Finite Group Databasebeta

← #10698062

Subgroups — #10698062

Order 768.

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

normal subgroups only

This group has 664 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
1C3×#2854C211yesmaximal, characteristic
2#82630C211yesmaximal, characteristic
3#82075C211yesmaximal, characteristic, complemented
4#71675C211yesmaximal, characteristic
5#71531C211yesmaximal, characteristic, complemented
6#69510C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#10323—31nomaximal, Sylow
9C2×C3×#123—21no—
10C2×C3×#121C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C2×#924C2211yescharacteristic
13C3×#390—21no—
14C3×#389—21no—
15C3×#388—22no—
16C3×#386—21no—
17C3×#380C2211yescharacteristic
18#5218—21no—
19#5212C2211yescharacteristic
20#5211—21no—
21#5210—21no—
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5041—22no—
25#5040—21no—
26#5034—21no—
27#4971—21no—
28#4933—22no—
29#4929—21no—
30#4928—21no—
31#4537—22no—
32#4524—22no—
33#4452—22no—
34#3187—31no—
35#2855—31no—
36#2854D311yescharacteristic
37#1817—31no—
38#1520—31no—
39#1519—31no—
40#1289—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C22×Dic6D411yescharacteristic
47C3×#123—42no—
48C3×#123—42no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—21no—
61C2×C3×C42—21no—
62C2×C3×C42—21no—
63C2×C3×C42—21no—
64C2×C3×C42—21no—
65C2×C3×C42—21no—
66C2×#217—21no—
67C2×#217—21no—
68C2×#214—22no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3D411yescharacteristic
71#924—21no—
72#924—21no—
73#924—21no—
74#924—21no—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—21no—
77C22×C3⋊C8—21no—
78C22×C3⋊C8D412yes—
79C3×#95—22no—
80C3×#95D412yes—
81#832—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85C3⋊#95D412yes—
86#830—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#810—22no—
91#808—22no—
92C2×#123—21no—
93C2×C22⋊Q8—31no—
94C2×#121C2×D311yescharacteristic
95C22×C42C2×D311yescharacteristic
96#483—61no—
97#481—61no—
98#465—62no—
99#450—61no—
100#448—61no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8—31no—
103#391—61no—
104#390—21no—
105#389—21no—
106#389—61no—
107#388—22no—
108#387—62no—
109#386—21no—
110#385—61no—
111#381—61no—
112#380—31no—
113#380C2×D311yescharacteristic
114#336—62no—
115#328—62no—
116#327—62no—
117C24×C3—21no—
118C24×C3C2×D411yescharacteristic
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4SD1612yescomplemented
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×Dic3—21no—
149C22×Dic3—22no—
150C22×Dic3—21no—
151C2×Dic6—21no—
152C2×Dic6—21no—
153C2×Dic6—42no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—42no—
157C3×C4⋊C4—44no—
158C3×C22⋊C4—44no—
159C3×C22⋊C4—44no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C42—41no—
165C3×C42—41no—
166C3×C42—41no—
167C3×C42—41no—
168#217—41no—
169#217—41no—
170#217—41no—
171#217—41no—
172#214—82no—
173C4⋊Dic3—21no—
174C4⋊Dic3—21no—
175C4⋊Dic3—42no—
176Dic3⋊C4—44no—
177Dic3⋊C4—41no—
178Dic3⋊C4—41no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—42no—
181C2×C3⋊C8—42no—
182C2×C3⋊C8—41no—
183C22×Q8—31no—
184C22×D4—21no—
185C22×D4#6711yescharacteristic
186C23×C4#6711yescharacteristic
187C23×C4#6711yescharacteristic
188C23×C4C22×D311yescharacteristic
189C2×SD16—62no—
190C22×C8—32no—
191C22×C8—61no—
192#123—42no—
193#123—42no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198#121—21no—
199#121—21no—
200#121—21no—
201#121—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—62no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4#6711yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221#106—62no—
222#103—62no—
223#102—62no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228#95—22no—
229#95—32no—
230#95—62no—
231#95—62no—
232#95—62no—
