Finite Group Databasebeta

← #10695960

Subgroups — #10695960

Order 768.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#82501C211yesmaximal, characteristic
2#82447C211yesmaximal, characteristic, complemented
3#79059C211yesmaximal, characteristic
4C3×#1487C211yesmaximal, characteristic, complemented
5#70793C211yesmaximal, characteristic, complemented
6#70756C211yesmaximal, characteristic
7#67713C211yesmaximal, characteristic, complemented
8#10002—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5231—21no—
15#5228—21no—
16#5131—21no—
17#5121—21no—
18#5031—21no—
19C4×#216—22no—
20#5021—21no—
21#4993—21no—
22#4989—22no—
23#4970—21no—
24#4939—21no—
25#4936C2211yescharacteristic
26#4928—21no—
27#4925C2211yescharacteristic
28#4847—22no—
29#4846—21no—
30#4842—21no—
31C3×Q8⋊C8—22no—
32#4483—22no—
33#4467—22no—
34#3104—31no—
35#3077—31no—
36#2580—31no—
37#1518—31no—
38#1487D311yescharacteristic, complemented
39#1478—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#216—22no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—22no—
57#926—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D411yescharacteristic
60#924—21no—
61#924—21no—
62C2×Dic3⋊C4—22no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67#870—21no—
68#869D411yescharacteristic
69#869D411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yes—
75C3×C22⋊C8—21no—
76C3×C4×C8—21no—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#830—21no—
82#821—22no—
83#821—22no—
84#818—22no—
85#817—22no—
86#810—22no—
87#807—22no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91C2×C4×Q8C2×D311yescharacteristic
92C2×C4×D4—31no—
93#484—61no—
94#478—62no—
95#477—61no—
96#476—61no—
97#474—61no—
98#462—61no—
99#451—61no—
100C4×Q16—62no—
101#431—31no—
102#431C2×D311yescharacteristic
103#428—61no—
104#423C2×D311yescharacteristic
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—22no—
114D4⋊C8—62no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12Q1612yescomplemented
132C2×C4×D3—41no—
133C2×C4×D3—41no—
134C2×C4×D3—82no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C3×M16—41no—
138C3×M16—41no—
139C2×C3×C8—22no—
140C2×C3×C8—41no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#217—41no—
152#216—44no—
153(C2×D3)⋊C4—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—82no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—41no—
164C4⋊Dic3—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167Dic3⋊C4—82no—
168C4×Dic3—41no—
169C2×C3⋊C8—22no—
170C22×Q8D1211yescharacteristic
171C22×D4—31no—
172C23×C4—61no—
173C2×Q16—62no—
174C2×M16—21no—
175C4⋊Q8—31no—
176C4⋊Q8—31no—
177#125—61no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190#117—61no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4D1211yescharacteristic
196C2×C22⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—62no—
199C2×C42—21no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#107—62no—
203#106—62no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—61no—
207C4⋊C8#6712yes—
208#103—62no—
209#103—62no—
210#102—62no—
211#102—62no—
212C22⋊C8—21no—
213C22⋊C8—61no—
214C4×C8—21no—
215C4×C8—32no—
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—
229C2×C3×C4—22no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12711yescharacteristic
243C2×C3×C4C2×Q1611yescharacteristic
244C2×C3×C4#13511yescharacteristic
245C2×C3×C4#13711yescharacteristic
246C2×C3×C4#13711yescharacteristic
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254D12—22no—
255D12—22no—
256D12—41no—
257D12—41no—
258C4×D3—82no—
259C4×D3—82no—
260Dic6—42no—
261Dic6—81no—
262C3×C8—42no—
263C3×C8—81no—
264C3⋊C8—42no—
265C24—61no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—42no—
269C2×Q8—61no—
270C2×Q8—61no—
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×C4C2×D1211yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287Q16—124no—
288M16—41no—
289M16—41no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—62no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—121no—
304C4⋊C4—121no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—242no—
308C22⋊C4—124no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—242no—
313C42—41no—
314C42—41no—
315C42—121no—
316C42#23511yescharacteristic
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
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#45111yescharacteristic
337C3×C4#45211yescharacteristic
338C3×C4#46711yescharacteristic
339C3×C4#47011yescharacteristic
340Dic3—82no—
341Dic3—81no—
342Dic3—81no—
343Dic3—81no—
344C23—62no—
345C23—122no—
346C23—122no—
347C23—121no—
348C23—121no—
349C23#88211yescharacteristic
350Q8—22no—
351Q8—22no—
352Q8—41no—
353Q8—41no—
354Q8—122no—
355Q8—241no—
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—122no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—122no—
375C2×C4—242no—
376C2×C4—242no—
377C2×C4#88211yescharacteristic
378C2×C4#88411yescharacteristic
379C2×C4#88411yescharacteristic
380C2×C4#91511yescharacteristic
381C2×C4#91611yescharacteristic
382C2×C4D3×Q1611yescharacteristic
383C2×C4#91811yescharacteristic
384C2×C4#93011yescharacteristic
385C2×C4#93711yescharacteristic
386C2×C4#94011yescharacteristic
387C8—42no—
388C8—81no—
389C8—122no—
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#480811yescharacteristic
404C22#480911yescharacteristic
405C22#496011yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—81no—
412C4—242no—
413C4—241no—
414C4—241no—
415C4—241no—
416C4#481311yescharacteristic
417C4#481611yescharacteristic
418C4#517311yescharacteristic
419C4#518611yescharacteristic
420C3#1000211yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—242no—
424C2#6999311yescharacteristic
425C2#7027311yescharacteristic
426C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1487C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1487D3
hypocenterC3#10002
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
2. #82447C2semidirect
4. C3×#1487C2semidirect
5. #70793C2semidirect
7. #67713C2semidirect
38. #1487D3semidirect
131. 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
#825012
#824472
#790592
C3×#14872
#707932
#707562
#677132
#100023