#217—41no—
149(C2×D3)⋊C4—42no—
150(C2×D3)⋊C4—42no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—81no—
154(C2×D3)⋊C4—81no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—81no—
162C4⋊Dic3—81no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167#208—41no—
168#208—41no—
169C2×C3⋊C8—22no—
170C22×D4—31no—
171C22×D4—61no—
172C23×C4C22×D311yescharacteristic
173C2×M16—21no—
174C2×M16—61no—
175#121—61no—
176#121—61no—
177#121—61no—
178#121—61no—
179#117—61no—
180#117—61no—
181#117—61no—
182#117—61no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
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×C42C4×D311yescharacteristic
201C2×C42C4×D311yescharacteristic
202C4⋊C8—64no—
203C22⋊C8—21no—
204C22⋊C8—32no—
205C22⋊C8—61no—
206C22⋊C8D1212yes—
207#97—62no—
208C4×C8—62no—
209#95—21no—
210#95—21no—
211#95—21no—
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×C3C2×C22⋊C411yescharacteristic
221C23×C3#12111yescharacteristic
222C23×C3#12111yescharacteristic
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
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—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—81no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12311yescharacteristic
251#67—82no—
252#67—82no—
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—81no—
265C2×Dic3—81no—
266C2×Dic3—81no—
267D12—81no—
268D12—81no—
269C3×C8—42no—
270C3⋊C8—42no—
271C24—61no—
272C24D3×D411yescharacteristic
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—241no—
280C2×D4—241no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—61no—
291C22×C4—121no—
292C22×C4C2×C4×D311yescharacteristic
293C22×C4C2×D1211yescharacteristic
294C22×C4#23511yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4#23911yescharacteristic
298M16—41no—
299M16—41no—
300M16—121no—
301M16—121no—
302C2×C8—22no—
303C2×C8—62no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—241no—
311C4⋊C4—241no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—61no—
316C22⋊C4—61no—
317C22⋊C4—61no—
318C22⋊C4—61no—
319C22⋊C4—122no—
320C22⋊C4—122no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C22⋊C4—241no—
326C22⋊C4—241no—
327C42—41no—
328C42—41no—
329C42—61no—
330C42—61no—
331C42—61no—
332C42—61no—
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#38611yescharacteristic
344C22×C3#41011yescharacteristic
345C22×C3#41011yescharacteristic
346C2×D3—81no—
347C2×D3—81no—
348C2×D3—81no—
349C2×D3—81no—
350C2×D3—81no—
351C2×D3—81no—
352C2×D3—81no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—42no—
358C3×C4—81no—
359C3×C4—161no—
360Dic3—41no—
361Dic3—41no—
362Dic3—41no—
363Dic3—41no—
364Dic3—81no—
365Dic3—161no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—121no—
371C23—121no—
372C23—121no—
373C23—121no—
374C23—121no—
375C23—121no—
376C23—121no—
377C23D3×C22⋊C411yescharacteristic
378C23#88411yescharacteristic
379C23#89511yescharacteristic
380D4—241no—
381D4—241no—
382D4—242no—
383D4—242no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
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×C4C4×D1211yescharacteristic
416C2×C4#88211yescharacteristic
417C2×C4#89311yescharacteristic
418C2×C4#89411yescharacteristic
419C8—42no—
420C8—122no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3#188411yescharacteristic
426C2×C3#189911yescharacteristic
427C2×C3#191011yescharacteristic
428D3—161no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—81no—
438C22—81no—
439C22—241no—
440C22—241no—
441C22—241no—
442C22—241no—
443C22—241no—
444C22—241no—
445C22—241no—
446C22#466511yescharacteristic
447C22#474011yescharacteristic
448C22#474011yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—42no—
454C4—81no—
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—481no—
468C2#6871311yescharacteristic
469C2#6877711yescharacteristic
470C2#6895411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9446
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
1. #78620C2semidirect
4. #69534C2semidirect
6. C3×#1284C2semidirect
7. #67616C2semidirect
20. #4939C4semidirect
21. #4939C4semidirect
22. #4936C4semidirect
23. #4935C4semidirect
38. #1284D3semidirect
462. C3#9446semidirect

Maximal subgroups

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

subgroupindex
#786202
#735152
#699782
#695342
#695312
C3×#12842
#676162
#94463