Finite Group Databasebeta

← #10695738

Subgroups — #10695738

Order 768.

830 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 111 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#82679C211yesmaximal, characteristic
2C2×#5040C211yesmaximal, characteristic
3C2×#4739C211yesmaximal, characteristic, complemented
4C2×C3×#327C211yesmaximal, characteristic
5#69424C211yesmaximal, characteristic, complemented
6C2×#4452C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9965—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×#214—21no—
12C2×#924C2211yescharacteristic
13C22×Dic12—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5222—21no—
17#5220—22no—
18C2×C3⋊#95—22no—
19C2×C3⋊#95C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×#816—21no—
22#5040—22no—
23#5040—22no—
24#4985—21no—
25C2×#810—21no—
26#4953—21no—
27#4947—21no—
28#4739—22no—
29#4739—22no—
30C3×#327—22no—
31#4452—22no—
32C2×#451—31no—
33C2×#448—31no—
34#2639—31no—
35C2×#328—31no—
36C2×#327D311yescharacteristic
37#1446—31no—
38#1280—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3D412yes—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—21no—
50C22×Dic6—44no—
51C22×Dic6D412yes—
52C22×Dic6D411yescharacteristic
53C22×C3×C8—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#217—22no—
62C2×#217—22no—
63C2×#214—24no—
64C2×#214—44no—
65C2×C4⋊Dic3D411yescharacteristic
66#924—21no—
67#924—21no—
68#924—21no—
69#924—21no—
70C2×Dic3⋊C4—21no—
71C22×C3⋊C8—21no—
72C2×Dic12—24no—
73C2×Dic12—44no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76C3×#95—21no—
77C3×#95—21no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—44no—
83#830—21no—
84#830D412yes—
85#816—24no—
86#810—24no—
87C23×Q8—31no—
88C22×Q16—61no—
89C22×SD16—61no—
90C2×C22⋊Q8—31no—
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⋊C8C2×D311yescharacteristic
100#400—61no—
101#397—61no—
102#394—61no—
103#393—61no—
104#386—62no—
105C2×#95—31no—
106C2×#95—62no—
107C2×#95C2×D311yescharacteristic
108#328—62no—
109#327—22no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—42no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—24no—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—42no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—42no—
143C22×Dic3—42no—
144C2×Dic6—24no—
145C2×Dic6—24no—
146C2×Dic6—22no—
147C2×Dic6—22no—
148C2×Dic6—21no—
149C2×Dic6—21no—
150C2×Dic6—44no—
151C2×Dic6—44no—
152C2×Dic6—44no—
153C2×Dic6—44no—
154C2×Dic6—42no—
155C2×Dic6—42no—
156C2×Dic6—41no—
157C2×Dic6—41no—
158C2×Dic6—42no—
159C2×C3×C8—22no—
160C2×C3×C8—42no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165#217—41no—
166#217—41no—
167#217—82no—
168#217—82no—
169#214—48no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172Dic3⋊C4—41no—
173Dic3⋊C4—41no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—42no—
176Dic12—48no—
177C22×Q8—31no—
178C22×Q8—32no—
179C22×Q8—61no—
180C22×Q8—61no—
181C22×Q8—61no—
182C22×Q8—61no—
183C22×Q8—61no—
184C22×Q8—124no—
185C22×D4—21no—
186C22×D4D1211yescharacteristic
187C23×C4—21no—
188C23×C4—32no—
189C23×C4—61no—
190C23×C4C22×D311yescharacteristic
191C2×Q16—64no—
192C2×Q16—124no—
193C2×SD16—64no—
194C2×SD16—124no—
195C22×C8—21no—
196C22×C8—61no—
197C22⋊Q8—61no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200C22⋊Q8—61no—
201#121—21no—
202#121—21no—
203#121—21no—
204#121—21no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4D1211yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212#103—64no—
213#103—64no—
214C22⋊C8—21no—
215C22⋊C8—32no—
216C22⋊C8—61no—
217C22⋊C8#6712yes—
218#95—21no—
219#95—21no—
220#95—62no—
221#95—62no—
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×D4—42no—
239C3×D4—81no—
240C3×D4—82no—
