Finite Group Databasebeta

← #10695734

Subgroups — #10695734

Order 768.

830 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 113 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×#2639C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic, complemented
3C2×#5040C211yesmaximal, characteristic
4C2×#4538C211yesmaximal, characteristic, complemented
5C2×#4537C211yesmaximal, characteristic
6#69424C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9965—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C22×#216—21no—
13C22×#214—21no—
14C3×#400—21no—
15C3×#397—21no—
16C3×#394—21no—
17C3×#393—21no—
18C3×#386—22no—
19C2×C3×#95C2211yescharacteristic
20C2×C3⋊#95—22no—
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5050—22no—
25#5050—22no—
26#5040—22no—
27#5040—22no—
28C2×#810—21no—
29C2×#810—21no—
30#4538—22no—
31#4537—22no—
32C2×#451—31no—
33C2×#448—31no—
34#2639D311yescharacteristic
35C2×#328—31no—
36C2×#327—31no—
37#1446—31no—
38#1280—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4D412yes—
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C23×Dic3—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—21no—
50C22×Dic6—21no—
51C22×Dic6—44no—
52C22×Dic6D412yes—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C3×#121—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×#216—24no—
69C2×#216—44no—
70C2×#214—24no—
71C2×#214—44no—
72C22×C3⋊C8—21no—
73C22×C3⋊C8—21no—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—21no—
77C3×#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—44no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#830D412yes—
85#810—24no—
86#810—24no—
87C23×Q8—31no—
88C22×Q16—61no—
89C22×SD16—61no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×#121C2×D311yescharacteristic
92#451—62no—
93#451—62no—
94#448—62no—
95#448—62no—
96C2×#103—61no—
97C2×#103—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8—31no—
100#400—21no—
101#397—21no—
102#394—21no—
103#393—21no—
104#386—22no—
105C2×#95—31no—
106C2×#95—62no—
107C2×#95C2×D311yescharacteristic
108#328—62no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—42no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—42no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4—42no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×Dic3—24no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—42no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—42no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—42no—
147C2×Dic6—24no—
148C2×Dic6—24no—
149C2×Dic6—22no—
150C2×Dic6—22no—
151C2×Dic6—44no—
152C2×Dic6—44no—
153C2×Dic6—44no—
154C2×Dic6—44no—
155C2×Dic6—42no—
156C2×Dic6—42no—
157C2×Dic6—41no—
158C2×Dic6—41no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—82no—
170C3×C22⋊C4—82no—
171#216—48no—
172#214—48no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—42no—
176C2×C3⋊C8—42no—
177C22×Q8—32no—
178C22×Q8—61no—
179C22×Q8—61no—
180C22×Q8—61no—
181C22×Q8—61no—
182C22×Q8—61no—
183C22×Q8—124no—
184C22×Q8#6711yescharacteristic
185C22×D4—21no—
186C22×D4#6711yescharacteristic
187C23×C4—61no—
188C23×C4—61no—
189C23×C4#6712yes—
190C23×C4C22×D311yescharacteristic
191C2×Q16—64no—
192C2×Q16—124no—
193C2×SD16—64no—
194C2×SD16—124no—
195C22×C8—61no—
196C22×C8—61no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204#121—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4#6711yescharacteristic
207C2×C4⋊C4#6711yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212#103—64no—
213#103—64no—
214C22⋊C8—32no—
215C22⋊C8—32no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—21no—
222#95—61no—
223#95—61no—
224#95—124no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—22no—
228C23×C3—41no—
229C23×C3—42no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—42no—
233C23×C3#12011yescharacteristic
234C23×C3#12111yescharacteristic
235C23×C3#12711yescharacteristic
236C23×C3#13512yes—
