Finite Group Databasebeta

← #10691259

Subgroups — #10691259

Order 768.

690 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 105 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#74031C211yesmaximal, characteristic
2#74004C211yesmaximal, characteristic, complemented
3#69944C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic, complemented
5#69509C211yesmaximal, characteristic, complemented
6C3×#1289C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9205—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C2×#882—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C3×#380C2211yescharacteristic
15#5212C2211yescharacteristic
16C2×#830C2211yescharacteristic
17#4991—22no—
18#4990—22no—
19#4958—21no—
20#4955—21no—
21Dic3×C22⋊C4—21no—
22#4674C412yescomplemented
23#4671C412yescomplemented
24#4670—22no—
25#4470—22no—
26#4468—22no—
27#4451—22no—
28#2617—31no—
29#2312—31no—
30#1519—31no—
31#1519—31no—
32C2×#323—31no—
33#1289—31no—
34#1289D311yescharacteristic, complemented
35C24×D3D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C22×#67—21no—
38C23×Dic3D411yescharacteristic
39C22×D12—22no—
40C22×D12C2×C411yescharacteristic
41C22×C3×C8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×#217—21no—
47#930—21no—
48#930—21no—
49#930—21no—
50#930—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4C2×C412yes—
61C2×(C2×D3)⋊C4D412yes—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×C4×Dic3—22no—
65C22×C3⋊C8—21no—
66#882—42no—
67#882—42no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95C2×C412yes—
71C3×#95D412yescomplemented
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95D412yescomplemented
75#830—21no—
76#830—21no—
77C2×#121—31no—
78C2×#120—61no—
79C22×C22⋊C4—31no—
80C2×C22⋊C8—31no—
81C2×C22⋊C8C2×D311yescharacteristic
82#397—32no—
83#394—62no—
84#392—32no—
85#388—62no—
86#388—62no—
87#386—61no—
88#380—31no—
89#380C2×D311yescharacteristic
90#380C2×D311yescharacteristic
91#379—61no—
92C4×C22⋊C4—61no—
93#327—62no—
94#327—62no—
95#323—62no—
96C24×C3C2×D411yescharacteristic
97C23×D3—21no—
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—81no—
105C23×D3—81no—
106C23×D3C22⋊C412yes—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—81no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C2×#67—41no—
117C2×#67—41no—
118C2×#67—41no—
119C2×#67—41no—
120C2×#67—41no—
121C2×#67—41no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—22no—
126C22×Dic3—81no—
127C22×Dic3C22⋊C412yes—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—44no—
131C2×D12—82no—
132C2×D12—82no—
133C2×D12—81no—
134C2×C3×C8—41no—
135C2×C3×C8—81no—
136C3×C4⋊C4—81no—
137C3×C4⋊C4—81no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142#217—41no—
143#217—41no—
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—22no—
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—22no—
148(C2×D3)⋊C4—44no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—82no—
152(C2×D3)⋊C4—82no—
153(C2×D3)⋊C4—82no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—81no—
157C4×Dic3—82no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—81no—
160C25—31no—
161C22×D4—31no—
162C22×D4—62no—
163C22×D4—61no—
164C23×C4—31no—
165C23×C4C22×D311yescharacteristic
166C22×C8—21no—
167C22×C8—61no—
168#121—61no—
169#121—61no—
170#121—61no—
171#121—61no—
172#120—122no—
173#120—122no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—32no—
181C2×C22⋊C4—32no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4—62no—
192C2×C42—62no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197#95—32no—
198#95—62no—
199#95—62no—
200#95C4×D312yes—
201#95D1212yescomplemented
202C23×C3—41no—
203C23×C3—41no—
204C23×C3C2×C22⋊C411yescharacteristic
205C23×C3#12011yescharacteristic
206C23×C3#12311yescharacteristic
207C23×C3#13511yescharacteristic
208C23×C3#13511yescharacteristic
209C23×C3AGL(1,ℤ/8)11yescharacteristic
210C23×C3AGL(1,ℤ/8)11yescharacteristic
211C22×D3—22no—
212C22×D3—22no—
213C22×D3—22no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3—42no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—42no—
223C22×D3—42no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—81no—
228C22×D3—81no—
229C22×D3—82no—
230C22×D3—82no—
231C22×D3—82no—
232C22×D3—82no—
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—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249#67—81no—
