Finite Group Databasebeta

← #10697674

Subgroups — #10697674

Order 768.

528 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3246C211yesmaximal, characteristic
2#83367C211yesmaximal, characteristic, complemented
3#83314C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic, complemented
5#71802C211yesmaximal, characteristic
6#70472C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#10257—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C2×#951—21no—
12C2×C4×Dic6C2211yescharacteristic
13C3×#476—21no—
14C3×#475—21no—
15C3×#465—22no—
16C3×#461—21no—
17C3×#460—21no—
18C2×C3×C4⋊C8C2211yescharacteristic
19#5232—21no—
20#5094—21no—
21#5091—21no—
22#5090—21no—
23#5075—21no—
24#5063—22no—
25#5054—22no—
26#5004—21no—
27#4995—21no—
28#4924C2211yescharacteristic
29#4924C2211yescharacteristic
30C2×C4×C3⋊C8C2211yescharacteristic
31#4550—22no—
32#4535—22no—
33D4⋊(C3⋊C8)—22no—
34#3246D311yescharacteristic
35#3078—31no—
36#3037—31no—
37#1984—31no—
38#1493—31no—
39#1492—31no—
40#1292—31no—
41C2×C3×#41—21no—
42C22×Dic6D411yescharacteristic
43C2×C3×SD16—22no—
44C22×C3×C8D412yes—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#127D411yescharacteristic
49C3×#125—21no—
50C3×#124—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×#117—21no—
60C2×C3×C42C2311yescharacteristic
61#951—42no—
62#951—42no—
63C2×#216—22no—
64C2×#214—22no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×Dic3⋊C4—21no—
67C2×C4×Dic3—21no—
68C2×#208—21no—
69C2×#208—21no—
70C22×C3⋊C8—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C4×Dic6—21no—
75C3×#106—22no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8—21no—
78C3×#103—22no—
79C3×#102—22no—
80#835—22no—
81#832—22no—
82#810—22no—
83#810—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)D412yes—
89C4⋊(C3⋊C8)D412yes—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92C2×#137—61no—
93#533C2×D311yescharacteristic
94#532C2×D311yescharacteristic
95C2×C4×Q8—31no—
96#479—62no—
97#477—61no—
98#476—21no—
99#475—21no—
100#475—61no—
101#465—22no—
102#464—62no—
103#462—61no—
104#461—21no—
105#460—21no—
106#460—61no—
107#428—61no—
108#423—31no—
109#423—31no—
110C2×C4⋊C8C2×D311yescharacteristic
111#419—61no—
112#417—61no—
113C2×C4×C8—31no—
114#336—62no—
115Q8⋊C8—62no—
116D4⋊C8—62no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—21no—
133C22×Dic3—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6SD1612yescomplemented
140C3×SD16—82no—
141C2×C3×C8—22no—
142C2×C3×C8—42no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159#216—44no—
160#214—44no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165C4×Dic3—41no—
166C4×Dic3—41no—
167#208—41no—
168#208—41no—
169#208—41no—
170#208—41no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×#41—21no—
177C22×Q8—31no—
178#137—122no—
179#137—122no—
180C2×Q16—62no—
181C2×SD16—22no—
182C2×SD16—62no—
183C2×M16—61no—
184C2×M16—61no—
185C22×C8—61no—
186C22×C8#6712yes—
187C4⋊Q8#6711yescharacteristic
188C4⋊Q8#6711yescharacteristic
189C4⋊Q8#6711yescharacteristic
190#127#6711yescharacteristic
191#125—21no—
192#124—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195#121—21no—
196#121—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—61no—
200C4×Q8—61no—
201C4×Q8—61no—
202C4×Q8—61no—
203C4×D4—21no—
204C4×D4—21no—
205#117—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C42—61no—
209C2×C42C22×D311yescharacteristic
210#107—62no—
211#106—22no—
212#106—62no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—32no—
216C4⋊C8—32no—
217C4⋊C8—61no—
218C4⋊C8—61no—
219#103—22no—
220#103—62no—
221#103—62no—
222#103—62no—
223#102—22no—
224#102—62no—
225C4×C8—61no—
226C4×C8—61no—
227C23×C3—41no—
228C23×C3#12111yescharacteristic
229C3×Q8—42no—
