Finite Group Databasebeta

← #10695507

Subgroups — #10695507

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×#3192C211yesmaximal, characteristic
2#82678C211yesmaximal, characteristic, complemented
3#82616C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic, complemented
6#69975C211yesmaximal, characteristic, complemented
7#67684C211yesmaximal, characteristic, complemented
8#9932—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C3×#481—21no—
13C3×#481—21no—
14C3×#469—22no—
15C3×#447—21no—
16C3×#447—21no—
17C3×#407C2211yescharacteristic
18#5222—22no—
19#5221—21no—
20C2×#834—22no—
21#5215—21no—
22#5212C2211yescharacteristic
23C4×#217C2211yescharacteristic
24#5207C2211yescharacteristic
25#5035—22no—
26#5034—21no—
27#5033—21no—
28#4972—21no—
29#4964—21no—
30#4953—21no—
31#4949—21no—
32#4555—22no—
33#4537—22no—
34#3192D311yescharacteristic
35#3094—31no—
36#2624—31no—
37#1964—31no—
38#1826—31no—
39#1507—31no—
40#1305—31no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C22×C3×D4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C2×C3×D8—22no—
47C2×C3×M16D412yes—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×#121—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217—22no—
63C2×#217—22no—
64C2×#217D412yes—
65#926—21no—
66#926—21no—
67#926—21no—
68#926—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4⋊Dic3D411yescharacteristic
72C2×Dic3⋊C4—22no—
73C2×C4×Dic3—21no—
74C2×C4×Dic3—21no—
75C2×#208—21no—
76C3×#106—22no—
77C3×#102—22no—
78C3×#102—22no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81#834—44no—
82#832—22no—
83#832—22no—
84C3⋊#95—22no—
85C3⋊#95—21no—
86C3⋊#95—21no—
87#830—21no—
88#830D412yes—
89#808—22no—
90#807—22no—
91C2×#121C2×D311yescharacteristic
92C2×#121C2×D311yescharacteristic
93C2×#117—31no—
94#482—62no—
95#481—21no—
96#481—21no—
97#481—61no—
98#480—61no—
99#469—22no—
100#447—21no—
101#447—21no—
102#428—61no—
103#418—61no—
104C2×#99—62no—
105#407—31no—
106#407C2×D311yescharacteristic
107#399—61no—
108#397—61no—
109#394—62no—
110#390—61no—
111#389—61no—
112#380—31no—
113C4×C22⋊C4—31no—
114#339—62no—
115#327—62no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3C2×D411yescharacteristic
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—82no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—42no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—41no—
145C3×D8—82no—
146C3×M16—42no—
147C2×C3×C8—22no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156#217—22no—
157#217—41no—
158#217—41no—
159#217—41no—
160#217—42no—
161#217—82no—
162#217—82no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—42no—
168Dic3⋊C4—82no—
169C4×Dic3—41no—
170C4×Dic3—41no—
171C4×Dic3—42no—
172#208—42no—
173C2×C3⋊C8—22no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4#6711yescharacteristic
177C22×D4#6711yescharacteristic
178C23×C4C22×D311yescharacteristic
179C2×D8—22no—
180C2×M16—61no—
181C2×M16#6712yes—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121—21no—
187#121—21no—
188#121—21no—
189#121—21no—
190#117—61no—
191#117—61no—
192#117—61no—
193#117—61no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4#6711yescharacteristic
199C2×C4⋊C4#6711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—32no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—62no—
207C2×C42—61no—
208C2×C42—61no—
209#107—62no—
210#106—22no—
211#106—62no—
212#102—22no—
213#102—22no—
214#102—62no—
215#102—62no—
216#99—124no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—32no—
220C22⋊C8—61no—
221#95—62no—
222#95—61no—
223#95—61no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—81no—
236C23×C3—81no—
237C23×C3#12011yescharacteristic
238C23×C3#12111yescharacteristic
239C23×C3#12711yescharacteristic
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—81no—
244C3×D4—82no—
245C3×D4—82no—
246C3×D4—82no—
247C3×D4—81no—
248C3×D4—81no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×Dic3—41no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275C3×C8—42no—
276C3⋊C8—42no—
277C24—21no—
278C24—21no—
279C24D3×D411yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—82no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—61no—
297C22×C4—62no—
298C22×C4—61no—
299C22×C4—121no—
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302C22×C4#23711yescharacteristic
303C22×C4C2×#6711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305C22×C4C2×#6711yescharacteristic
306D8—82no—
307M16—42no—
308M16—122no—
309C2×C8—22no—
310C2×C8—62no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—122no—
320C4⋊C4—242no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—62no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C22⋊C4—122no—
330C22⋊C4—242no—
331C22⋊C4—242no—
332C42—122no—
333C42—121no—
334C42—121no—
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—81no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—162no—
354C22×C3#39211yescharacteristic
355C22×C3#45311yescharacteristic
356C22×C3#45411yescharacteristic
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—81no—
361C3×C4—81no—
362C3×C4#45812yes—
363Dic3—82no—
364Dic3—82no—
365Dic3—81no—
366Dic3—81no—
367C23—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—81no—
379C23—81no—
380C23#93711yescharacteristic
381C23#93911yescharacteristic
382C23#94011yescharacteristic
383D4—42no—
384D4—42no—
385D4—42no—
386D4—81no—
387D4—82no—
388D4—82no—
389D4—82no—
390D4—81no—
391D4—81no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—122no—
403C2×C4—121no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—121no—
407C2×C4—122no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—241no—
413C2×C4—241no—
414C2×C4#93711yescharacteristic
415C2×C4#93811yescharacteristic
416C2×C4#93911yescharacteristic
417C2×C4#95311yescharacteristic
418C8—42no—
419C8—122no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—161no—
425C2×C3—161no—
426C2×C3#200611yescharacteristic
427C2×C3#200811yescharacteristic
428C2×C3#202211yescharacteristic
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—81no—
445C22—81no—
446C22—81no—
447C22—162no—
448C22#519811yescharacteristic
449C22#520111yescharacteristic
450C22#522311yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—81no—
455C4—81no—
456C4—242no—
457C4—242no—
458C4—241no—
459C4—241no—
460C4#524312yes—
461C3#993211yescharacteristic, complemented, Sylow
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—161no—
467C2—161no—
468C2#7184911yescharacteristic
469C2#7209811yescharacteristic
470C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3192C2
socleC22×C3#392
hypercenter#3192D3
hypocenterC3#9932
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3192

Complements

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

NH ≅ G/Ntype
2. #82678C2semidirect
3. #82616C2semidirect
4. #71686C2semidirect
5. #71673C2semidirect
6. #69975C2semidirect
7. #67684C2semidirect
461. C3#9932semidirect

Maximal subgroups

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

subgroupindex
C3×#31922
#826782
#826162
#716862
#716732
#699752
#676842
#99323