Finite Group Databasebeta

← #34004219

Subgroups — #34004219

Order 18688.

1030 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 152 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#22625788C211yesmaximal, characteristic
2#22625788C211yesmaximal, characteristic
3C2×#17989446C214yesmaximal, complemented
4C2×C4⋊Q8×D73C211yesmaximal, characteristic
5C2×#14556658C2212yes—
6C2×#14556658C2212yes—
7C2×#14556648—21no—
8C2×#14556648—21no—
9#17989453—21no—
10#17989453—21no—
11#17989446—28no—
12C22×Q8×D73—21no—
13C2×#14556591C411yescharacteristic
14C2×#14556590—21no—
15C2×#14556590C2212yes—
16C2×#14556589—21no—
17C4⋊Q8×D73—28no—
18C2×C4⋊C4×D73—21no—
19C2×C4⋊C4×D73C412yes—
20C2×C42×D73C2211yescharacteristic
21C2×C4⋊Q8×C73C411yescharacteristic
22C22×#10596833—22no—
23C22×#10596833—22no—
24#14556658—24no—
25#14556658—24no—
26#14556648—42no—
27#14556648—42no—
28C22×Dic146—22no—
29C22×Dic146—21no—
30C2×Q8×D73—28no—
31C2×Q8×D73—44no—
32C2×Q8×D73—42no—
33C2×Q8×D73—42no—
34C22×C4×D73—22no—
35C22×C4×D73C2×C411yescharacteristic
36C22×C4×D73D411yescharacteristic
37C22×C4×D73D411yescharacteristic
38#14556591—24no—
39#14556590—28no—
40#14556590—24no—
41#14556590D418yes—
42#14556589—28no—
43C2×C4⋊Dic73—21no—
44C2×C4⋊Dic73C2×C412yes—
45C2×#11924007—22no—
46C2×#11924007—22no—
47C4⋊C4×D73—28no—
48C4⋊C4×D73—24no—
49C4⋊C4×D73—24no—
50C2×C4×Dic73D411yescharacteristic
51C2×C4×Dic73D411yescharacteristic
52C2×C4×Dic73C2311yescharacteristic
53C42×D73—24no—
54C22×Q8×C73—21no—
55C4⋊Q8×C73—24no—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73C2×C412yes—
58C2×C42×C73C2×C411yescharacteristic
59C2×#10596833—42no—
60C2×#10596833—42no—
61C2×#10596833—42no—
62C2×#10596833—42no—
63C2×#10596833—42no—
64C2×#10596833—42no—
65C23×D73C22⋊C411yescharacteristic
66C2×Dic146—24no—
67C2×Dic146—28no—
68C2×Dic146—42no—
69C2×Dic146—42no—
70C2×Dic146—42no—
71C2×Dic146—42no—
72C2×Dic146—44no—
73Q8×D73—416no—
74Q8×D73—416no—
75C22×Dic73—22no—
76C22×Dic73C22⋊C411yescharacteristic
77C22×Dic73C2×D411yescharacteristic
78C22×Dic73C2×D411yescharacteristic
79C2×C4×D73—22no—
80C2×C4×D73—22no—
81C2×C4×D73—22no—
82C2×C4×D73—22no—
83C2×C4×D73—22no—
84C2×C4×D73—22no—
85C2×C4×D73—22no—
86C2×C4×D73—28no—
87C2×C4×D73—44no—
88C2×C4×D73—44no—
89C2×C4×D73D812yes—
90C2×C4×D73SD1612yes—
91C2×C4×D73SD1612yes—
92C2×C4×D73Q1612yes—
93C4⋊Dic73—24no—
94C4⋊Dic73—22no—
95C4⋊Dic73C22⋊C414yes—
96#11924007—28no—
97#11924007—28no—
98C4×Dic73—22no—
99C4×Dic73—22no—
100C4×Dic73C2×D414yes—
101C2×Q8×C73—24no—
102C2×Q8×C73—42no—
103C2×Q8×C73—42no—
104C22×C4×C73—22no—
105C22×C4×C73C22⋊C411yescharacteristic
106C22×C4×C73C22⋊C411yescharacteristic
107C22×C4×C73C22×C411yescharacteristic
108C4⋊C4×C73—24no—
109C4⋊C4×C73—22no—
110C4⋊C4×C73C22⋊C414yes—
111C42×C73—22no—
112C23×C73C2×C22⋊C411yescharacteristic