233#95#6712yes—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3#12011yescharacteristic
242C23×C3#12011yescharacteristic
243C23×C3#12111yescharacteristic
244C23×C3C2×SD1611yescharacteristic
245C23×C3#13511yescharacteristic
246C23×C3AGL(1,ℤ/8)11yescharacteristic
247C23×C3#13711yescharacteristic
248C3×D4—42no—
249C3×D4—81no—
250C3×D4—81no—
251C3×D4—81no—
252C3×D4—81no—
253C3×D4—81no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12711yescharacteristic
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—42no—
291Dic6—42no—
292Dic6—81no—
293C3⋊C8—42no—
294C3⋊C8—81no—
295C24—21no—
296C24C2×#6711yescharacteristic
297C2×Q8—61no—
298C2×Q8—61no—
299C2×Q8—122no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—62no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4#21412yescomplemented
326C22×C4D3×D411yescharacteristic
327C22×C4D3×D411yescharacteristic
328C22×C4D3×D411yescharacteristic
329C22×C4#23711yescharacteristic
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332SD16—242no—
333C2×C8—62no—
334C2×C8—122no—
335C2×C8—122no—
336C2×C8—121no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—61no—
342C4⋊C4—61no—
343C4⋊C4—124no—
344C4⋊C4—122no—
345C4⋊C4—121no—
346C4⋊C4—121no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—121no—
354C22⋊C4—121no—
355C22⋊C4—121no—
356C22⋊C4—121no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—82no—
372C22×C3—82no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3#39211yescharacteristic
377C22×C3#44811yescharacteristic
378C22×C3#44911yescharacteristic
379C22×C3#45011yescharacteristic
380C22×C3#45211yescharacteristic
381C22×C3#46511yescharacteristic
382C22×C3#46811yescharacteristic
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—42no—
387C3×C4—42no—
388C3×C4—82no—
389C3×C4—81no—
390C3×C4#45812yes—
391Dic3—82no—
392Dic3—81no—
393Dic3—81no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23#93211yescharacteristic
402C23C2×#21411yescharacteristic
403C23#93711yescharacteristic
404C23#93911yescharacteristic
405C23#95011yescharacteristic
406C23#95111yescharacteristic
407C23#95311yescharacteristic
408Q8—122no—
409Q8—241no—
410D4—42no—
411D4—81no—
412D4—81no—
413D4—81no—
414D4—81no—
415D4—81no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—122no—
449C2×C4#93711yescharacteristic
450C2×C4#93811yescharacteristic
451C2×C4#93911yescharacteristic
452C2×C4#94011yescharacteristic
453C8—122no—
454C8—241no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—161no—
462C2×C3#200011yescharacteristic
463C2×C3#201011yescharacteristic
464C2×C3#202211yescharacteristic
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—82no—
476C22—82no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22#503711yescharacteristic
481C22#504011yescharacteristic
482C22#504111yescharacteristic
483C22#504211yescharacteristic
484C22#521811yescharacteristic
485C22#522311yescharacteristic
486C22#523911yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—82no—
493C4—81no—
494C4—242no—
495C4—241no—
496C4—241no—
497C4#505312yes—
498C3#1032311yescharacteristic, complemented, Sylow
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—161no—
506C2#7168511yescharacteristic
507C2#7170611yescharacteristic
508C2#7171511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2854C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#2854D3
hypocenterC3#10323
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2854

Complements

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

NH ≅ G/Ntype
3. #82075C2semidirect
5. #71531C2semidirect
6. #69510C2semidirect
7. #67625C2semidirect
141. C22×C3×C4SD16semidirect
325. C22×C4#214semidirect
498. C3#10323semidirect

Maximal subgroups

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

subgroupindex
C3×#28542
#826302
#820752
#716752
#715312
#695102
#676252
#103233