Finite Group Databasebeta

← #10695516

Subgroups — #10695516

Order 768.

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

normal subgroups only

This group has 568 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×#2625C211yesmaximal, characteristic, complemented
2#82738C211yesmaximal, characteristic
3#82217C211yesmaximal, characteristic
4#71688C211yesmaximal, characteristic, complemented
5#71590C211yesmaximal, characteristic, complemented
6#69528C211yesmaximal, characteristic
7#67626C211yesmaximal, characteristic, complemented
8#9933—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C3×#400—21no—
13C3×#398—21no—
14C3×#397—22no—
15C3×#391—21no—
16C3×#389—21no—
17C3×#380C2211yescharacteristic
18#5227—21no—
19C4×#217C2211yescharacteristic
20#5207C2211yescharacteristic
21#5207C2211yescharacteristic
22#5049—22no—
23#5046—21no—
24#5044—21no—
25#5003—22no—
26#5001—21no—
27#4995—21no—
28#4968—21no—
29#4967—21no—
30#4967—21no—
31C2×#806—22no—
32#4538—22no—
33#4532—22no—
34#3193—31no—
35#3094—31no—
36#2625D311yescharacteristic, complemented
37#1964—31no—
38#1826—31no—
39#1508—31no—
40#1305—31no—
41C22×C3×Q8D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×D12D411yescharacteristic
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×#117—21no—
50C3×#117—21no—
51C3×#117—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—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⋊C4D412yes—
62C2×C3×C42—21no—
63C2×#217—21no—
64C2×#215—22no—
65#930—21no—
66#930—21no—
67#930—21no—
68#930—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4×Dic3—21no—
72C2×#208—21no—
73C2×#208D412yes—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—22no—
77C3×#95—21no—
78#835—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#807—22no—
89#807—22no—
90#806—44no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C2×#121—31no—
93C2×#117C2×D311yescharacteristic
94#484—61no—
95#482—62no—
96#481—61no—
97#480—61no—
98#480—61no—
99#468—62no—
100#450—61no—
101#448—61no—
102#428—61no—
103#418—61no—
104C2×#99—62no—
105#407—31no—
106#407—31no—
107#400—21no—
108#398—21no—
109#397—22no—
110#391—21no—
111#389—21no—
112#380C2×D311yescharacteristic
113C4×C22⋊C4—31no—
114#339—62no—
115#328—62no—
116C24×C3C2×D411yescharacteristic
117C23×D3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C2×#67—42no—
134C2×#67—42no—
135C2×#67—41no—
136C2×#67—41no—
137C22×Dic3—21no—
138C22×Dic3—41no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—42no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—42no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—82no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—42no—
157C3×C42—41no—
158C3×C42—41no—
159#217—22no—
160#217—41no—
161#215—82no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4×Dic3—42no—
167#208—42no—
168#208—42no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C22×Q8#6711yescharacteristic
172C22×D4—31no—
173C22×D4—61no—
174C23×C4C22×D311yescharacteristic
175C2×SD16—62no—
176C2×M16—32no—
177C2×M16—61no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186#117—21no—
187#117—21no—
188#117—21no—
189#117—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C4⋊C4#6711yescharacteristic
196C2×C4⋊C4#6711yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4#6712yes—
202C2×C42—21no—
203C2×C42—61no—
204#107—62no—
205#107—62no—
206#106—62no—
207#103—62no—
208#102—62no—
209#102—62no—
210#102—62no—
211#99—124no—
212C22⋊C8—32no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95—61no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3#12011yescharacteristic
226C23×C3#12111yescharacteristic
227C23×C3#12711yescharacteristic
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—81no—
231C22×D3—81no—
232C3×Q8—42no—
233C3×Q8—81no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12111yescharacteristic
257#67—82no—
258#67—81no—
259#67—81no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—41no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265D12—42no—
266D12—42no—
267C3⋊C8—42no—
268C3⋊C8—42no—
269C24—61no—
270C24C2×#6711yescharacteristic
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—122no—
277C2×D4—122no—
278C2×D4—122no—
279C2×D4—121no—
280C2×D4—121no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—61no—
288C22×C4—121no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4#23711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295SD16—242no—
296M16—122no—
297M16—122no—
298C2×C8—62no—
299C2×C8—62no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—82no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—62no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C42—41no—
322C42—41no—
323C42—122no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#39211yescharacteristic
335C22×C3#45311yescharacteristic
336C22×C3#45411yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—162no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—82no—
345C3×C4—81no—
346C3×C4—81no—
347C3×C4—81no—
348C3×C4—81no—
349C3×C4#45812yes—
350Dic3—82no—
351Dic3—81no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—122no—
357C23—121no—
358C23—241no—
359C23—241no—
360C23#93811yescharacteristic
361C23#94011yescharacteristic
362C23#95311yescharacteristic
363Q8—42no—
364Q8—81no—
365D4—122no—
366D4—122no—
367D4—242no—
368D4—241no—
369D4—241no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—121no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4#93711yescharacteristic
395C2×C4#93711yescharacteristic
396C2×C4#93911yescharacteristic
397C2×C4#93911yescharacteristic
398C8—122no—
399C8—122no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3#200611yescharacteristic
405C2×C3#200911yescharacteristic
406C2×C3#202211yescharacteristic
407D3—161no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—81no—
417C22—81no—
418C22—241no—
419C22—241no—
420C22—241no—
421C22—482no—
422C22#522311yescharacteristic
423C22#524111yescharacteristic
424C22#524511yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—82no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4—242no—
434C4—241no—
435C4#505312yes—
436C3#993311yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—481no—
442C2#7170611yescharacteristic
443C2#7186911yescharacteristic
444C2#7187911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2625C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#2625D3
hypocenterC3#9933
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2625

Complements

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

NH ≅ G/Ntype
1. C3×#2625C2semidirect
4. #71688C2semidirect
5. #71590C2semidirect
7. #67626C2semidirect
36. #2625D3semidirect
436. C3#9933semidirect

Maximal subgroups

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

subgroupindex
C3×#26252
#827382
#822172
#716882
#715902
#695282
#676262
#99333