Finite Group Databasebeta

← #10697682

Subgroups — #10697682

Order 768.

526 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 101 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#83373C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3#79059C211yesmaximal, characteristic
4C3×#1493C211yesmaximal, characteristic, complemented
5#69787C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#10258—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5131—21no—
15#5121—21no—
16C2×(C2×D3)⋊C8—22no—
17#5091—21no—
18#5090—21no—
19#5087—21no—
20#5084—21no—
21#5066—22no—
22#5051—22no—
23#5031—21no—
24#5021—21no—
25#5001—21no—
26#4995—21no—
27#4924C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4847—22no—
30#4846—21no—
31#4842—21no—
32C3×Q8⋊C8—22no—
33#4456—22no—
34#3249—31no—
35#3077—31no—
36#3077—31no—
37#1954—31no—
38#1493—31no—
39#1493D311yescharacteristic, complemented
40#1292—31no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×M16—21no—
44C22×C3×C8—21no—
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×#117—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#216—22no—
55C2×(C2×D3)⋊C4—21no—
56#926—21no—
57C2×C4⋊Dic3D411yescharacteristic
58#924—21no—
59#924—21no—
60C2×#208—21no—
61C22×C3⋊C8D412yes—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75#835—22no—
76#821—22no—
77#821—22no—
78(C2×D3)⋊C8—44no—
79#818—22no—
80#817—22no—
81#810—22no—
82#809—22no—
83#809—22no—
84#808—22no—
85#807—22no—
86#807—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)D412yes—
91#533—31no—
92#533C2×D311yescharacteristic
93C2×C4×D4—31no—
94#478—62no—
95#478—62no—
96#477—61no—
97#477—61no—
98#476—61no—
99#476—61no—
100#474—61no—
101#474—61no—
102#467—62no—
103#462—61no—
104#462—61no—
105#428—61no—
106#428—61no—
107#423—31no—
108#423C2×D311yescharacteristic
109C2×C4⋊C8—31no—
110#418—61no—
111#418—61no—
112C2×C22⋊C8—62no—
113C2×C4×C8C2×D311yescharacteristic
114Q8⋊C8—22no—
115Q8⋊C8—62no—
116C23×D3—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12Q1612yescomplemented
129C2×C4×D3—42no—
130C2×C4×D3—42no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
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×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153#217—41no—
154#216—82no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165C4×Dic3—41no—
166#208—41no—
167#208—41no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—42no—
171C22×D4—31no—
172C23×C4—61no—
173C2×Q16—62no—
174C2×M16—21no—
175C2×M16—61no—
176C22×C8—21no—
177C22×C8—32no—
178C4⋊Q8—31no—
179C4⋊Q8—31no—
180C4⋊Q8D1211yescharacteristic
181C4⋊Q8D1211yescharacteristic
182#125—21no—
183#125—61no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8—61no—
187C22⋊Q8—61no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—61no—
191C4×Q8—61no—
192C4×D4—61no—
193C4×D4—61no—
194C4×D4—61no—
195C4×D4—61no—
196#117—21no—
197#117—61no—
198C2×C4⋊C4—31no—
199C2×C22⋊C4—61no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#107—62no—
203#107—62no—
204#106—62no—
205#106—62no—
206C4⋊C8—21no—
207C4⋊C8—32no—
208C4⋊C8—61no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211C4⋊C8#6712yes—
212#103—62no—
213#103—62no—
214#102—62no—
215#102—62no—
216#102—62no—
217#102—62no—
218C22⋊C8—124no—
219C4×C8—21no—
220C4×C8—21no—
221C23×C3#12111yescharacteristic
222C22×D3—22no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3—41no—
226C22×D3—41no—
227C3×Q8—42no—
228C3×Q8—41no—
229C3×Q8—41no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12711yescharacteristic
241C2×C3×C4C2×Q1611yescharacteristic
242C2×C3×C4#13511yescharacteristic