113C22×D73—21no—
114C22×D73—21no—
115C22×D73—21no—
116C22×D73—21no—
117C22×D73#10112yes—
118C22×D73#10212yes—
119C22×D73#10312yes—
120Q8×C73—48no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—24no—
128C2×C4×C73—42no—
129C2×C4×C73—42no—
130C2×C4×C73#10212yes—
131C2×C4×C73#10312yes—
132C2×C4×C73C2×C22⋊C412yes—
133C2×Dic73—21no—
134C2×Dic73—21no—
135C2×Dic73—24no—
136C2×Dic73—42no—
137C2×Dic73—42no—
138C2×Dic73#10112yes—
139C2×Dic73#12012yes—
140C2×Dic73#12012yes—
141C2×Dic73C2×D811yescharacteristic
142C2×Dic73C2×SD1611yescharacteristic
143C2×Dic73C2×SD1611yescharacteristic
144C2×Dic73C2×Q1611yescharacteristic
145C2×Dic73AGL(1,ℤ/8)12yes—
146C2×Dic73#13712yes—
147C4×D73—28no—
148C4×D73—24no—
149C4×D73—24no—
150C4×D73—24no—
151C4×D73—24no—
152C4×D73—416no—
153Dic146—48no—
154Dic146—48no—
155Dic146—48no—
156#10596833—82no—
157#10596833—82no—
158C22×C73#37911yescharacteristic
159C22×C73C2×#10111yescharacteristic
160C22×C73C2×#10111yescharacteristic
161C22×C73C2×#10211yescharacteristic
162C22×C73C2×#10311yescharacteristic
163C22×C73#41811yescharacteristic
164C22×C73#41911yescharacteristic
165C2×D73—22no—
166C2×D73—22no—
167C2×D73—22no—
168C2×D73—22no—
169C2×D73—22no—
170C2×D73—22no—
171C2×D73#33214yescomplemented
172C4×C73—22no—
173C4×C73—22no—
174C4×C73—22no—
175C4×C73—44no—
176Dic73—22no—
177Dic73—44no—
178Dic73#44712yes—
179Dic73#44812yes—
180Dic73#45012yes—
181Dic73#45112yes—
182#8340—731nomaximal, Sylow
183C2×C73C2×#33211yescharacteristic
184C2×C73#154611yescharacteristic
185C2×C73#154911yescharacteristic
186C2×C73#155611yescharacteristic
187C2×C73#178411yescharacteristic
188C2×C73#178611yescharacteristic
189C2×C73#178711yescharacteristic
190D73—24no—
191C22×C4⋊Q8—731no—
192#1797—731no—
193#1797—731no—
194C2×#332—734no—
195C73#834011yescharacteristic, complemented, Sylow
196C23×Q8—1461no—
197C2×C4⋊Q8—732no—
198C2×C4⋊Q8—731no—
199C2×C4⋊Q8—1461no—
200C2×C4⋊Q8—1461no—
201C2×C4⋊Q8—1468no—
202C2×C4⋊Q8#6440811yescharacteristic
203C22×C4⋊C4—732no—
204C22×C4⋊C4—1461no—
205C22×C42—731no—
206C2×C4⋊C8—732no—
207C2×C4⋊C8—732no—
208C2×C22⋊C8—1461no—
209C2×C22⋊C8—1461no—
210#336—1461no—
211#336—1461no—
212#332—1468no—
213C22×Q8—21no—
214C22×Q8—2922no—
215C22×Q8—2922no—
216C22×Q8—2924no—
217C22×Q8—1462no—
218C22×Q8—1468no—
219C22×Q8—1461no—
220C23×C4—731no—
221C23×C4—731no—
222C23×C4—731no—
223C23×C4—1462no—
224C22×C8—1462no—
225C22×C8—1462no—
226C4⋊Q8—24no—
227C4⋊Q8—738no—
228C4⋊Q8—1464no—
229C4⋊Q8—1464no—
230C4⋊Q8—1468no—
231C4⋊Q8—1468no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—732no—
234C2×C4⋊C4—1464no—
235C2×C4⋊C4—1468no—
236C2×C4⋊C4—1462no—
237C2×C4⋊C4—1462no—
238C2×C4⋊C4—1461no—
239C2×C4⋊C4—1464no—
240C2×C4⋊C4C2×#6440812yes—
241C2×C42—731no—
242C2×C42—731no—
243C2×C42—731no—
244C2×C42—1464no—
