Finite Group Databasebeta

← #10714613

Subgroups — #10714613

Order 768.

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

normal subgroups only

This group has 496 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#82888C211yesmaximal, characteristic
2C3×#1890C211yesmaximal, characteristic
3C2×#4539C211yesmaximal, characteristic
4#71655C211yesmaximal, characteristic
5#71647C211yesmaximal, characteristic
6C2×#4472C211yesmaximal, characteristic
7#67695C211yesmaximal, characteristic
8#13421—31nomaximal, Sylow
9C2×C3×#123C2211yescharacteristic
10C3×#518C2211yescharacteristic
11C2×#935C2211yescharacteristic
12C2×C3×#99C2211yescharacteristic
13C3×#381—21no—
14#5221C2211yescharacteristic
15C2×#834C2211yescharacteristic
16C2×#833C2211yescharacteristic
17#5210—21no—
18#4995—21no—
19#4982—21no—
20#4976—21no—
21#4952—21no—
22#4952—21no—
23#4946—21no—
24#4943—21no—
25C3×#342—21no—
26#4539—22no—
27#4532—21no—
28#4526—21no—
29#4472—22no—
30#2724—31no—
31C2×#352—31no—
32C2×#352—31no—
33#1890—31no—
34#1890D311yescharacteristic
35#1856—31no—
36#1393—31no—
37C22×C3×D4C2311yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×Dic3D411yescharacteristic
40C3×#123—21no—
41C3×#123—21no—
42C3×#123—21no—
43C3×#123—21no—
44C3×C4×D4—41no—
45C3×C4×D4—41no—
46C3×#117—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4D411yescharacteristic
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217D411yescharacteristic
58#935—21no—
59#935—21no—
60#935—21no—
61#935—21no—
62C2×C4⋊Dic3D411yescharacteristic
63C2×Dic3⋊C4—21no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—21no—
67C2×C4×Dic3D411yescharacteristic
68C2×#208—21no—
69C3×#99—21no—
70C3×#99—21no—
71C3×#95—41no—
72C3×#95—41no—
73#834—21no—
74#834—21no—
75#833—21no—
76#833D412yes—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—41no—
83C3⋊#95—41no—
84#808—41no—
85#807—41no—
86C2×#123—31no—
87C2×#123C2×D311yescharacteristic
88#518C2×D311yescharacteristic
89#428—61no—
90C2×#100—31no—
91C2×#99—31no—
92C2×#99C2×D311yescharacteristic
93#398—61no—
94#397—61no—
95#397—61no—
96#390—31no—
97#389—61no—
98#385—61no—
99#385—61no—
100#381—21no—
101#381—61no—
102#352—62no—
103#352—62no—
104#342—21no—
105#342—61no—
106#339—61no—
107C24×C3C2×D411yescharacteristic
108C24×C3C2×D411yescharacteristic
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4C2×D411yescharacteristic
112C2×C3×D4#4111yescharacteristic
113C2×C3×D4#4111yescharacteristic
114C2×C3×D4#4111yescharacteristic
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—41no—
134C22×Dic3—41no—
135C22×Dic3—41no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—21no—
141C3×C22⋊C4—21no—
142C3×C22⋊C4—41no—
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×C42—41no—
150#217—21no—
151#217—21no—
152#217—42no—
153#217—42no—
154#217—41no—
155#217—41no—
156#217—41no—
157#217—41no—
158C4⋊Dic3—41no—
159Dic3⋊C4—42no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162Dic3⋊C4—81no—
163Dic3⋊C4—81no—
164C4×Dic3—41no—
165#208—22no—
166#208—41no—
167C2×C3⋊C8—41no—
168C22×D4C22×D311yescharacteristic
169C23×C4—31no—
170C23×C4#6711yescharacteristic
171C2×M16—61no—
172#123—21no—
173#123—21no—
174#123—21no—
175#123—21no—
176#123—61no—
177#123—61no—
178#123—61no—
179#123—61no—
180C4×D4—41no—
181C4×D4—41no—
182#117—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—31no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4#6711yescharacteristic
199C2×C42—31no—
200#107—121no—
201#106—121no—
202#100—32no—
203#100—61no—
204#99—21no—
205#99—21no—
206#99—61no—
207#99—61no—
208#95—41no—
209#95—41no—
210#95—61no—
211#95—61no—
212#95—61no—
213#95—61no—
214#95—61no—
215#95—121no—
216#95—121no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3#12011yescharacteristic
226C23×C3#12111yescharacteristic
227C23×C3#12111yescharacteristic
228C23×C3#12311yescharacteristic
229C23×C3#12311yescharacteristic
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—41no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
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—81no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12411yescharacteristic
255C2×Dic3—22no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
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—
272C3⋊C8—42no—
273C24D3×D411yescharacteristic
274C24C2×#6711yescharacteristic
275C2×D4—41no—
276C2×D4—41no—
277C2×D4#23511yescharacteristic
278C2×D4#23511yescharacteristic
279C2×D4D3×D411yescharacteristic
280C2×D4#23711yescharacteristic
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4—121no—
301C22×C4D3×D411yescharacteristic
302M16—62no—
303M16—121no—
304C2×C8—121no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—121no—
310C4⋊C4—122no—
311C4⋊C4—121no—
312C4⋊C4—121no—
313C4⋊C4—241no—
314C4⋊C4—241no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—61no—
325C22⋊C4—61no—
326C22⋊C4—122no—
327C22⋊C4—122no—
328C22⋊C4—121no—
329C22⋊C4—121no—
330C22⋊C4—121no—
331C22⋊C4—121no—
332C42—41no—
333C42—121no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3#39411yescharacteristic
351C22×C3B(4,𝔽2)11yescharacteristic
352C22×C3#45811yescharacteristic
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—81no—
356C3×C4—81no—
357C3×C4—81no—
358C3×C4—81no—
359C3×C4—161no—
360Dic3—42no—
361Dic3—81no—
362Dic3—81no—
363Dic3—81no—
364Dic3—161no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23#88311yescharacteristic
374C23#88411yescharacteristic
375C23#92911yescharacteristic
376C23#93711yescharacteristic
377C23#93911yescharacteristic
378D4—41no—
379D4—41no—
380D4—41no—
381D4—41no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—62no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4—122no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—241no—
415C2×C4—241no—
416C2×C4—241no—
417C2×C4—241no—
418C2×C4#88411yescharacteristic
419C2×C4#88511yescharacteristic
420C8—122no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3#204411yescharacteristic
428C2×C3#219811yescharacteristic
429C2×C3#220111yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—81no—
445C22—81no—
446C22#495911yescharacteristic
447C22#503211yescharacteristic
448C22#505311yescharacteristic
449C4—21no—
450C4—21no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—122no—
456C4—161no—
457C4—241no—
458C4—241no—
459C4—241no—
460C4—481no—
461C3#1342111yescharacteristic, complemented, Sylow
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—41no—
467C2—41no—
468C2#6959911yescharacteristic
469C2#7100211yescharacteristic
470C2#7100311yescharacteristic
subgroupG/H
centerC22#5032
commutator, prime radical, cosocleC22×C3×D4C23
FrattiniC22×D4C22×D3
FittingC3×#1890C2
socleC22×C3B(4,𝔽2)
hypercenter#1890D3
hypocenterC3#13421
LCSG ▷ C22×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1890

Complements

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

NH ≅ G/Ntype
461. C3#13421semidirect

Maximal subgroups

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

subgroupindex
#828882
C3×#18902
C2×#45392
#716552
#716472
C2×#44722
#676952
#134213