Finite Group Databasebeta

← #10711997

Subgroups — #10711997

Order 768.

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

normal subgroups only

This group has 516 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
1#74684C211yesmaximal, characteristic
2#70069C211yesmaximal, characteristic
3#70021C211yesmaximal, characteristic
4C2×#4473C211yesmaximal, characteristic
5#69597C211yesmaximal, characteristic
6C3×#1390C211yesmaximal, characteristic
7#67698C211yesmaximal, characteristic
8#12967—31nomaximal, Sylow
9#5798C2211yescharacteristic
10C2×#936C2211yescharacteristic
11C2×C3×#100C2211yescharacteristic
12C2×C3×#99C2211yescharacteristic
13C3×#386C2211yescharacteristic
14#5220C2211yescharacteristic
15C2×#834C2211yescharacteristic
16#5210—21no—
17#5210—21no—
18#5120—21no—
19#4950—21no—
20#4946—21no—
21#4945—21no—
22#4944—21no—
23#4650—21no—
24#4646—21no—
25#4555—21no—
26#4527—21no—
27#4527—21no—
28#4473C414yescomplemented
29#2379—31no—
30C2×#354—31no—
31#1885—31no—
32#1867—31no—
33#1867—31no—
34#1390—31no—
35#1390D311yescharacteristic
36C22×C3×D4C2311yescharacteristic
37C23×C3×C4D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×Dic6D411yescharacteristic
40C2×C3×M16—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4D411yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50C2×#217D411yescharacteristic
51#936—21no—
52#936—21no—
53#936C2×C412yes—
54#936Q812yes—
55D4×Dic3—41no—
56D4×Dic3—41no—
57#926—21no—
58C2×C4⋊Dic3—41no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—41no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3D411yescharacteristic
65C3×#100—21no—
66C3×#100—21no—
67C3×#99—21no—
68C3×#99C2×C412yes—
69C3×#95—21no—
70#834—21no—
71#834C2×C412yes—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—41no—
78C3⋊#95—41no—
79#816—41no—
80#816—41no—
81C2×#124—31no—
82#518—31no—
83#419—61no—
84C2×#100C2×D311yescharacteristic
85C2×#99—31no—
86C2×#99C2×D311yescharacteristic
87#391—61no—
88#388—61no—
89#388—61no—
90#386—31no—
91#386C2×D311yescharacteristic
92#385—61no—
93#385—61no—
94#385—61no—
95#385—61no—
96#382—61no—
97#354—34no—
98#342—61no—
99#342—61no—
100#339—61no—
101C24×C3C2×D411yescharacteristic
102C24×C3C2×D411yescharacteristic
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4C22×C411yescharacteristic
106C2×C3×D4C2×Q811yescharacteristic
107C2×C3×D4#4111yescharacteristic
108C2×C3×D4#4111yescharacteristic
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—81no—
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—41no—
124C22×Dic3—41no—
125C22×Dic3—81no—
126C2×Dic6—21no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6C4⋊C412yescomplemented
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—41no—
133C3×C22⋊C4—22no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—81no—
137#217—22no—
138#217—22no—
139#217—22no—
140#217—22no—
141#217—22no—
142#217—41no—
143#217—41no—
144#217—41no—
145#217—41no—
146#217—41no—
147#217—81no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—82no—
151Dic3⋊C4—81no—
152Dic3⋊C4—81no—
153Dic3⋊C4—81no—
154Dic3⋊C4—82no—
155C4×Dic3—21no—
156C4×Dic3—41no—
157C4×Dic3—81no—
158C4×Dic3C4⋊C412yescomplemented
159C22×Q8—31no—
160C22×D4C22×D311yescharacteristic
161C23×C4—31no—
162C23×C4D1211yescharacteristic
163C2×M16—21no—
164#124—32no—
165#124—32no—
166#124—61no—
167#124—61no—
168C4×D4—121no—
169C4×D4—121no—
170#117—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—121no—
175C2×C4⋊C4—121no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—31no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4D1211yescharacteristic
186C2×C42—31no—
187C2×C42—61no—
188#103—121no—
189#103—121no—
190#100—21no—