237C23×C3#13712yes—
238C3×Q8—42no—
239C3×Q8—81no—
240C3×D4—42no—
241C3×D4—81no—
242C3×D4—82no—
243C3×D4—81no—
244C3×D4—81no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—44no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—82no—
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×Dic3—24no—
268C2×Dic3—42no—
269C2×Dic3—44no—
270C2×Dic3—42no—
271C2×Dic3—44no—
272C2×Dic3—44no—
273C2×Dic3—42no—
274C2×Dic3—82no—
275C2×Dic3—81no—
276C2×Dic3—81no—
277Dic6—24no—
278Dic6—24no—
279Dic6—42no—
280Dic6—42no—
281Dic6—44no—
282Dic6—44no—
283C3⋊C8—42no—
284C3⋊C8—42no—
285C24—21no—
286C24C2×#6711yescharacteristic
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—42no—
290C2×Q8—64no—
291C2×Q8—64no—
292C2×Q8—62no—
293C2×Q8—62no—
294C2×Q8—122no—
295C2×Q8—122no—
296C2×Q8—124no—
297C2×Q8—124no—
298C2×Q8—124no—
299C2×Q8—124no—
300C2×Q8—121no—
301C2×Q8—121no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—64no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—122no—
322C22×C4—121no—
323C22×C4—121no—
324C22×C4—122no—
325C22×C4—121no—
326C22×C4—121no—
327C22×C4—122no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4D3×D411yescharacteristic
331C22×C4#23711yescharacteristic
332C22×C4C2×#6711yescharacteristic
333C22×C4C2×#6711yescharacteristic
334Q16—128no—
335SD16—128no—
336C2×C8—62no—
337C2×C8—62no—
338C2×C8—122no—
339C2×C8—122no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3#39211yescharacteristic
366C22×C3#45212yes—
367C22×C3#45511yescharacteristic
368C22×C3#45611yescharacteristic
369C22×C3B(4,𝔽2)12yes—
370C22×C3#46412yes—
371C22×C3#49511yescharacteristic
372C22×C3#49711yescharacteristic
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—82no—
378C3×C4—81no—
379C3×C4—81no—
380C3×C4—81no—
381Dic3—44no—
382Dic3—82no—
383C23—21no—
384C23—22no—
385C23—21no—
386C23—41no—
387C23—42no—
388C23—41no—
389C23—42no—
390C23—41no—
391C23#93811yescharacteristic
392C23#94011yescharacteristic
393C23#95012yes—
394C23#95112yes—
395C23#95311yescharacteristic
396Q8—42no—
397Q8—64no—
398Q8—64no—
399Q8—81no—
400Q8—122no—
401Q8—122no—
402Q8—124no—
403Q8—124no—
404D4—42no—
405D4—81no—
406D4—82no—
407D4—81no—
408D4—81no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—64no—
427C2×C4—82no—
428C2×C4—122no—
429C2×C4—124no—
430C2×C4—122no—
431C2×C4—124no—
432C2×C4—122no—
433C2×C4—124no—
434C2×C4—241no—
435C2×C4—242no—
436C2×C4—241no—
437C2×C4#93711yescharacteristic
438C2×C4#93711yescharacteristic
439C2×C4#93911yescharacteristic
440C2×C4#93911yescharacteristic
441C8—122no—
442C8—122no—
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—161no—
447C2×C3#160312yes—
448C2×C3#160612yes—
449C2×C3#200711yescharacteristic
450C2×C3#201511yescharacteristic
451C2×C3#202111yescharacteristic
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—82no—
464C22—82no—
465C22#505412yes—
466C22#505911yescharacteristic
467C22#506011yescharacteristic
468C22#522311yescharacteristic
469C22#523912yes—
470C22#524211yescharacteristic
471C22#524412yes—
472C22#524611yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—82no—
481C4—124no—
482C4—242no—
483C3#996511yescharacteristic, complemented, Sylow
484C2—21no—
485C2—22no—
486C2—21no—
487C2—161no—
488C2#7172911yescharacteristic
489C2#7185111yescharacteristic
490C2#7188111yescharacteristic
491C2#7188512yes—
492C2#7188912yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2639C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenter#2639D3
hypocenterC3#9965
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2639

Complements

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

NH ≅ G/Ntype
2. C2×#5050C2semidirect
4. C2×#4538C2semidirect
6. #69424C2semidirect
7. #67613C2semidirect
483. C3#9965semidirect

Maximal subgroups

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

subgroupindex
C3×#26392
C2×#50502
C2×#50402
C2×#45382
C2×#45372
#694242
#676132
#99653