Finite Group Databasebeta

← #10695956

Subgroups — #10695956

Order 768.

548 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3104C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3#82501C211yesmaximal, characteristic
4#71838C211yesmaximal, characteristic, complemented
5#71622C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic
7#67738C211yesmaximal, characteristic, complemented
8#10002—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#484—21no—
13C3×#477—21no—
14C3×#462—21no—
15C3×#451—21no—
16C3×C4×Q16—22no—
17C3×#431C2211yescharacteristic
18#5231—21no—
19#5087—21no—
20#5084—21no—
21#5066—22no—
22#5001—21no—
23#4995—21no—
24#4993—21no—
25#4989—22no—
26#4939—21no—
27#4936C2211yescharacteristic
28#4928—21no—
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4538—22no—
32#4535—22no—
33#4459—22no—
34#3104D311yescharacteristic, complemented
35#3077—31no—
36#2580—31no—
37#1518—31no—
38#1487—31no—
39#1478—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×Q16—22no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×#117—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—22no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×Dic3⋊C4—22no—
67C2×#208—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C3×#107—22no—
73C3×C4⋊C8—21no—
74C3×#103—22no—
75C3×#103—22no—
76C3×C22⋊C8—21no—
77C3×C4×C8D412yes—
78C3⋊#95—22no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#830—21no—
82#809—22no—
83#809—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)D412yes—
89C4×C3⋊C8—21no—
90#533C2×D311yescharacteristic
91C2×C4×Q8C2×D311yescharacteristic
92C2×C4×D4—31no—
93#484—21no—
94#478—62no—
95#477—21no—
96#476—61no—
97#474—61no—
98#462—21no—
99#451—21no—
100C4×Q16—22no—
101#431—31no—
102#431C2×D311yescharacteristic
103#428—61no—
104#423—31no—
105#418—61no—
106#399—61no—
107#393—61no—
108#390—61no—
109#389—61no—
110#385—62no—
111#382—31no—
112#328—62no—
113Q8⋊C8—62no—
114D4⋊C8—62no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—22no—
127C22×Dic3—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12Q1612yescomplemented
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—82no—
137C3×Q16—44no—
138C2×C3×C8—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42—41no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—82no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—41no—
164C4⋊Dic3—41no—
165Dic3⋊C4—82no—
166#208—41no—
167#208—41no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—41no—
170C22×Q8#6711yescharacteristic
171C22×D4—31no—
172C23×C4—61no—
173C2×Q16—22no—
174C2×M16—61no—
175C4⋊Q8#6711yescharacteristic
176C4⋊Q8#6711yescharacteristic
177#125—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190#117—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4#6711yescharacteristic
196C2×C22⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—62no—
199C2×C42—21no—
200C2×C42C22×D311yescharacteristic
201#107—22no—
202#107—62no—
203#106—62no—
204C4⋊C8—21no—
205C4⋊C8—32no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#103—22no—
209#103—22no—
210#102—62no—
211#102—62no—
212C22⋊C8—21no—
213C22⋊C8—61no—
214C4×C8—61no—
215C4×C8#6712yes—
216#95—32no—
217#95—62no—
218#95—61no—
219C23×C3#12011yescharacteristic
220C22×D3—22no—
221C22×D3—42no—
222C22×D3—42no—
223C22×D3—41no—
224C22×D3—41no—
225C3×Q8—22no—
226C3×Q8—22no—
227C3×Q8—41no—
228C3×Q8—41no—
229C3×Q8—42no—
230C3×Q8—81no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12711yescharacteristic
247C2×C3×C4C2×Q1611yescharacteristic
248C2×C3×C4#13511yescharacteristic
249C2×C3×C4#13711yescharacteristic
250C2×C3×C4#13711yescharacteristic
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256D12—22no—
257D12—22no—
258D12—41no—
259D12—41no—
260C4×D3—82no—
261C4×D3—82no—
262C3×C8—42no—
263C3⋊C8—42no—
264C3⋊C8—81no—
265C24—61no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—42no—
271C2×D4—32no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—62no—
280C22×C4—61no—
281C22×C4—121no—
282C22×C4—121no—
283C22×C4—242no—
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4C2×#6711yescharacteristic
287Q16—44no—
288M16—121no—
289M16—121no—
290C2×C8—22no—
291C2×C8—62no—
292C2×C8—121no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—242no—
308C22⋊C4—41no—
309C22⋊C4—124no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—242no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42D3×D411yescharacteristic
317C42#23711yescharacteristic
318C42C2×#6711yescharacteristic
319C42C2×#6711yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3#39211yescharacteristic
323C22×C3#45111yescharacteristic
324C22×C3#45211yescharacteristic
325C2×D3—44no—
326C2×D3—42no—
327C2×D3—82no—
328C2×D3—81no—
329C2×D3—81no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—42no—
335C3×C4—81no—
336C3×C4—81no—
337C3×C4—81no—
338C3×C4#45111yescharacteristic
339C3×C4#45211yescharacteristic
340C3×C4#46711yescharacteristic
341C3×C4#47011yescharacteristic
342Dic3—82no—
343Dic3—81no—
344C23—62no—
345C23—122no—
346C23—122no—
347C23—121no—
348C23—121no—
349C23#93711yescharacteristic
350Q8—22no—
351Q8—22no—
352Q8—41no—
353Q8—41no—
354Q8—42no—
355Q8—81no—
356D4—62no—
357D4—62no—
358D4—121no—
359D4—121no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—121no—
375C2×C4—242no—
376C2×C4—242no—
377C2×C4D3×Q1611yescharacteristic
378C2×C4#91811yescharacteristic
379C2×C4#91811yescharacteristic
380C2×C4#91911yescharacteristic
381C2×C4#93711yescharacteristic
382C2×C4#93811yescharacteristic
383C2×C4#93911yescharacteristic
384C2×C4#93911yescharacteristic
385C2×C4#94011yescharacteristic
386C2×C4#95311yescharacteristic
387C8—42no—
388C8—122no—
389C8—241no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3#200111yescharacteristic
393C2×C3#200811yescharacteristic
394C2×C3#201011yescharacteristic
395D3—82no—
396C22—21no—
397C22—21no—
398C22—124no—
399C22—122no—
400C22—242no—
401C22—241no—
402C22—241no—
403C22#518611yescharacteristic
404C22#518711yescharacteristic
405C22#522311yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—242no—
415C4—241no—
416C4#518611yescharacteristic
417C4#518711yescharacteristic
418C4#518811yescharacteristic
419C4#518911yescharacteristic
420C3#1000211yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—242no—
424C2#7171511yescharacteristic
425C2#7207111yescharacteristic
426C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3104C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3104D3
hypocenterC3#10002
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
1. C3×#3104C2semidirect
4. #71838C2semidirect
5. #71622C2semidirect
7. #67738C2semidirect
34. #3104D3semidirect
133. C2×D12Q16semidirect
420. C3#10002semidirect

Maximal subgroups

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

subgroupindex
C3×#31042
#833502
#825012
#718382
#716222
#707572
#677382
#100023