Finite Group Databasebeta

← #10692221

Subgroups — #10692221

Order 768.

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

normal subgroups only

This group has 576 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#78641C211yesmaximal, characteristic
2#74014C211yesmaximal, characteristic, complemented
3#69978C211yesmaximal, characteristic, complemented
4#69977C211yesmaximal, characteristic, complemented
5#69528C211yesmaximal, characteristic, complemented
6C3×#1305C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9343—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5211C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5135—22no—
17#4990—21no—
18#4989—21no—
19#4957—21no—
20#4956—21no—
21C2×#806—21no—
22C2×#806—21no—
23#4735—21no—
24#4735C412yescomplemented
25#4732—21no—
26#4732C412yescomplemented
27#4665—21no—
28#4653—21no—
29#4470—22no—
30#4470—21no—
31#4470—21no—
32#3192—31no—
33#2377—31no—
34#1964—31no—
35#1510—31no—
36#1507—31no—
37#1305D311yescharacteristic, complemented
38#1284—31no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×#67—21no—
42C22×D12C2×C411yescharacteristic, complemented
43C22×D12D411yescharacteristic
44C22×C4×D3—21no—
45C2×C3×M16—21no—
46C2×C3×M16—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C22⋊C4D411yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×#217—21no—
51C2×#217—21no—
52#930—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56#930—21no—
57#930—21no—
58#930C2×C412yes—
59#930D412yes—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×C4⋊Dic3C2×C411yescharacteristic, complemented
66C2×C4⋊Dic3D411yescharacteristic
67C2×Dic3⋊C4—21no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C2×D24—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8C2×C412yescomplemented
75C3×#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D411yescharacteristic
81C3⋊#95D411yescharacteristic
82#821—22no—
83#821—22no—
84#817—22no—
85#806—42no—
86#806—42no—
87C2×#121—31no—
88C2×#121—31no—
89#481—32no—
90#481—61no—
91#469—62no—
92#447—32no—
93#447—61no—
94C2×#99—61no—
95C2×#99—61no—
96#407C2×D311yescharacteristic
97#407C2×D311yescharacteristic
98#390—61no—
99#388—61no—
100#387—61no—
101#386—61no—
102#386—61no—
103#385—61no—
104#381—31no—
105#380C2×D311yescharacteristic
106C4×C22⋊C4—31no—
107#327—61no—
108#327—61no—
109#327—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C23×D3—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—81no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—22no—
123C2×#67—22no—
124C2×#67—41no—
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—81no—
131C2×#67—81no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—41no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—81no—
141C2×D12—81no—
142C2×D12C22⋊C412yes—
143C2×C4×D3—81no—
144C2×C4×D3—81no—
145C2×C4×D3—81no—
146C3×M16—81no—
147C3×M16—81no—
148C2×C3×C8—22no—
149C2×C3×C8—41no—
150C3×C4⋊C4—81no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153#217—22no—
154#217—41no—
155#217—81no—
156(C2×D3)⋊C4—22no—
157(C2×D3)⋊C4—22no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—81no—
161(C2×D3)⋊C4—81no—
162(C2×D3)⋊C4—81no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3C22⋊C412yes—
167Dic3⋊C4—81no—
168C4×Dic3—81no—
169C4×Dic3—81no—
170D24—82no—
171C22×D4—31no—
172C22×D4—31no—
173C22×D4—61no—
174C22×D4—61no—
175C23×C4—61no—
176C23×C4C22×D311yescharacteristic
177C2×D8—62no—
178C2×M16—21no—
179C2×M16—21no—
180#121—32no—
181#121—32no—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186#121—61no—
187#121—61no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—61no—
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×C22⋊C4—61no—
199C2×C22⋊C4D1211yescharacteristic
200C2×C22⋊C4D1211yescharacteristic
201C2×C42—61no—
202C2×C42—61no—
203#106—62no—
204#102—62no—
205#102—62no—
206#99—122no—
207#99—122no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8C4×D312yescomplemented
212#95—21no—
213#95—31no—
214#95—31no—
215#95—61no—
216#95—61no—
217#95—61no—
218#95—61no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C4×D411yescharacteristic
224C23×C3#12011yescharacteristic
225C23×C3#12111yescharacteristic
