Finite Group Databasebeta

← #10695745

Subgroups — #10695745

Order 768.

804 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×C3×#451C211yesmaximal, characteristic
2C2×#5228C211yesmaximal, characteristic
3#82789C211yesmaximal, characteristic
4#72134C211yesmaximal, characteristic
5C2×#4538C211yesmaximal, characteristic
6C2×#4467C211yesmaximal, characteristic
7#67674C211yesmaximal, characteristic
8#9967—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×Q16—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×#216—21no—
13C2×#924C2211yescharacteristic
14C3×#451—22no—
15C3×#451—22no—
16C2×C3×#103—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5230—21no—
19#5228—22no—
20#5228—22no—
21C2×#835—21no—
22C2×C3⋊#95—22no—
23C2×C3⋊#95C2211yescharacteristic
24C2×C3⋊#95C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#4983—21no—
27#4981—21no—
28#4979—21no—
29#4942—22no—
30#4538—22no—
31#4467—22no—
32C2×#451—31no—
33C2×#451D311yescharacteristic
34#2659—31no—
35C2×#328—31no—
36C2×#328—31no—
37#1446—31no—
38#1280—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—44no—
45C22×C3×Q8D411yescharacteristic
46C22×C3×Q8D412yes—
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C23×Dic3—21no—
50C23×Dic3D412yes—
51C22×Dic6D411yescharacteristic
52C2×C3×Q16—24no—
53C2×C3×Q16—44no—
54C22×C3×C8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C22⋊C4—21no—
62C2×#217—21no—
63C2×#216—24no—
64C2×#216—44no—
65C2×C4⋊Dic3D411yescharacteristic
66#924—21no—
67#924—21no—
68#924—21no—
69#924—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C22×C3⋊C8—21no—
74C3×#103—24no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D412yes—
77#835—24no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95—44no—
85#830—21no—
86#830D412yes—
87C23×Q8C2×D311yescharacteristic
88C22×Q16—21no—
89C22×Q16—61no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92#451—22no—
93#451—22no—
94#451—62no—
95#451—62no—
96C2×#103—21no—
97C2×#103—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8C2×D311yescharacteristic
100#398—61no—
101#398—61no—
102#395—61no—
103#395—61no—
104#380—62no—
105C2×#95—31no—
106C2×#95—31no—
107C2×#95—62no—
108#328—62no—
109#328—62no—
110C24×C3C2×D411yescharacteristic
111C2×C3×Q8—24no—
112C2×C3×Q8—24no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—44no—
118C2×C3×Q8—42no—
119C2×C3×Q8—42no—
120C2×C3×Q8—44no—
121C2×C3×Q8—42no—
122C2×C3×Q8—44no—
123C2×C3×Q8—44no—
124C2×C3×Q8—41no—
125C2×C3×Q8—41no—
126C22×C3×C4—24no—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—42no—
146C22×Dic3—42no—
147C22×Dic3—42no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—42no—
151C22×Dic3Q1614yescomplemented
152C2×Dic6—21no—
153C2×Dic6—21no—
154C2×Dic6—42no—
155C3×Q16—48no—
156C2×C3×C8—22no—
157C2×C3×C8—42no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164#217—41no—
165#217—41no—
166#216—48no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171Dic3⋊C4—82no—
172Dic3⋊C4—82no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—42no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—31no—
181C22×Q8—44no—
182C22×Q8#6711yescharacteristic
183C22×Q8#6712yes—
184C23×C4—21no—
185C23×C4—32no—
186C23×C4—61no—
187C23×C4C22×D311yescharacteristic
188C2×Q16—24no—
189C2×Q16—44no—
190C2×Q16—64no—
191C2×Q16—124no—
192C22×C8—21no—
193C22×C8—61no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200C22⋊Q8—61no—
201C22⋊Q8—61no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4—62no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—61no—
210#103—24no—
211#103—64no—
212C22⋊C8—21no—
213C22⋊C8—32no—
214C22⋊C8—61no—
215C22⋊C8#6712yes—
216#95—62no—
217#95—62no—
218#95—61no—
219#95—61no—
220#95—61no—
221#95—61no—
222#95—124no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—22no—
226C23×C3#12011yescharacteristic
227C23×C3#12111yescharacteristic
228C23×C3#12711yescharacteristic
