Finite Group Databasebeta

← #10695509

Subgroups — #10695509

Order 768.

610 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#2624C211yesmaximal, characteristic, complemented
2#82627C211yesmaximal, characteristic
3#82217C211yesmaximal, characteristic, complemented
4#71673C211yesmaximal, characteristic, complemented
5#71590C211yesmaximal, characteristic, complemented
6#69528C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9932—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C3×#399—21no—
13C3×#397—21no—
14C3×#394—22no—
15C3×#390—21no—
16C3×#389—21no—
17C3×#380C2211yescharacteristic
18#5217—21no—
19C4×#217C2211yescharacteristic
20#5207C2211yescharacteristic
21#5207C2211yescharacteristic
22#5039—22no—
23#5036—21no—
24#5034—21no—
25#5003—22no—
26#5001—21no—
27#4995—21no—
28#4968—21no—
29#4968—21no—
30#4967—21no—
31C2×#806—22no—
32#4537—22no—
33#4532—22no—
34#3192—31no—
35#3094—31no—
36#2624D311yescharacteristic, complemented
37#1964—31no—
38#1826—31no—
39#1507—31no—
40#1305—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×D12D411yescharacteristic
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#117—21no—
51C3×#117—21no—
52C3×#117—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4D412yes—
64C2×C3×C42—21no—
65C2×#217—21no—
66C2×#213—22no—
67#930—21no—
68#930—21no—
69#930—21no—
70#930—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×C4×Dic3—21no—
74C2×#208—21no—
75C2×#208D412yes—
76C3×#95—22no—
77C3×#95—21no—
78#832—22no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#809—22no—
85#809—22no—
86#809—22no—
87#808—22no—
88#808—22no—
89#807—22no—
90#806—44no—
91C2×#121—31no—
92C2×#121C2×D311yescharacteristic
93C2×#117C2×D311yescharacteristic
94#482—62no—
95#481—61no—
96#481—61no—
97#481—61no—
98#480—61no—
99#469—62no—
100#447—61no—
101#447—61no—
102#428—61no—
103#418—61no—
104C2×#99—62no—
105#407—31no—
106#407—31no—
107#399—21no—
108#397—21no—
109#394—22no—
110#390—21no—
111#389—21no—
112#380C2×D311yescharacteristic
113C4×C22⋊C4—31no—
114#339—62no—
115#327—62no—
116C24×C3—21no—
117C24×C3C2×D411yescharacteristic
118C23×D3—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—42no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—82no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—42no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C2×#67—42no—
137C2×#67—42no—
138C2×#67—41no—
139C2×#67—41no—
140C22×Dic3—21no—
141C22×Dic3—41no—
142C2×D12—21no—
143C2×D12—21no—
144C2×D12—42no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4—82no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—42no—
158C3×C22⋊C4—82no—
159C3×C22⋊C4—82no—
160C3×C42—41no—
161C3×C42—41no—
162#217—22no—
163#217—41no—
164#213—82no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4×Dic3—42no—
170#208—42no—
171#208—42no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C22×D4—21no—
175C22×D4—31no—
176C22×D4—61no—
177C22×D4#6711yescharacteristic
178C23×C4C22×D311yescharacteristic
179C2×D8—62no—
180C2×M16—32no—
181C2×M16—61no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121—61no—
187#121—61no—
188#121—61no—
189#121—61no—
190#117—21no—
191#117—21no—
192#117—21no—
193#117—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4#6711yescharacteristic
198C2×C4⋊C4#6711yescharacteristic
199C2×C4⋊C4#6711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4#6712yes—
207C2×C42—21no—
208C2×C42—61no—
209#107—62no—
210#106—62no—
211#106—62no—
212#102—62no—
213#102—62no—
214#102—62no—
215#102—62no—
216#99—124no—
217C22⋊C8—32no—
218C22⋊C8—61no—
219C22⋊C8—61no—
220C22⋊C8—61no—
221#95—22no—
222#95—21no—
223#95—61no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3#12011yescharacteristic
234C23×C3#12111yescharacteristic
235C23×C3#12711yescharacteristic
236C22×D3—41no—
237C22×D3—42no—
238C22×D3—81no—
239C22×D3—81no—
240C3×D4—42no—
241C3×D4—81no—
242C3×D4—82no—
243C3×D4—82no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265#67—82no—
266#67—81no—
267#67—81no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—41no—
271C2×Dic3—81no—
272C2×Dic3—81no—
273D12—42no—
274D12—42no—
275C3⋊C8—42no—
276C3⋊C8—42no—
277C24—21no—
278C24—61no—
279C24C2×#6711yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—82no—
288C2×D4—122no—
289C2×D4—122no—
290C2×D4—122no—
291C2×D4—121no—
292C2×D4—121no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—42no—
298C22×C4—61no—
299C22×C4—121no—
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302C22×C4D3×D411yescharacteristic
303C22×C4#23711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305C22×C4C2×#6711yescharacteristic
306D8—242no—
307M16—122no—
308M16—122no—
309C2×C8—62no—
310C2×C8—62no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—82no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—62no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—121no—
330C22⋊C4—121no—
331C22⋊C4—121no—
332C42—41no—
333C42—41no—
334C42—122no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—82no—
346C22×C3—82no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3#39211yescharacteristic
351C22×C3#45311yescharacteristic
352C22×C3#45411yescharacteristic
353C2×D3—81no—
354C2×D3—81no—
355C2×D3—81no—
356C2×D3—162no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—82no—
361C3×C4—81no—
362C3×C4—81no—
363C3×C4—81no—
364C3×C4#45812yes—
365Dic3—82no—
366Dic3—81no—
367C23—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—122no—
377C23—121no—
378C23—241no—
379C23—241no—
380C23#93811yescharacteristic
381C23#94011yescharacteristic
382C23#95311yescharacteristic
383D4—42no—
384D4—81no—
385D4—82no—
386D4—82no—
387D4—122no—
388D4—122no—
389D4—242no—
390D4—241no—
391D4—241no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—121no—
412C2×C4—241no—
413C2×C4—241no—
414C2×C4#93711yescharacteristic
415C2×C4#93711yescharacteristic
416C2×C4#93911yescharacteristic
417C2×C4#93911yescharacteristic
418C8—122no—
419C8—122no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—161no—
425C2×C3#200611yescharacteristic
426C2×C3#200811yescharacteristic
427C2×C3#202211yescharacteristic
428D3—161no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—81no—
438C22—81no—
439C22—82no—
440C22—82no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—241no—
445C22—241no—
446C22—241no—
447C22—482no—
448C22#522311yescharacteristic
449C22#524111yescharacteristic
450C22#524511yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—82no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—242no—
459C4—241no—
460C4#505312yes—
461C3#993211yescharacteristic, complemented, Sylow
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—161no—
467C2—481no—
468C2#7170611yescharacteristic
469C2#7185811yescharacteristic
470C2#7187911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2624C2
socleC22×C3#392
prime radical#82627C2
hypercenter#2624D3
hypocenterC3#9932
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2624

Complements

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

NH ≅ G/Ntype
1. C3×#2624C2semidirect
3. #82217C2semidirect
4. #71673C2semidirect
5. #71590C2semidirect
6. #69528C2semidirect
7. #67626C2semidirect
36. #2624D3semidirect
461. C3#9932semidirect

Maximal subgroups

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

subgroupindex
C3×#26242
#826272
#822172
#716732
#715902
#695282
#676262
#99323