Finite Group Databasebeta

← #10693247

Subgroups — #10693247

Order 768.

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

normal subgroups only

This group has 538 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#78624C211yesmaximal, characteristic
2#74058C211yesmaximal, characteristic
3#69978C211yesmaximal, characteristic
4#69975C211yesmaximal, characteristic, complemented
5#69534C211yesmaximal, characteristic, complemented
6C3×#1306C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9525—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#381C2211yescharacteristic
14#5211C2211yescharacteristic
15C4×#217C2211yescharacteristic
16#5131—21no—
17#5121—21no—
18#4957—21no—
19#4956—21no—
20#4737—22no—
21#4736C412yescomplemented
22#4735C412yescomplemented
23#4669—21no—
24#4666—21no—
25#4665—21no—
26#4653—21no—
27#4555—21no—
28#4555—21no—
29#4527—21no—
30#4527—21no—
31#4470—22no—
32#3094—31no—
33#2333—31no—
34#1826—31no—
35#1820—31no—
36#1510—31no—
37#1306D311yescharacteristic, complemented
38#1284—31no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×M16—21no—
44C2×C3×M16—21no—
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C22⋊C4D411yescharacteristic
47C2×#217—21no—
48C2×#217—21no—
49C2×#217—21no—
50#930—21no—
51#930—21no—
52#930C2×C412yes—
53#930D412yes—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58#926—21no—
59#926—21no—
60#926—21no—
61#926—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic, complemented
63C2×C4⋊Dic3C2×C411yescharacteristic, complemented
64C2×C4⋊Dic3D411yescharacteristic
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×C4×Dic3D411yescharacteristic
69C2×C4×Dic3D411yescharacteristic
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8C2×C412yescomplemented
74C3×#95—21no—
75C3×#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#821—22no—
82#821—22no—
83#818—22no—
84#817—22no—
85C2×#121—31no—
86C2×#117—31no—
87#482—62no—
88#481—32no—
89#480—32no—
90#428—61no—
91#418—61no—
92#407C2×D311yescharacteristic
93#407C2×D311yescharacteristic
94#390—61no—
95#388—61no—
96#387—61no—
97#386—61no—
98#386—61no—
99#381—31no—
100#381C2×D311yescharacteristic
101C4×C22⋊C4—31no—
102C4×C22⋊C4—61no—
103#342—61no—
104#342—61no—
105#339—61no—
106#339—61no—
107#327—62no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×#67—22no—
121C2×#67—22no—
122C2×#67—81no—
123C2×#67—81no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—41no—
130C2×D12—21no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12C22⋊C412yes—
134C2×C4×D3—42no—
135C2×C4×D3—81no—
136C2×C4×D3—81no—
137C3×M16—41no—
138C3×M16—41no—
139C3×M16—41no—
140C3×M16—41no—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145#217—42no—
146#217—41no—
147#217—41no—
148#217—41no—
149#217—41no—
150(C2×D3)⋊C4—22no—
151(C2×D3)⋊C4—22no—
152(C2×D3)⋊C4—42no—
153(C2×D3)⋊C4—81no—
154(C2×D3)⋊C4—81no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3C22⋊C412yes—
161Dic3⋊C4—81no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165C4×Dic3—41no—
166C4×Dic3—81no—
167C22×D4—31no—
168C22×D4—61no—
169C23×C4—61no—
170C23×C4C22×D311yescharacteristic
171C2×M16—21no—
172C2×M16—21no—
173#121—32no—
174#121—32no—
175#121—61no—
176#121—61no—
177#117—61no—
178#117—61no—
179#117—61no—
180#117—61no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4D1211yescharacteristic
193C2×C22⋊C4D1211yescharacteristic
194C2×C42—31no—
195C2×C42—31no—
196C2×C42—61no—
197C2×C42—61no—
198#107—62no—
199#106—62no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8C4×D312yescomplemented
206#95—21no—
207#95—21no—
208#95—61no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3C4×D411yescharacteristic
218C23×C3#12011yescharacteristic
219C23×C3#12111yescharacteristic
220C22×D3—22no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4C2×C22⋊C411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12311yescharacteristic
245#67—82no—
246#67—82no—
247C2×Dic3—22no—
248C2×Dic3—22no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261C2×Dic3—81no—
262C2×Dic3—81no—
263D12—42no—
264D12—81no—
265C4×D3—84no—
266C3×C8—42no—
267C3×C8—81no—
268C24—61no—
269C24C2×D1211yescharacteristic
270C2×D4—32no—
271C2×D4—62no—
272C2×D4—62no—
273C2×D4—61no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—241no—
277C2×D4—241no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—122no—
288C22×C4—121no—
289C22×C4—241no—
290C22×C4—241no—
291C22×C4C2×C4×D311yescharacteristic
292C22×C4#23511yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4D3×D411yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4#23911yescharacteristic
297M16—41no—
298M16—41no—
299M16—41no—
300M16—41no—
301C2×C8—22no—
302C2×C8—41no—
303C4⋊C4—32no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—241no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—122no—
315C22⋊C4—122no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—241no—
321C22⋊C4—241no—
322C42—121no—
323C42—121no—
324C42—121no—
325C42—121no—
326C42—241no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3#38611yescharacteristic
338C22×C3#45811yescharacteristic
339C22×C3#45811yescharacteristic
340C2×D3—42no—
341C2×D3—42no—
342C2×D3—42no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C2×D3—81no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—81no—
352C3×C4—81no—
353Dic3—42no—
354Dic3—42no—
355Dic3—81no—
356Dic3—81no—
357Dic3—81no—
358Dic3—161no—
359C23—21no—
360C23—21no—
361C23—41no—
362C23—41no—
363C23—62no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23—121no—
368C23—121no—
369C23—121no—
370C23C4×D1211yescharacteristic
371C23#88211yescharacteristic
372C23#89511yescharacteristic
373D4—122no—
374D4—241no—
375D4—242no—
376D4—242no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—244no—
403C2×C4—241no—
404C2×C4—241no—
405C2×C4—241no—
406C2×C4—241no—
407C2×C4—241no—
408C2×C4D3×C22⋊C411yescharacteristic
409C2×C4#88411yescharacteristic
410C2×C4#89311yescharacteristic
411C2×C4#89411yescharacteristic
412C8—42no—
413C8—81no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3#189011yescharacteristic
419C2×C3#190411yescharacteristic
420C2×C3#190611yescharacteristic
421D3—82no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—81no—
431C22—81no—
432C22—122no—
433C22—122no—
434C22—122no—
435C22—241no—
436C22—241no—
437C22—241no—
438C22—241no—
439C22#466511yescharacteristic
440C22#474211yescharacteristic
441C22#474211yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—81no—
447C4—81no—
448C4—122no—
449C4—122no—
450C4—241no—
451C4—241no—
452C4—241no—
453C4—481no—
454C3#952511yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—41no—
458C2—41no—
459C2—242no—
460C2#6896311yescharacteristic
461C2#6902711yescharacteristic
462C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1306C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1306D3
hypocenterC3#9525
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
4. #69975C2semidirect
5. #69534C2semidirect
6. C3×#1306C2semidirect
7. #67682C2semidirect
21. #4736C4semidirect
22. #4735C4semidirect
37. #1306D3semidirect
62. C2×C4⋊Dic3C2×C4semidirect
63. C2×C4⋊Dic3C2×C4semidirect
73. C3×C22⋊C8C2×C4semidirect
205. C22⋊C8C4×D3semidirect
454. C3#9525semidirect

Maximal subgroups

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

subgroupindex
#786242
#740582
#699782
#699752
#695342
C3×#13062
#676822
#95253