241C3×D4—81no—
242C3×D4—81no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—44no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—82no—
254C2×C3×C4—81no—
255C2×C3×C4—81no—
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×Dic3—24no—
261C2×Dic3—42no—
262C2×Dic3—44no—
263C2×Dic3—42no—
264C2×Dic3—44no—
265C2×Dic3—44no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—41no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—82no—
275Dic6—24no—
276Dic6—24no—
277Dic6—42no—
278Dic6—42no—
279Dic6—44no—
280Dic6—44no—
281Dic6—42no—
282Dic6—81no—
283C3×C8—42no—
284C3⋊C8—42no—
285C24—21no—
286C24D3×D411yescharacteristic
287C2×Q8—64no—
288C2×Q8—64no—
289C2×Q8—62no—
290C2×Q8—61no—
291C2×Q8—62no—
292C2×Q8—61no—
293C2×Q8—122no—
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—42no—
307C2×D4—42no—
308C2×D4—41no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—64no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—61no—
320C22×C4—61no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—122no—
324C22×C4—121no—
325C22×C4—121no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4C2×D1211yescharacteristic
329C22×C4#23511yescharacteristic
330C22×C4D3×D411yescharacteristic
331C22×C4D3×D411yescharacteristic
332C22×C4D3×D411yescharacteristic
333C22×C4C2×#6711yescharacteristic
334Q16—128no—
335SD16—128no—
336C2×C8—22no—
337C2×C8—42no—
338C2×C8—62no—
339C2×C8—122no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—61no—
343C4⋊C4—61no—
344C4⋊C4—121no—
345C4⋊C4—121no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—121no—
349C22⋊C4—121no—
350C22⋊C4—242no—
351C22⋊C4—242no—
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—
379Dic3—44no—
380Dic3—82no—
381Dic3—81no—
382Dic3—81no—
383C23—22no—
384C23—21no—
385C23—21no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—41no—
390C23—41no—
391C23#88211yescharacteristic
392C23#88411yescharacteristic
393C23#91212yes—
394C23#91512yes—
395C23#94011yescharacteristic
396Q8—64no—
397Q8—64no—
398Q8—122no—
399Q8—122no—
400Q8—124no—
401Q8—124no—
402Q8—122no—
403Q8—241no—
404D4—42no—
405D4—81no—
406D4—81no—
407D4—81no—
408D4—82no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—64no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—122no—
424C2×C4—124no—
425C2×C4—122no—
426C2×C4—124no—
427C2×C4—124no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—121no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—242no—
437C2×C4#88211yescharacteristic
438C2×C4#88411yescharacteristic
439C2×C4#93011yescharacteristic
440C2×C4#93711yescharacteristic
441C8—42no—
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—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—81no—
461C22—81no—
462C22—82no—
463C22—82no—
464C22—81no—
465C22#474112yes—
466C22#476712yes—
467C22#478312yes—
468C22#482511yescharacteristic
469C22#482611yescharacteristic
470C22#496011yescharacteristic
471C22#515511yescharacteristic
472C22#516711yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—82no—
478C4—81no—
479C4—124no—
480C4—241no—
481C4—242no—
482C4—241no—
483C3#996511yescharacteristic, complemented, Sylow
484C2—21no—
485C2—21no—
486C2—22no—
487C2—161no—
488C2#6828712yes—
489C2#6829412yes—
490C2#7003711yescharacteristic
491C2#7021511yescharacteristic
492C2#7039311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenterC2×#327D3
hypocenterC3#9965
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
3. C2×#4739C2semidirect
5. #69424C2semidirect
6. C2×#4452C2semidirect
7. #67674C2semidirect
483. C3#9965semidirect

Maximal subgroups

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

subgroupindex
#826792
C2×#50402
C2×#47392
C2×C3×#3272
#694242
C2×#44522
#676742
#99653