Finite Group Databasebeta

← #10696001

Subgroups — #10696001

Order 768.

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

normal subgroups only

This group has 654 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×#2540C211yesmaximal, characteristic
2#82949C211yesmaximal, characteristic, complemented
3#82315C211yesmaximal, characteristic, complemented
4#71718C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic, complemented
6#69493C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#10008—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C3×C22⋊Q8—21no—
11C2×C3×#121—21no—
12C2×C3×C4×D4C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#399—21no—
15C3×#392—21no—
16C3×#390—22no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5228—21no—
19#5218—21no—
20#5218—21no—
21#5060—21no—
22#5055—21no—
23#5054—22no—
24#5019—21no—
25#5018—22no—
26#5007—21no—
27#4970—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4540—22no—
32D4⋊(C3⋊C8)—22no—
33#4523—22no—
34#3215—31no—
35#3110—31no—
36#2540D311yescharacteristic
37#1986—31no—
38#1527—31no—
39#1490—31no—
40#1285—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4—22no—
43C22×C3×D4D411yescharacteristic
44C22×C3×D4D411yescharacteristic
45C23×C3×C4—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×C22⋊Q8—44no—
51C3×#121—44no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C42D412yes—
65C2×C3×C42C2311yescharacteristic
66C2×#216—22no—
67C2×#214—22no—
68C2×#214—22no—
69#926—21no—
70#924—21no—
71#924—21no—
72C2×#208—21no—
73#870—21no—
74#869D411yescharacteristic, complemented
75#869D411yescharacteristic, complemented
76C4×Dic6—21no—
77C4×Dic6—21no—
78C3×#95—21no—
79#835—22no—
80#832—22no—
81#832—22no—
82#830—21no—
83#830—21no—
84#810—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yescomplemented
88#803—21no—
89#803—21no—
90#803D412yescomplemented
91C4×C3⋊C8—21no—
92#533—31no—
93C2×#124C2×D311yescharacteristic
94C2×C22⋊Q8—21no—
95C2×#121—21no—
96C2×C4×D4C2×D311yescharacteristic
97#497—61no—
98#489—61no—
99#484—61no—
100#474—61no—
101#464—62no—
102#461—61no—
103#451—61no—
104#450—61no—
105#450—61no—
106#440—62no—
107#432—31no—
108#432—31no—
109C4×M16—31no—
110#399—21no—
111#392—21no—
112#390—22no—
113C4×C4⋊C4C2×D311yescharacteristic
114#339—62no—
115#329—62no—
116D4⋊C8—62no—
117C24×C3—21no—
118C24×C3—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—42no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—42no—
127C2×C3×D4—44no—
128C2×C3×D4—42no—
129C2×C3×D4—44no—
130C2×C3×D4—82no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C2×Dic6—21no—
142C2×Dic6—21no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—44no—
147C3×C22⋊C4—44no—
148C3×C22⋊C4—44no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C42—42no—
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42#4111yescharacteristic
160#217—41no—
161#216—82no—
162#214—44no—
163#214—82no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170C4×Dic3—41no—
171#208—22no—
172#208—41no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—41no—
176C22×Q8#6711yescharacteristic
177C22×D4—22no—
178C22×D4#6711yescharacteristic
179C22×D4#6711yescharacteristic
180C23×C4—21no—
181C2×Q16—62no—
182C2×SD16—62no—
183C2×SD16—62no—
184C2×M16—61no—
185C4⋊Q8—31no—
186C4⋊Q8—31no—
187#125—61no—
188#124—21no—
189#124—21no—
190#124—21no—
191#124—21no—
192C22⋊Q8—44no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195#121—44no—
196C4×Q8—61no—
197C4×Q8—61no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202#117—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4#6711yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42#6712yes—
212C2×C42C22×D311yescharacteristic
213#107—62no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216#103—62no—
217#103—62no—
218#102—62no—
219#102—62no—
220C22⋊C8—61no—
221C22⋊C8—61no—
222#97—32no—
223#97—61no—
224#97—61no—
225C4×C8—61no—
226#95—21no—
227C23×C3—22no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3#12011yescharacteristic
240C3×Q8—42no—
241C3×Q8—81no—
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—42no—
247C3×D4—81no—
248C3×D4—88no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—82no—
262C2×C3×C4—82no—
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12711yescharacteristic
269C2×Dic3—41no—
270C2×Dic3—41no—
271Dic6—42no—
272Dic6—42no—
273C3⋊C8—42no—
274C3⋊C8—42no—
275C3⋊C8—42no—
276C24—21no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—42no—
281C2×Q8—61no—
282C2×Q8—61no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—42no—
288C2×D4—44no—
289C2×D4—42no—
290C2×D4—44no—
291C2×D4—82no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C2×#6711yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302Q16—242no—
303SD16—124no—
304SD16—242no—
305M16—62no—
306M16—121no—
307C2×C8—62no—
308C2×C8—62no—
309C2×C8—121no—
310C4⋊C4—24no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—44no—
314C4⋊C4—61no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—121no—
319C4⋊C4—121no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—121no—
329C42—42no—
330C42—121no—
331C42D3×D411yescharacteristic
332C42D3×D411yescharacteristic
333C42D3×D411yescharacteristic
334C42#23711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3#39211yescharacteristic
350C22×C3#45511yescharacteristic
351C22×C3#45611yescharacteristic
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—44no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4#45312yes—
361Dic3—81no—
362Dic3—81no—
363C23—22no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23#95311yescharacteristic
376Q8—42no—
377Q8—81no—
378Q8—122no—
379Q8—122no—
380D4—22no—
381D4—22no—
382D4—41no—
383D4—41no—
384D4—42no—
385D4—81no—
386D4—88no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4#93711yescharacteristic
404C2×C4#93711yescharacteristic
405C2×C4#93811yescharacteristic
406C2×C4#93911yescharacteristic
407C2×C4#93911yescharacteristic
408C2×C4#94011yescharacteristic
409C8—122no—
410C8—122no—
411C8—122no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—82no—
415C2×C3—161no—
416C2×C3#200211yescharacteristic
417C2×C3#200711yescharacteristic
418C2×C3#201011yescharacteristic
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—81no—
425C22—81no—
426C22—82no—
427C22—82no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22#522311yescharacteristic
434C22#524211yescharacteristic
435C22#524611yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—44no—
442C4—81no—
443C4—81no—
444C4—241no—
445C4—241no—
446C4#507712yes—
447C3#1000811yescharacteristic, complemented, Sylow
448C2—21no—
449C2—21no—
450C2—82no—
451C2—161no—
452C2#7171511yescharacteristic
453C2#7176911yescharacteristic
454C2#7188111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2540C2
socleC22×C3#392
hypercenter#2540D3
hypocenterC3#10008
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2540

Complements

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

NH ≅ G/Ntype
2. #82949C2semidirect
3. #82315C2semidirect
4. #71718C2semidirect
5. #71599C2semidirect
6. #69493C2semidirect
7. #67621C2semidirect
74. #869D4semidirect
75. #869D4semidirect
87. C4⋊(C3⋊C8)D4semidirect
90. #803D4semidirect
447. C3#10008semidirect

Maximal subgroups

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

subgroupindex
C3×#25402
#829492
#823152
#717182
#715992
#694932
#676212
#100083