243C2×C3×C4#13711yescharacteristic
244C2×C3×C4#13711yescharacteristic
245C2×Dic3—42no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249D12—22no—
250D12—22no—
251D12—41no—
252D12—41no—
253C4×D3—82no—
254C4×D3—81no—
255C4×D3—81no—
256Dic6—42no—
257Dic6—41no—
258Dic6—41no—
259C3×C8—42no—
260C3×C8—42no—
261C3⋊C8—42no—
262C3⋊C8—42no—
263C24—61no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×D4—32no—
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—61no—
274C22×C4—61no—
275C22×C4—122no—
276C22×C4—122no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4#23511yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282Q16—242no—
283M16—41no—
284M16—41no—
285M16—121no—
286M16—121no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—62no—
292C2×C8—62no—
293C2×C8—122no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—41no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C42—41no—
313C42—121no—
314C42C2×D1211yescharacteristic
315C42D3×D411yescharacteristic
316C42D3×D411yescharacteristic
317C42C2×#6711yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3#39211yescharacteristic
321C22×C3#46711yescharacteristic
322C22×C3#47011yescharacteristic
323C2×D3—44no—
324C2×D3—42no—
325C2×D3—82no—
326C2×D3—81no—
327C2×D3—81no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—81no—
332C3×C4—81no—
333C3×C4#45111yescharacteristic
334C3×C4#45111yescharacteristic
335C3×C4#45211yescharacteristic
336C3×C4#45211yescharacteristic
337C3×C4#45311yescharacteristic
338C3×C4#45411yescharacteristic
339Dic3—81no—
340Dic3—81no—
341Dic3—81no—
342C23—62no—
343C23—122no—
344C23—122no—
345C23—121no—
346C23—121no—
347C23#88411yescharacteristic
348Q8—42no—
349Q8—41no—
350Q8—41no—
351Q8—122no—
352Q8—121no—
353Q8—121no—
354D4—62no—
355D4—62no—
356D4—121no—
357D4—121no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—122no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—242no—
368C2×C4—241no—
369C2×C4—241no—
370C2×C4#88211yescharacteristic
371C2×C4#88211yescharacteristic
372C2×C4#88411yescharacteristic
373C2×C4#91511yescharacteristic
374C2×C4#91611yescharacteristic
375C2×C4D3×Q1611yescharacteristic
376C2×C4#91811yescharacteristic
377C2×C4#93011yescharacteristic
378C2×C4#93711yescharacteristic
379C2×C4#94011yescharacteristic
380C8—42no—
381C8—42no—
382C8—122no—
383C8—122no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3#200111yescharacteristic
387C2×C3#200611yescharacteristic
388C2×C3#201711yescharacteristic
389D3—82no—
390C22—21no—
391C22—21no—
392C22—124no—
393C22—122no—
394C22—242no—
395C22—241no—
396C22—241no—
397C22#481311yescharacteristic
398C22#481611yescharacteristic
399C22#496011yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—81no—
404C4—81no—
405C4—241no—
406C4—241no—
407C4—241no—
408C4#480811yescharacteristic
409C4#480911yescharacteristic
410C4#482211yescharacteristic
411C4#482711yescharacteristic
412C4#517311yescharacteristic
413C4#518611yescharacteristic
414C3#1025811yescharacteristic, complemented, Sylow
415C2—21no—
416C2—21no—
417C2—242no—
418C2#7001311yescharacteristic
419C2#7027311yescharacteristic
420C2#7039211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1493D3
hypocenterC3#10258
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
1. #83373C2semidirect
4. C3×#1493C2semidirect
6. #69738C2semidirect
7. #67711C2semidirect
39. #1493D3semidirect
128. C2×D12Q16semidirect
414. C3#10258semidirect

Maximal subgroups

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

subgroupindex
#833732
#833502
#790592
C3×#14932
#697872
#697382
#677112
#102583