191#100—21no—
192#99—21no—
193#99—32no—
194#99—61no—
195#99C4×D312yes—
196#95—21no—
197#95—61no—
198#95—61no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95—121no—
203#95—121no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3C4×D411yescharacteristic
213C23×C3C4×D411yescharacteristic
214C23×C3#12011yescharacteristic
215C23×C3C22⋊Q811yescharacteristic
216C23×C3C22⋊Q811yescharacteristic
217C3×D4—41no—
218C3×D4—41no—
219C3×D4—41no—
220C3×D4—41no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4C2×C4⋊C411yescharacteristic
235C2×C3×C4#12311yescharacteristic
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258C2×Dic3—81no—
259Dic6—42no—
260Dic6—81no—
261C3×C8—81no—
262C24C2×D1211yescharacteristic
263C24D3×D411yescharacteristic
264C2×Q8—32no—
265C2×Q8—61no—
266C2×Q8—121no—
267C2×Q8—121no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4C2×C4×D311yescharacteristic
271C2×D4#23511yescharacteristic
272C2×D4#23711yescharacteristic
273C2×D4D3×Q811yescharacteristic
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—81no—
287C22×C4—121no—
288C22×C4—121no—
289C22×C4—241no—
290C22×C4D3×D411yescharacteristic
291M16—41no—
292M16—41no—
293C2×C8—41no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—241no—
297C4⋊C4—241no—
298C4⋊C4—241no—
299C4⋊C4—242no—
300C4⋊C4—242no—
301C22⋊C4—22no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—62no—
305C22⋊C4—62no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—81no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—241no—
316C42—32no—
317C42—61no—
318C42—121no—
319C42—241no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3#38511yescharacteristic
337C22×C3B(4,𝔽2)11yescharacteristic
338C22×C3#45811yescharacteristic
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—81no—
342C3×C4—82no—
343C3×C4—81no—
344Dic3—42no—
345Dic3—42no—
346Dic3—81no—
347Dic3—82no—
348Dic3—81no—
349Dic3—81no—
350Dic3—81no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23C4×D1211yescharacteristic
360C23#67⋊C411yescharacteristic
361C23#88211yescharacteristic
362C23#89611yescharacteristic
363C23#89711yescharacteristic
364Q8—122no—
365Q8—241no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—122no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—122no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4—241no—
402C2×C4—241no—
403C2×C4—241no—
404C2×C4—241no—
405C2×C4—241no—
406C2×C4#88311yescharacteristic
407C2×C4D3×C4⋊C411yescharacteristic
408C8—81no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3#186711yescharacteristic
416C2×C3#219911yescharacteristic
417C2×C3#220111yescharacteristic
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—81no—
433C22—81no—
434C22#466811yescharacteristic
435C22#474111yescharacteristic
436C22#474211yescharacteristic
437C4—21no—
438C4—21no—
439C4—81no—
440C4—82no—
441C4—81no—
442C4—122no—
443C4—122no—
444C4—241no—
445C4—242no—
446C4—241no—
447C4—241no—
448C4—241no—
449C3#1296711yescharacteristic, complemented, Sylow
450C2—21no—
451C2—21no—
452C2—41no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2#6896111yescharacteristic
457C2#6913111yescharacteristic
458C2#6914111yescharacteristic
subgroupG/H
centerC22#4741
commutatorC2×C3×D4C22×C4
FrattiniC22×D4C22×D3
FittingC3×#1390C2
socleC22×C3B(4,𝔽2)
prime radical, cosocleC22×C3×D4C23
hypercenter#1390D3
hypocenterC3#12967
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
28. #4473C4semidirect
129. C2×Dic6C4⋊C4semidirect
158. C4×Dic3C4⋊C4semidirect
449. C3#12967semidirect

Maximal subgroups

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

subgroupindex
#746842
#700692
#700212
C2×#44732
#695972
C3×#13902
#676982
#129673