245C2×C42C2×#6440811yescharacteristic
246C4⋊C8—1464no—
247C4⋊C8—1464no—
248C22⋊C8—2922no—
249C22⋊C8—2922no—
250C24—731no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—2922no—
255C2×Q8—2922no—
256C2×Q8—2922no—
257C2×Q8—2924no—
258C2×Q8—29216no—
259C2×Q8—29216no—
260C2×Q8—2922no—
261C2×Q8—1468no—
262C2×Q8—1464no—
263C22×C4—22no—
264C22×C4—2924no—
265C22×C4—2924no—
266C22×C4—731no—
267C22×C4—731no—
268C22×C4—731no—
269C22×C4—732no—
270C22×C4—732no—
271C22×C4—732no—
272C22×C4—732no—
273C22×C4—1468no—
274C22×C4—1462no—
275C22×C4—1462no—
276C22×C4—1462no—
277C22×C4—1462no—
278C22×C4—1462no—
279C22×C4—1462no—
280C22×C4—1462no—
281C22×C4—1462no—
282C22×C4C22⋊#6440811yescharacteristic
283C22×C4C22⋊#6440811yescharacteristic
284C22×C4C22×#6440811yescharacteristic
285C2×C8—2922no—
286C2×C8—2922no—
287C2×C8—2922no—
288C2×C8—2922no—
289C2×C8—2922no—
290C2×C8—2922no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—734no—
294C4⋊C4—1468no—
295C4⋊C4—1468no—
296C4⋊C4—1464no—
297C4⋊C4—1462no—
298C4⋊C4C22⋊#6440814yes—
299C42—22no—
300C42—734no—
301C42—1462no—
302C42—1462no—
303C23—732no—
304C23—732no—
305C23—732no—
306C23—1461no—
307C23—1461no—
308C23—1461no—
309C23—1461no—
310C23C2×C22⋊#6440811yescharacteristic
311Q8—48no—
312Q8—2928no—
313Q8—2928no—
314Q8—2928no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—29216no—
325C2×C4—2922no—
326C2×C4—2922no—
327C2×C4—732no—
328C2×C4—732no—
329C2×C4—731no—
330C2×C4—731no—
331C2×C4—731no—
332C2×C4—731no—
333C2×C4—732no—
334C2×C4—732no—
335C2×C4—732no—
336C2×C4—1464no—
337C2×C4—1464no—
338C2×C4—1464no—
339C2×C4—1464no—
340C2×C4—1468no—
341C2×C4—1464no—
342C2×C4—1461no—
343C2×C4—1461no—
344C2×C4#1455665312yes—
345C2×C4#1455665412yes—
346C2×C4C2×C22⋊#6440812yes—
347C8—5842no—
348C8—5842no—
349C22—734no—
350C22—1462no—
351C22—1462no—
352C22—1462no—
353C22—1462no—
354C22—1462no—
355C22—1462no—
356C22#1798952811yescharacteristic
357C22D73⋊#10111yescharacteristic
358C22D73⋊#10111yescharacteristic
359C22C2×#1455665311yescharacteristic
360C22C2×#1455665411yescharacteristic
361C22#1798957911yescharacteristic
362C22#1798958011yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—44no—
367C4—2924no—
368C4—732no—
369C4—732no—
370C4—732no—
371C4—732no—
372C4—1462no—
373C2—1464no—
374C2D73⋊#33211yescharacteristic
375C2#2262544111yescharacteristic
376C2#2262544711yescharacteristic
377C2#2262546111yescharacteristic
378C2#2262576611yescharacteristic
379C2#2262576911yescharacteristic
380C2#2262577011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#8340
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#17989446C2semidirect
171. C2×D73#332semidirect
195. C73#8340semidirect

Maximal subgroups

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

subgroupindex
#226257882
C2×#179894462
C2×C4⋊Q8×D732
#834073