226C22×D3—22no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—81no—
237C22×D3—81no—
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—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4C2×C22⋊C411yescharacteristic
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12311yescharacteristic
254#67—82no—
255#67—82no—
256#67—81no—
257#67—81no—
258#67—81no—
259#67—81no—
260C2×Dic3—22no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275D12—42no—
276D12—42no—
277D12—81no—
278D12—81no—
279C4×D3—84no—
280C3×C8—42no—
281C3×C8—81no—
282C24—61no—
283C24—61no—
284C24C2×D1211yescharacteristic
285C2×D4—32no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—121no—
292C2×D4—121no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4—121no—
296C2×D4—121no—
297C2×D4—241no—
298C2×D4—241no—
299C2×D4—241no—
300C2×D4—241no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—81no—
308C22×C4—121no—
309C22×C4—241no—
310C22×C4—241no—
311C22×C4—241no—
312C22×C4C2×C4×D311yescharacteristic
313C22×C4#23511yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4#23911yescharacteristic
318D8—242no—
319M16—81no—
320M16—81no—
321C2×C8—22no—
322C2×C8—41no—
323C4⋊C4—32no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—81no—
328C4⋊C4—241no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—241no—
338C22⋊C4—241no—
339C22⋊C4—241no—
340C22⋊C4—241no—
341C42—241no—
342C42—241no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3#38611yescharacteristic
354C22×C3#45311yescharacteristic
355C22×C3#45411yescharacteristic
356C2×D3—42no—
357C2×D3—42no—
358C2×D3—42no—
359C2×D3—81no—
360C2×D3—81no—
361C2×D3—81no—
362C2×D3—81no—
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—161no—
367C2×D3—161no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—81no—
373C3×C4—81no—
374Dic3—42no—
375Dic3—81no—
376Dic3—81no—
377Dic3—82no—
378Dic3—81no—
379C23—21no—
380C23—21no—
381C23—41no—
382C23—41no—
383C23—62no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23—241no—
394C23—241no—
395C23C4×D1211yescharacteristic
396C23#88211yescharacteristic
397C23#89511yescharacteristic
398D4—122no—
399D4—122no—
400D4—241no—
401D4—241no—
402D4—242no—
403D4—242no—
404D4—241no—
405D4—241no—
406D4—241no—
407D4—241no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—62no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—122no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—244no—
434C2×C4—241no—
435C2×C4—241no—
436C2×C4D3×C22⋊C411yescharacteristic
437C2×C4#88411yescharacteristic
438C2×C4#89311yescharacteristic
439C2×C4#89411yescharacteristic
440C8—42no—
441C8—81no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3#188311yescharacteristic
447C2×C3#190411yescharacteristic
448C2×C3#191311yescharacteristic
449D3—82no—
450D3—161no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—81no—
460C22—81no—
461C22—122no—
462C22—122no—
463C22—122no—
464C22—241no—
465C22—241no—
466C22—241no—
467C22—241no—
468C22—241no—
469C22—241no—
470C22—241no—
471C22—481no—
472C22—481no—
473C22#466511yescharacteristic
474C22#474711yescharacteristic
475C22#475311yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—81no—
481C4—81no—
482C4—122no—
483C4—241no—
484C4—241no—
485C4—242no—
486C4—241no—
487C3#934311yescharacteristic, complemented, Sylow
488C2—21no—
489C2—21no—
490C2—41no—
491C2—41no—
492C2—242no—
493C2—481no—
494C2#6896411yescharacteristic
495C2#6900511yescharacteristic
496C2#6902711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1305C2
socleC22×C3#386
prime radical#78641C2
cosocleC23×C3×C4C23
hypercenter#1305D3
hypocenterC3#9343
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
2. #74014C2semidirect
3. #69978C2semidirect
4. #69977C2semidirect
5. #69528C2semidirect
6. C3×#1305C2semidirect
7. #67682C2semidirect
24. #4735C4semidirect
26. #4732C4semidirect
37. #1305D3semidirect
42. C22×D12C2×C4semidirect
65. C2×C4⋊Dic3C2×C4semidirect
74. C3×C22⋊C8C2×C4semidirect
211. C22⋊C8C4×D3semidirect
487. C3#9343semidirect

Maximal subgroups

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

subgroupindex
#786412
#740142
#699782
#699772
#695282
C3×#13052
#676822
#93433