Finite Group Databasebeta

← #10698122

Subgroups — #10698122

Order 768.

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

normal subgroups only

This group has 914 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
1C3×#2435C211yesmaximal, characteristic
2#84050C211yesmaximal, characteristic
3#82742C211yesmaximal, characteristic, complemented
4#72129C211yesmaximal, characteristic
5#71688C211yesmaximal, characteristic, complemented
6#69445C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#10333—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×#124—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#936—21no—
13C2×#924C2211yescharacteristic
14C3×#394—22no—
15C3×#390—21no—
16C3×#379—22no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5247—22no—
19#5228—21no—
20#5222—22no—
21#5221—21no—
22C2×#833—22no—
23C4×#217C2211yescharacteristic
24#5207C2211yescharacteristic
25#5207C2211yescharacteristic
26#5052—22no—
27#5050—21no—
28#5043—21no—
29#4971—21no—
30#4538—22no—
31#4452—22no—
32#3195—31no—
33#2435—31no—
34#2435D311yescharacteristic
35#1783—31no—
36#1508—31no—
37#1508—31no—
38#1297—31no—
39C25×C3D412yes—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—21no—
47C22×C3×D4—44no—
48C23×C3×C4C2311yescharacteristic
49C22×Dic6D411yescharacteristic
50C3×#124—44no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C42—21no—
62C2×#217—24no—
63C2×#217—21no—
64C2×#217—21no—
65C2×#217—24no—
66C2×#216—22no—
67#936—44no—
68C2×C4⋊Dic3D411yescharacteristic
69#924—21no—
70#924—21no—
71#924—21no—
72#924—21no—
73C2×Dic3⋊C4—21no—
74C2×C4×Dic3—21no—
75C2×#208—21no—
76C2×#208D412yes—
77C3×#95D412yes—
78#835—22no—
79#833—44no—
80C3⋊#95D412yes—
81#830—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85#810—22no—
86#808—22no—
87C23×D4C2×D311yescharacteristic
88C2×#124—21no—
89C2×#124—61no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92#483—61no—
93#483—61no—
94#470—62no—
95#451—61no—
96#451—61no—
97C2×#100—62no—
98#407—31no—
99#407—31no—
100#394—22no—
101#394—62no—
102#390—21no—
103#390—61no—
104#379—22no—
105#379—62no—
106C4×C22⋊C4—31no—
107C4×C22⋊C4C2×D311yescharacteristic
108#328—62no—
109#328—62no—
110C24×C3—22no—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—44no—
114C24×C3—44no—
115C24×C3—44no—
116C24×C3C2×D411yescharacteristic
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—44no—
127C2×C3×D4—44no—
128C2×C3×D4—44no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—42no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×Dic3—22no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C2×Dic6—21no—
147C2×Dic6—21no—
148C2×Dic6—42no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C22⋊C4—42no—
154C3×C22⋊C4—48no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—84no—
158C3×C42—42no—
159#217—48no—
160#217—42no—
161#217—41no—
162#217—41no—
163#217—84no—
164#216—82no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169C4×Dic3—42no—
170#208—42no—
171#208—42no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—41no—
174C25#6712yes—
175C22×Q8—31no—
176C22×Q8#6711yescharacteristic
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—44no—
184C23×C4C22×D311yescharacteristic
185C2×Q16—62no—
186C2×M16—32no—
187C2×M16—61no—
188#124—44no—
189#124—124no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4#6711yescharacteristic
202C2×C22⋊C4—24no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—64no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—64no—
210C2×C42—21no—
211C2×C42—61no—
212#106—62no—
213#103—62no—
214#103—62no—
215#100—124no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C22⋊C8—61no—
220#95—32no—
221#95#6712yes—
222C23×C3—22no—
223C23×C3—22no—
224C23×C3—24no—
225C23×C3—24no—
226C23×C3—24no—
227C23×C3—41no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3—44no—
232C23×C3—44no—
233C23×C3—42no—
234C23×C3—44no—
235C23×C3—44no—
236C23×C3—82no—
237C23×C3—82no—
238C23×C3—816no—
239C23×C3#12011yescharacteristic
240C23×C3#12011yescharacteristic
241C23×C3#12111yescharacteristic
242C3×Q8—42no—
243C3×Q8—81no—
244C3×D4—44no—
245C3×D4—44no—
246C3×D4—44no—
247C3×D4—44no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12711yescharacteristic
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272Dic6—42no—
273Dic6—81no—
274C3⋊C8—42no—
275C3⋊C8—81no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—44no—
280C24—44no—
281C24—44no—
282C24C2×#6711yescharacteristic
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—122no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—62no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4D3×D411yescharacteristic
310C22×C4D3×D411yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4#23711yescharacteristic
313C22×C4C2×#6711yescharacteristic
314C22×C4C2×#6711yescharacteristic
315Q16—242no—
316M16—122no—
317M16—122no—
318C2×C8—62no—
319C2×C8—121no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—121no—
327C4⋊C4—121no—
328C22⋊C4—48no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—84no—
333C22⋊C4—128no—
334C22⋊C4—122no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—244no—
338C42—42no—
339C42—122no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—48no—
349C22×C3—48no—
350C22×C3—44no—
351C22×C3—84no—
352C22×C3—84no—
353C22×C3—84no—
354C22×C3—84no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3#39211yescharacteristic
358C22×C3#45512yes—
359C22×C3#45512yes—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—82no—
364C3×C4—81no—
365C3×C4—81no—
366C3×C4B(4,𝔽2)12yes—
367Dic3—82no—
368Dic3—81no—
369Dic3—81no—
370C23—22no—
371C23—22no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—44no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23—82no—
385C23—82no—
386C23—816no—
387C23#93711yescharacteristic
388C23#93911yescharacteristic
389C23#95311yescharacteristic
390Q8—42no—
391Q8—81no—
392Q8—122no—
393Q8—241no—
394D4—44no—
395D4—44no—
396D4—44no—
397D4—44no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4#93711yescharacteristic
419C2×C4#93811yescharacteristic
420C2×C4#93911yescharacteristic
421C2×C4#94011yescharacteristic
422C8—122no—
423C8—241no—
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—41no—
427C2×C3—84no—
428C2×C3#200411yescharacteristic
429C2×C3#200411yescharacteristic
430C2×C3#202111yescharacteristic
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—44no—
439C22—48no—
440C22—48no—
441C22—44no—
442C22—84no—
443C22—84no—
444C22—84no—
445C22—84no—
446C22—82no—
447C22—82no—
448C22#506112yes—
449C22#522311yescharacteristic
450C22#524212yes—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—82no—
455C4—81no—
456C4—81no—
457C4—242no—
458C4—241no—
459C4—241no—
460C4#503212yes—
461C3#1033311yescharacteristic, complemented, Sylow
462C2—22no—
463C2—22no—
464C2—41no—
465C2—84no—
466C2#7167111yescharacteristic
467C2#7173111yescharacteristic
468C2#7185711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2435C2
socleC22×C3#392
prime radicalC23×C3×D4C22
hypercenter#2435D3
hypocenterC3#10333
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
3. #82742C2semidirect
5. #71688C2semidirect
6. #69445C2semidirect
7. #67618C2semidirect
461. C3#10333semidirect

Maximal subgroups

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

subgroupindex
C3×#24352
#840502
#827422
#721292
#716882
#694452
#676182
#103333