250#67—81no—
251#67—81no—
252#67—81no—
253C2×Dic3—22no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—81no—
263D12—81no—
264D12—81no—
265D12—88no—
266C3×C8—81no—
267C3⋊C8—81no—
268C24—32no—
269C24—61no—
270C24—61no—
271C24—61no—
272C24—61no—
273C24—61no—
274C24—61no—
275C24—61no—
276C24—241no—
277C24—241no—
278C24D3×D411yescharacteristic
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—124no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—242no—
289C2×D4—242no—
290C2×D4—241no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—32no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—62no—
298C22×C4—81no—
299C22×C4—241no—
300C22×C4C2×C4×D311yescharacteristic
301C22×C4C2×D1211yescharacteristic
302C22×C4#23511yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4#23911yescharacteristic
306C2×C8—41no—
307C2×C8—81no—
308C2×C8—121no—
309C2×C8—241no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4—241no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—62no—
320C22⋊C4—62no—
321C22⋊C4—62no—
322C22⋊C4—62no—
323C22⋊C4—124no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22⋊C4—242no—
329C22⋊C4—242no—
330C22⋊C4—242no—
331C42—242no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3#38611yescharacteristic
343C22×C3#41611yescharacteristic
344C22×C3#41811yescharacteristic
345C22×C3#44911yescharacteristic
346C22×C3#44911yescharacteristic
347C22×C3#48211yescharacteristic
348C22×C3#48211yescharacteristic
349C2×D3—44no—
350C2×D3—42no—
351C2×D3—42no—
352C2×D3—42no—
353C2×D3—82no—
354C2×D3—82no—
355C2×D3—82no—
356C2×D3—82no—
357C2×D3—82no—
358C2×D3—82no—
359C2×D3—82no—
360C2×D3—81no—
361C2×D3—81no—
362C2×D3—81no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—82no—
368C3×C4—82no—
369Dic3—42no—
370Dic3—82no—
371Dic3—81no—
372C23—41no—
373C23—41no—
374C23—62no—
375C23—62no—
376C23—62no—
377C23—122no—
378C23—122no—
379C23—122no—
380C23—122no—
381C23—122no—
382C23—122no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—241no—
391C23—241no—
392C23—242no—
393C23—242no—
394C23—242no—
395C23—242no—
396C23D3×C22⋊C411yescharacteristic
397C23#88211yescharacteristic
398C23#89411yescharacteristic
399C23#91111yescharacteristic
400C23#91411yescharacteristic
401C23#91611yescharacteristic
402C23#91911yescharacteristic
403D4—241no—
404D4—241no—
405D4—248no—
406D4—241no—
407D4—241no—
408D4—241no—
409D4—241no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—62no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—122no—
431C2×C4—241no—
432C2×C4C4×D1211yescharacteristic
433C2×C4#88411yescharacteristic
434C2×C4#89311yescharacteristic
435C2×C4#89511yescharacteristic
436C8—81no—
437C8—241no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3#187511yescharacteristic
445C2×C3#191111yescharacteristic
446C2×C3#191311yescharacteristic
447D3—82no—
448D3—161no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—124no—
460C22—122no—
461C22—122no—
462C22—122no—
463C22—242no—
464C22—242no—
465C22—242no—
466C22—242no—
467C22—242no—
468C22—242no—
469C22—242no—
470C22—241no—
471C22—241no—
472C22—241no—
473C22#466511yescharacteristic
474C22#477011yescharacteristic
475C22#478111yescharacteristic
476C22#480411yescharacteristic
477C22#481011yescharacteristic
478C22#486611yescharacteristic
479C22#488511yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—82no—
485C4—82no—
486C4—122no—
487C4—242no—
488C4—241no—
489C3#920511yescharacteristic, complemented, Sylow
490C2—21no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—242no—
497C2—481no—
498C2#6870511yescharacteristic
499C2#6888111yescharacteristic
500C2#6896411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#386
prime radical#74031C2
cosocleC23×C3×C4C23
hypercenter#1289D3
hypocenterC3#9205
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #74004C2semidirect
3. #69944C2semidirect
4. C2×#4468C2semidirect
5. #69509C2semidirect
6. C3×#1289C2semidirect
7. #67625C2semidirect
22. #4674C4semidirect
23. #4671C4semidirect
34. #1289D3semidirect
71. C3×#95D4semidirect
74. C3⋊#95D4semidirect
201. #95D12semidirect
489. C3#9205semidirect

Maximal subgroups

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

subgroupindex
#740312
#740042
#699442
C2×#44682
#695092
C3×#12892
#676252
#92053