229C23×C3C2×Q1612yes—
230C23×C3#13712yes—
231C3×Q8—24no—
232C3×Q8—24no—
233C3×Q8—42no—
234C3×Q8—42no—
235C3×Q8—44no—
236C3×Q8—44no—
237C3×Q8—42no—
238C3×Q8—81no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—24no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—44no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—44no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×Dic3—24no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—44no—
263C2×Dic3—44no—
264C2×Dic3—42no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—42no—
270C2×Dic3—82no—
271C2×Dic3—82no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274Dic6—42no—
275Dic6—81no—
276C3×C8—42no—
277C3⋊C8—42no—
278C24D3×D411yescharacteristic
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—21no—
285C2×Q8—44no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—44no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—61no—
295C2×Q8—61no—
296C2×Q8—122no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—34no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—122no—
312C22×C4—122no—
313C22×C4—122no—
314C22×C4—121no—
315C22×C4—122no—
316C22×C4—121no—
317C22×C4D3×D411yescharacteristic
318C22×C4D3×D411yescharacteristic
319C22×C4#23711yescharacteristic
320C22×C4C2×#6711yescharacteristic
321C22×C4C2×#6711yescharacteristic
322C22×C4C2×#6711yescharacteristic
323Q16—48no—
324Q16—128no—
325C2×C8—22no—
326C2×C8—42no—
327C2×C8—62no—
328C2×C8—122no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—61no—
334C4⋊C4—61no—
335C4⋊C4—121no—
336C4⋊C4—121no—
337C4⋊C4—242no—
338C4⋊C4—242no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3#39211yescharacteristic
352C22×C3#45112yes—
353C22×C3#45511yescharacteristic
354C22×C3#45611yescharacteristic
355C22×C3B(4,𝔽2)12yes—
356C22×C3#46212yes—
357C22×C3#49411yescharacteristic
358C22×C3#49711yescharacteristic
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—44no—
364C3×C4—81no—
365C3×C4—81no—
366Dic3—44no—
367Dic3—82no—
368Dic3—81no—
369Dic3—81no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23D3×Q1612yes—
374C23#91812yes—
375C23#93711yescharacteristic
376C23#93911yescharacteristic
377C23#94011yescharacteristic
378Q8—24no—
379Q8—24no—
380Q8—42no—
381Q8—42no—
382Q8—44no—
383Q8—44no—
384Q8—42no—
385Q8—81no—
386Q8—122no—
387Q8—241no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—64no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—124no—
408C2×C4—124no—
409C2×C4—122no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—122no—
415C2×C4—242no—
416C2×C4—242no—
417C2×C4—241no—
418C2×C4—241no—
419C2×C4#93711yescharacteristic
420C2×C4#93811yescharacteristic
421C2×C4#93911yescharacteristic
422C2×C4#95311yescharacteristic
423C8—42no—
424C8—122no—
425C2×C3—21no—
426C2×C3—22no—
427C2×C3—21no—
428C2×C3#160212yes—
429C2×C3#160512yes—
430C2×C3#200711yescharacteristic
431C2×C3#201611yescharacteristic
432C2×C3#202111yescharacteristic
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22#503212yes—
442C22#518212yes—
443C22#518311yescharacteristic
444C22#518612yes—
445C22#519011yescharacteristic
446C22#520211yescharacteristic
447C22#520511yescharacteristic
448C22#522311yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—44no—
454C4—81no—
455C4—81no—
456C4—124no—
457C4—242no—
458C4—241no—
459C4—241no—
460C3#996711yescharacteristic, complemented, Sylow
461C2—21no—
462C2—22no—
463C2—21no—
464C2#7098012yes—
465C2#7098312yes—
466C2#7167111yescharacteristic
467C2#7207311yescharacteristic
468C2#7211011yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#451C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenterC2×#451D3
hypocenterC3#9967
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
151. C22×Dic3Q16semidirect
460. C3#9967semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4512
C2×#52282
#827892
#721342
C2×#45382
C2×#44672
#676742
#99673