230C3×Q8—41no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×D4—42no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—41no—
238C3×D4—41no—
239C3×D4—81no—
240C3×D4—81no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12711yescharacteristic
255C2×C3×C4C2×SD1611yescharacteristic
256C2×C3×C4#13511yescharacteristic
257C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
258C2×C3×C4#13711yescharacteristic
259C2×Dic3—22no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263C2×Dic3—41no—
264Dic6—22no—
265Dic6—22no—
266Dic6—41no—
267Dic6—41no—
268C3×C8—42no—
269C3⋊C8—42no—
270C3⋊C8—42no—
271C3⋊C8—42no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—32no—
280C2×Q8—61no—
281C2×Q8—61no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—41no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4D3×D411yescharacteristic
293C22×C4#23711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295Q16—124no—
296SD16—82no—
297SD16—124no—
298M16—121no—
299M16—121no—
300M16—121no—
301M16—121no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—62no—
307C2×C8—121no—
308C2×C8—121no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—41no—
325C42—121no—
326C42—121no—
327C42D3×D411yescharacteristic
328C42D3×D411yescharacteristic
329C42C2×#6711yescharacteristic
330C42C2×#6711yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3#39211yescharacteristic
337C22×C3#46511yescharacteristic
338C22×C3#46811yescharacteristic
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—81no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4#44811yescharacteristic
346C3×C4#44911yescharacteristic
347C3×C4#45011yescharacteristic
348C3×C4#45211yescharacteristic
349C3×C4#45511yescharacteristic
350C3×C4#45611yescharacteristic
351Dic3—42no—
352Dic3—81no—
353C23—41no—
354C23#93811yescharacteristic
355Q8—42no—
356Q8—41no—
357Q8—41no—
358Q8—41no—
359Q8—41no—
360Q8—62no—
361Q8—62no—
362Q8—121no—
363Q8—121no—
364D4—42no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—81no—
370D4—81no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—62no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—122no—
383C2×C4—121no—
384C2×C4D3×SD1611yescharacteristic
385C2×C4#91411yescharacteristic
386C2×C4#91511yescharacteristic
387C2×C4#91611yescharacteristic
388C2×C4#93711yescharacteristic
389C2×C4#93711yescharacteristic
390C2×C4#93911yescharacteristic
391C2×C4#93911yescharacteristic
392C2×C4#94011yescharacteristic
393C2×C4#95311yescharacteristic
394C8—42no—
395C8—122no—
396C8—122no—
397C8—122no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—161no—
401C2×C3#200011yescharacteristic
402C2×C3#200711yescharacteristic
403C2×C3#201711yescharacteristic
404C22—21no—
405C22—21no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22#517111yescharacteristic
410C22#517411yescharacteristic
411C22#522311yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—81no—
416C4—81no—
417C4—81no—
418C4—122no—
419C4—241no—
420C4#516911yescharacteristic
421C4#517011yescharacteristic
422C4#517211yescharacteristic
423C4#517311yescharacteristic
424C4#524211yescharacteristic
425C4#524611yescharacteristic
426C3#1025711yescharacteristic, complemented, Sylow
427C2—21no—
428C2—21no—
429C2—161no—
430C2#7188111yescharacteristic
431C2#7205411yescharacteristic
432C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3246C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3246D3
hypocenterC3#10257
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
2. #83367C2semidirect
4. #71826C2semidirect
6. #70472C2semidirect
7. #67705C2semidirect
139. C2×Dic6SD16semidirect
426. C3#10257semidirect

Maximal subgroups

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

subgroupindex
C3×#32462
#833672
#833142
#718262
#718022
#704722
#677052
#102573