Finite Group Databasebeta

← #10714639

Subgroups — #10714639

Order 768.

508 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 73 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#82989C211yesmaximal, characteristic
2C3×#1885C211yesmaximal, characteristic
3C2×#4541C211yesmaximal, characteristic
4#71657C211yesmaximal, characteristic
5#71655C211yesmaximal, characteristic
6C2×#4476C211yesmaximal, characteristic
7#67698C211yesmaximal, characteristic
8#13425—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C3×#518C2211yescharacteristic
11C2×#935C2211yescharacteristic
12C3×#419—21no—
13C2×C3×#100C2211yescharacteristic
14#5230—21no—
15#5220C2211yescharacteristic
16C2×#834C2211yescharacteristic
17C2×#834C2211yescharacteristic
18#4952—21no—
19#4952—21no—
20#4946—21no—
21#4946—21no—
22#4943—21no—
23#4942—21no—
24#4933—21no—
25C3×#339—21no—
26#4541—22no—
27#4526—21no—
28#4526—21no—
29#4476—22no—
30#2717—31no—
31C2×#354—31no—
32C2×#352—31no—
33#1890—31no—
34#1885D311yescharacteristic
35#1867—31no—
36#1390—31no—
37C22×C3×Q8D411yescharacteristic
38C22×C3×D4C2311yescharacteristic
39C23×Dic3D411yescharacteristic
40C23×Dic3D411yescharacteristic
41C2×C3×M16—21no—
42C3×#124—21no—
43C3×#124—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×C4×D4—41no—
47C3×C4×D4—41no—
48C3×#117—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C42D411yescharacteristic
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217D411yescharacteristic
59C2×#217D411yescharacteristic
60#935—21no—
61#935—21no—
62#935—21no—
63#935—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—41no—
69C3×#103—41no—
70C3×#103—41no—
71C3×#100—21no—
72C3×#100—21no—
73#834—21no—
74#834—21no—
75#834—21no—
76#834D412yes—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—41no—
84C3⋊#95—41no—
85C3⋊#95—41no—
86C2×#124C2×D311yescharacteristic
87C2×#123—31no—
88#518C2×D311yescharacteristic
89#419—21no—
90C2×#100C2×D311yescharacteristic
91C2×#99—31no—
92C2×#99—31no—
93#397—61no—
94#397—61no—
95#395—61no—
96#387—61no—
97#386—31no—
98#385—61no—
99#385—61no—
100#381—61no—
101#380—61no—
102#354—62no—
103#352—62no—
104#342—61no—
105#342—61no—
106#339—21no—
107C24×C3C2×D411yescharacteristic
108C24×C3C2×D411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4C2×D411yescharacteristic
116C2×C3×D4#4111yescharacteristic
117C2×C3×D4#4111yescharacteristic
118C2×C3×D4#4111yescharacteristic
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—42no—
133C22×Dic3—41no—
134C22×Dic3—41no—
135C22×Dic3—41no—
136C22×Dic3—41no—
137C22×Dic3—41no—
138C3×M16—41no—
139C3×M16—41no—
140C2×C3×C8—41no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42—41no—
153#217—22no—
154#217—21no—
155#217—21no—
156#217—42no—
157#217—41no—
158#217—41no—
159#217—41no—
160#217—41no—
161#217—41no—
162#217—41no—
163Dic3⋊C4—42no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—81no—
167Dic3⋊C4—81no—
168Dic3⋊C4—82no—
169C22×Q8#6711yescharacteristic
170C22×D4C22×D311yescharacteristic
171C23×C4—31no—
172C23×C4—31no—
173C2×M16—21no—
174#124—21no—
175#124—21no—
176#124—21no—
177#124—21no—
178#123—61no—
179#123—61no—
180#123—61no—
181#123—61no—
182C4×D4—41no—
183C4×D4—41no—
184#117—21no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—121no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—31no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C42#6711yescharacteristic
201#103—41no—
202#103—41no—
203#100—21no—
204#100—21no—
205#99—32no—
206#99—61no—
207#99—61no—
208#99—61no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—121no—
216#95—121no—
217#95—121no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3#12011yescharacteristic
227C23×C3#12111yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12311yescharacteristic
230C23×C3#12411yescharacteristic
231C3×Q8—42no—
232C3×Q8—81no—
233C3×D4—41no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—41no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12311yescharacteristic
253C2×Dic3—22no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—42no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—81no—
269C2×Dic3—81no—
270C2×Dic3—81no—
271C2×Dic3—81no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275C3×C8—81no—
276C24D3×D411yescharacteristic
277C24D3×D411yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4#23511yescharacteristic
285C2×D4#23511yescharacteristic
286C2×D4D3×D411yescharacteristic
287C2×D4#23711yescharacteristic
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—122no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4—121no—
306C22×C4C2×#6711yescharacteristic
307M16—41no—
308M16—41no—
309C2×C8—41no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—122no—
313C4⋊C4—121no—
314C4⋊C4—121no—
315C4⋊C4—241no—
316C4⋊C4—241no—
317C4⋊C4—242no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—61no—
326C22⋊C4—62no—
327C22⋊C4—61no—
328C22⋊C4—122no—
329C22⋊C4—121no—
330C22⋊C4—121no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C42—21no—
336C42—21no—
337C42—41no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3#39411yescharacteristic
355C22×C3B(4,𝔽2)11yescharacteristic
356C22×C3#45811yescharacteristic
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—81no—
360C3×C4—81no—
361C3×C4—81no—
362C3×C4—81no—
363Dic3—42no—
364Dic3—81no—
365Dic3—81no—
366Dic3—82no—
367Dic3—81no—
368Dic3—161no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23#88311yescharacteristic
378C23#88411yescharacteristic
379C23#88411yescharacteristic
380C23#88511yescharacteristic
381C23#93711yescharacteristic
382Q8—42no—
383Q8—81no—
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—62no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—122no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—241no—
418C2×C4—241no—
419C2×C4—241no—
420C2×C4—241no—
421C2×C4—241no—
422C2×C4—241no—
423C2×C4—241no—
424C2×C4#92911yescharacteristic
425C2×C4#93911yescharacteristic
426C8—81no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3#204411yescharacteristic
434C2×C3#219711yescharacteristic
435C2×C3#220111yescharacteristic
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—81no—
451C22—81no—
452C22#495911yescharacteristic
453C22#503211yescharacteristic
454C22#505311yescharacteristic
455C4—21no—
456C4—21no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—122no—
462C4—241no—
463C4—241no—
464C4—242no—
465C4—241no—
466C4—481no—
467C3#1342511yescharacteristic, complemented, Sylow
468C2—21no—
469C2—21no—
470C2—41no—
471C2—41no—
472C2—41no—
473C2—41no—
474C2#6959911yescharacteristic
475C2#7099011yescharacteristic
476C2#7099211yescharacteristic
subgroupG/H
centerC22#5032
commutator, prime radical, cosocleC22×C3×D4C23
FrattiniC22×D4C22×D3
FittingC3×#1885C2
socleC22×C3B(4,𝔽2)
hypercenter#1885D3
hypocenterC3#13425
LCSG ▷ C22×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1885

Complements

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

NH ≅ G/Ntype
467. C3#13425semidirect

Maximal subgroups

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

subgroupindex
#829892
C3×#18852
C2×#45412
#716572
#716552
C2×#44762
#676982
#134253